Jekyll2024-08-23T07:15:59-07:00https://dzackgarza.com/feed.xmlD. Zack GarzaPersonal website and Mathematics blog.D. Zack Garzadzackgarza@gmail.comSome notes on Krantz’s “A Mathematician’s survival guide”2022-12-31T19:05:00-08:002022-12-31T19:05:00-08:00https://dzackgarza.com/some-notes-on-krantz-s-a-mathematician-s-survival-guide<p>The following are just some random snippets I extracted while reading “A Mathematician’s Survival Guide: Graduate School and Early Career Development” by Steven Krantz. I don’t <em>necessarily</em> endorse any of this advice in particular, I just found these snippets to be interesting things to remember and think about.</p>
<ul>
<li>
<p><strong>It is of the essence that you work in a subject area, and on a thesis problem, that you like and can develop some enthusiasm for</strong>. Just as an athlete will say, “No pain, no gain.”, so a Ph.D. student in mathematics might say, “No passion, no thesis.” Just so, you also want a thesis advisor who can light a fire under you and can find a problem that will absorb you. You want a mentor who will inspire you to strive for excellence and achievement. You also want a thesis advisor with whom you can work comfortably and have a congenial relationship.</p>
</li>
<li>
<p><strong>Be courteous and friendly to the staff. They are the people who hold the department together</strong> and are there to help you with your work. If they do something especially nice for you, bring them cookies or flowers.</p>
</li>
<li>
<p>I have covered the chapter and verse of how to teach, and in particular how to teach recitation sections, in the book <em>KRA1</em>. I shall not repeat those insights here. Let me just conclude by noting that you must take your teaching duties seriously. <strong>Learn your students’ names. Show them that you care. Make yourself available outside of class</strong>. Be fair and even-handed. <strong>Your department depends on you to do a good job, the math department’s reputation around campus hinges on you and the other math teachers</strong>.</p>
</li>
<li>
<p>Am I Supposed to Work All of the Time? <strong>Definitely not</strong>. Fred Almgren, my friend and faculty mentor in graduate school, liked to say that <strong>graduate students should work four hours per day. What they did beyond that was their own business</strong>. Now one should bear in mind here that Fred had extraordinary powers of concentration. Four hours of work for Fred was like ten hours of work for anyone else.</p>
</li>
<li>
<p>…problems on the qual, then you had better get seven or eight of them almost entirely correct. <strong>If some of the questions ask you to state theorems or definitions, then you had better get them letter perfect</strong>-with proper English and all the quantifiers in the right places. <strong>What you are learning is a discipline</strong> and your work had better manifest that discipline.</p>
</li>
<li>
<p>As I’ve stated previously, a qualifying exam is not like a calculus test. <strong>You will not pass a qualifying exam on partial credit alone</strong>. The examiners are trying to determine whether you are qualified to do thesis work. Can you see to the heart of a problem? Can you write a proof well? Can you recognize correct mathematics and incorrect mathematics? Can you think critically?
<strong>As a result, what the examiners want to see on your qual is a substantial number of questions answered substantially correctly.</strong></p>
</li>
<li>
<p>Well, you don’t want to flunk too many quals and you don’t want to flunk them too often. Every program will give you at least two tries, and different people mature (intellectually) at different rates. The quals are not like the thesis. <strong>Qualifying exams are just a basic learning situation, one at which you have excelled all of your life. If you apply yourself and follow the advice given here (and of course follow the advice of your advisor), then you will certainly get through the quals.</strong></p>
</li>
<li>
<p>…the questions is terrific training. Learning to answer them is an even better regimen. And you can work out the answers together. <strong>Learning to talk about mathematics is an essential part of any graduate education</strong>.</p>
</li>
<li>So how do you study for an exam that contains such questions? Typically, you are not going to find questions like these in books. And, even if you do, you are not going to find the answers laid out for you. Let me put it this way: <strong>When you study for a qual, it is not enough to learn just the statements of the theorems and the proofs. Of course you must do at least that much</strong>. And at some schools this basic effort may be sufficient for a low pass. But the new thing that you must learn to do in graduate school is to ask yourself questions. <strong>Turn the ideas over in your mind and ask</strong>,
<ul>
<li>Why is the theorem stated this way?</li>
<li>Why is this hypothesis really needed?</li>
<li>What happens if we change the conclusion from this to that?</li>
<li>What would be a counterexample?</li>
<li>Why does the proof go like this?</li>
</ul>
</li>
<li>
<p>I will conclude this section by enunciating <strong>a very important principle</strong> (which will be repeated often in this text) of getting an advanced education. You are no longer learning calculus or another trivial subject where it is sufficient to read the text and do the homework. You are now doing the toughest thing you will ever have done in your life. <strong>It is essential that you talk to people-all the time. In this way you can orient yourself, keep to your course, be sure you are doing the right thing, and have a constant reality check.</strong> It is also an important part of being a mathematician to be able to communicate - not just technical mathematics but also information about mathematics, about teaching, and about the profession. You are now not simply learning mathematics - you are learning to create it. <strong>So my advice is to talk to your fellow students and to the faculty (and to the staff) about everything</strong>. Eat lunch with a group, socialize, talk to your office mates. This is your new life.</p>
</li>
<li>
<p>One thing that you begin to realize while you are a graduate student is that <strong>learning does not have to be a formalized process</strong>. You do not need to take a course, with a teacher, homework, and a grade, in order to learn a new subject. By the time you reach an established level in the academic world, you probably will not have the patience to sit through courses; instead, you will learn things entirely on your own. An intermediary step to that lofty position is to develop the habit of auditing courses. One of the things we were taught right away in the Princeton graduate program is that <strong>we should think of the courses just as we think of the books in the library: these are resources that you can drop in on, and drop out of</strong>. You don’t need to register. Just access them as your interests and your studies dictate.</p>
</li>
<li>
<p>Always remember that the <strong>qualifying exams are not the point of graduate school</strong>. They are just a step along the way. <strong>The main thing is to write a good thesis</strong>. So your short-term goal, at the beginning of your graduate career, is just to get through those quals. <strong>The quals are a zero-one game. Once you have passed them, then you need never look back</strong>. It’s time to write the thesis.</p>
</li>
<li><strong>Make a point of getting to know some of your classmates (your peers) and also some of the graduate students who are ahead of you in the program</strong>. The latter bunch will be full of a lot of gossip and a lot of baloney, but they also have passed the quals and they are familiar with how the program works. They know which classes to take, and the various hoops that a graduate student must jump through. Pick their brains. They can tell you which quals are hard and which are easy. Who writes the quals and who grades them. How quickly you are expected to get through the qualifying exams. They will know who the good instructors are, who gives good courses, and who the good thesis advisors are. This is vital information that you must know and understand.
<ul>
<li>(Ask them) where the graduate students hang out, what is expected of you. Where the coffee pot is. Do the same with the Department Chairperson.</li>
</ul>
</li>
<li>
<p>(If) this is your first day in graduate school, I admonish you to take charge of the situation. Figure out where the math building is, go there, and <strong>introduce yourself to the Chairperson’s secretary and the Graduate secretary</strong>. If the Graduate Chairperson is around, shake hands and introduce yourself.</p>
</li>
<li>
<p>As your program develops, <strong>keep in touch with the Graduate Director and with your thesis advisor to make sure you are making good progress</strong>. The rest should take care of itself.</p>
</li>
<li>
<p>It is my fervent belief, well-supported by experience, that the <strong>main reason that people often fail at tasks or programs that they set for themselves is that they never figure out what it was that they were supposed to be doing</strong>. The Ph.D. program in mathematics is a multistep, fairly complex process. There are many junctures at which one could lose track and not get the right mentoring or advice. The purpose of this book is to provide some objective reference material, presented in an accessible but authoritative tone, to aid in the graduate education process.</p>
</li>
<li>
<p>Of course <strong>a collaboration is like a marriage and you must manage it with the same delicacy</strong>. Some very fine collaborations have fallen by the wayside because of priority disputes or personal differences; this is just deplorable. Read [KRA2] and also [KRA6] to find out more about how mathematical collaborations function.</p>
</li>
<li>
<p><strong>How do you establish such an international reputation</strong>? Well, you must publish, and in good journals. Talk to people. Go to conferences. Give talks. Share your ideas. Collaborate with people. The strategy is to <strong>let people know who you are and what you have to offer</strong>. You want the established people - around the country and around the world - to think of you as an “up-and-coming person”, one whom they are happy to assess and praise. You want to be a person whose papers are read and quoted.</p>
</li>
<li>
<p>Of course I don’t need to tell you that you must be on top of your mathematics. The people interviewing you want to know what kind of mathematician you are. Be prepared to discuss your thesis and what your current research interests are. <strong>If you can get involved in a serious mathematical “chalk talk” with people -standing at the blackboard and doing mathematics - then you are sure to make a good impression.</strong></p>
</li>
<li>
<p><strong>What really makes a terrific impression in your job application dossier</strong>-at least to a research institution-<strong>is getting some letters from faculty outside of your university</strong>. I often invite colloquium speakers who would be of interest to my graduate students and arrange for the students to spend some time with the guests. This frequently results in the outside faculty member becoming familiar with my student’s work, perhaps even reading the thesis. Then this person can write a really nice letter on behalf of my student. More than one of my Ph.D. students has written a paper with a mathematician at another university - even before graduation.</p>
</li>
<li>For the most part, you must go through the formality of submitting job applications. This includes
<ul>
<li>Completing an AMS Cover Sheet (available from the American Mathematical Society web site or in issues of the Notices of the AMS),</li>
<li>Putting together a Curriculum Vitae (see [KRA2] for some advice on how to write your vita),</li>
<li>Getting letters of recommendation (usually three),</li>
<li>Writing a Teaching Statement, and</li>
<li>Writing a Research Statement.</li>
</ul>
</li>
<li>
<p>(To which) journal should you submit? The book [KRA2] contains extensive detail on how to write up a paper and on the process of submitting the paper to a journal.</p>
</li>
<li>
<p><strong>You may have been a child prodigy, but now you are just another mathematician</strong>. Your progress, achievements, and contributions are probably like everyone else’s. On the other hand, mathematics is one of the finest and most erudite achievements of the human mind. The scholarly standard in the mathematics profession is one of the highest in the academic world. It is something to be proud of. And you are part of it.</p>
</li>
<li>
<p>After you have graduated and you have been in the profession for a while, it will appear that</p>
<ul>
<li>
<p>Everyone else is getting invited to all of the big conferences;</p>
</li>
<li>
<p>Everyone else has lots of research grants;</p>
</li>
<li>
<p>Everyone else is winning Sloan Fellowships;</p>
</li>
<li>
<p>Everyone else is getting invited to speak at the International Congress of Mathematicians;</p>
</li>
<li>
<p>Everyone else is getting the plum jobs.</p>
<ul>
<li>…at that time was for the advisors to just shuffle us off to the plum jobs - the Moore Instructorships at MIT, the Dickson Instructorships at Chicago, the Peirce Instructorships at Harvard. It was a halcyon time…</li>
</ul>
</li>
<li>
<p>Other prestigious appointments:</p>
<ul>
<li>
<p>Elected to the National Academy of Sciences;</p>
</li>
<li>
<p>A Chair Professorship at Harvard;</p>
</li>
<li>
<p>Winning the Wolf Prize;</p>
</li>
<li>
<p>Getting the Steele Prize.</p>
</li>
</ul>
</li>
</ul>
</li>
<li>
<p><strong>(Experience) failure and disappointment every day, and try to find ways to surmount it</strong>. For the sake of success in mathematical research, <strong>it is much more important to be able to cope with the frustration inherent in the process than it is to be “quick” or “brilliant”</strong>. To be sure, the top people are often successful, lucky, ….</p>
</li>
<li>The only way to get the writing process going is to begin. <strong>Start small. Generate a broad outline of your thesis</strong>. The first pass can be a very vague adumbration, with just the key topics laid out. Then <strong>develop the outline by gradually adding detail</strong>. Show it to your thesis advisor just to make sure that you are headed in the right direction.</li>
<li>
<p>Now create a very detailed outline. <strong>This draft should actually list every definition, every lemma, every theorem, and every example. Don’t write them all out</strong>. Just indicate each one with a couple of words: “the covering lemma” or “the spectral sequence argument” or “the completeness axiom”.</p>
</li>
<li>
<p><strong>The more you do mathematics, the more you will treasure concrete examples</strong>. I would say that <strong>most of my best work is based on just a few examples that I return to over and over again</strong>. Always remember that <strong>we learn inductively (going from the specific to the general)</strong> rather than deductively (going from the general to the specific). <strong>The deductive mode is highly appropriate for recording mathematics, but it does not work for discovering mathematics. You discover, and create, mathematics by starting small, by doing little calculations, and then working up to more substantial and meaningful calculations. The aggregate of many calculations can become an insight and that insight might turn into a conjecture. Even more effort might transmogrify that conjecture into a theorem</strong>. It is a fantastic process.</p>
</li>
<li>Learn good work habits.
<ul>
<li><strong>When a calculation finally works out, write it up carefully</strong>, number the pages, date it, and put your name on it. File it away in an organized manner so that you can find it again.</li>
<li><strong>Keep a daily journal</strong>. Record in it what you have tried, what works, and what does not.</li>
<li>Be aware of the fact that, when you do a calculation or make a discovery, it will seem as plain as day and something you will never forget. Sadly, you will. <strong>And you will forget it most surely exactly when you need it. So write it down.</strong></li>
<li>Often things that you tried and did not quite work out are just as valuable as things that are beautiful and glow in the dark. <strong>Keep a record of everything you try</strong>. This will be valuable both for psychological support and also as an archive of your efforts.</li>
</ul>
</li>
<li>
<p>Let me assure you that, in working on your thesis, <strong>you will not get better every day</strong>. Some days you will seem to learn something new and get closer to the goal. Other days you will have to tear up what you did the day before and try again. There will be days when you seem to be blocked and others when it appears as though you are on a fool’s errand. Don’t let it get you down. This is the life of a mathematician. <strong>Your entire career is going to be like this</strong>. At least now you have senior mentors to guide you.</p>
</li>
<li>(On how to finish a thesis) The short answer to the question is: <strong>You work on your thesis problem by sitting in a quiet place and calculating. You try things and then modify them and jiggle them and then try them again. Fill dozens and dozens of pages with your speculations and trials and scribblings and conjectures, then throw them away and start again</strong>. During this process, you are constantly talking to people, going to seminars, writing e-mails, and asking questions. You immerse yourself in the problem and swim around in it until you find something that floats - something to latch onto. Gradually, you develop that handhold into a thesis.
<ul>
<li>This is neither a trivial nor a naive question. <strong>The answer is not, “Just sit down and do it.”</strong> I had a friend in college who did all his homework assignments, and his takehome exams as well, by poring through every book in the library until he found something that looked like a solution to the exercise on which he was working. This worked fairly well. He received reasonably decent grades but he did not really learn much of anything, because he never put his mettle to the test. <strong>You develop your brain by banging it against things, by stretching it, and by challenging it. This means that you must do the work yourself.</strong></li>
</ul>
</li>
<li>
<p>If you are stuck on what you are doing, or if you are discouraged by what you are doing, or feel that you are not making any progress, then please say sol! Your professor is not a mindreader.</p>
</li>
<li>
<p><strong>When an experienced, senior mathematician gives you a thesis problem, then s/he is doing you a tremendous favor</strong>. Your advisor is, in effect, saying, “Here is something worth doing and it is at your level. It is a doable problem and you will get a publishable paper out of it. People are interested in this topic and you will begin to make your reputation by solving this problem. Moreover, working on this problem will lead you to other worthwhile things later on.” This is something that very few young mathematicians can do for themselves. <strong>You are not well enough read and not sufficiently networked in the mathematical community to know which problems are interesting and which are not. You certainly do not know which problems are tractable and which are not, and you do not know which problems fit your abilities. When things do not work out, you do not know what else to try, or how to adjust the problem to make it more feasible.</strong></p>
</li>
<li>
<p>(On Krantz’s advisor) The thesis problems that he gave me were completely unfamiliar and they were all quite difficult. But they were good, they were the kinds of problems that you could jump in and start working on right away, and they had connections to many other things. In other words, they were ideal thesis problems.</p>
</li>
<li>
<p>A more typical scenario is for the advisor to say, “Why don’t you read this paper of mine and see whether it gives you any ideas?” or “Why don’t you read this paper and we’ll talk about it?” or “Why don’t you proofread the chapters of my new book? It’s a good way to learn the subject and there are a lot of good problems posed in there.” This is fine, and you should do what you are told. Again, don’t be bashful. <strong>Talk about your reading with everyone</strong> - with your fellow students, with other faculty, in seminars, and of course with your advisor. It is definitely not a zero-one game. <strong>It is not as though you can only speak when you have a theorem to show. You can speak any time you like, for almost any reason; and you should.</strong></p>
</li>
<li>When I advise undergraduates applying to graduate school in mathematics, I always tell them the same thing: “<strong>Go to the best school you can get into, find the hottest, smartest professor around, sign up to work with that professor, and do anything he or she tells you to do</strong>.” That is what I did, and it worked like a charm. <strong>Of course this is not necessarily the best advice for everyone</strong>. Some students do not want to go to an extremely competitive program…</li>
</ul>D. Zack Garzadzackgarza@gmail.comRecommendations: Undergraduate Resources2021-04-11T14:18:00-07:002021-04-11T14:18:00-07:00https://dzackgarza.com/recommendations-undergraduate-resources<h1 id="what-to-do-as-an-undergraduate">What To Do as an Undergraduate</h1>
<p>I defer to this excellent page from the math department at UC Irvine: <a href="https://www.math.uci.edu/math-majors/math-grad-school-resources /">https://www.math.uci.edu/math-majors/math-grad-school-resources /</a></p>
<h1 id="general-notesremarks">General Notes/Remarks</h1>
<ul>
<li>
<p>Anything lower-division: <a href="https://www.khanacademy.org/">Khan Academy</a>. Just be sure to actually work the problems!</p>
</li>
<li>
<p>On Youtube videos: <em>many</em> lectures and talks are being posted online these days, and these can vary in quality. This is slightly mitigated by the fact that you can watch them at 2x speed.</p>
</li>
<li>
<p>On video lectures in general: they are a crutch! Use them to supplement and enrich your understanding, but passively watching a lecture is a low-utility activity when it comes to learning.</p>
<p>The best resources are always materials from your own instructors and courses. If you do passively watch videos, it helps to engage yourself: write questions in your notes, timestamps/title of important topics, etc.</p>
</li>
<li>
<p><a href="https://ocw.mit.edu/index.htm">MIT OCW</a>: the video lectures (when available) are generally of very high quality. These cover a quite a few areas, and are also a good source of practice problems and exams.</p>
</li>
<li>
<p>Look (or ask) for book recommendations for your topic/class on <a href="https://math.stackexchange.com/">Math StackExchange (MSE)</a> or <a href="https://mathoverflow.net/">MathOverflow (MO)</a>.
Don’t be afraid to consult multiple books on the same topic!
I’ve generally had good luck following the <a href="https://math.stackexchange.com/questions/tagged/reference-request">reference-request tags</a>.</p>
</li>
<li>
<p>If you find an author/lecturer/general source that you particularly like or learn well from, see what other content they have! If they cover other topics, be open to learning those – you never quite know what will be useful where.</p>
</li>
<li>
<p>For lower division or introductory courses, try to <a href="https://lmgtfy.app/?q=libgen">obtain a textbook</a> paired with a solutions manual. This is a great way to drill problems, check your answers, and identify your weaknesses. Beware typos!</p>
</li>
<li>
<p>Don’t bother with sites like Chegg. The solutions there are <em>often</em> incorrect, and this can lead to academic integrity issues.</p>
</li>
<li>
<p>For computational or engineering courses: Schaum’s Outlines can be useful, but the quality is <em>very</em> subject-dependent.</p>
</li>
<li>
<p>Learn how to use a Computer Algebra System (CAS). Don’t use it as a crutch - just learn enough syntax so that you can quickly run “sanity checks” on your computations. <a href="http://www.wolframalpha.com/">Wolfram Alpha</a> and <a href="https://www.symbolab.com/">Symbolab</a> are good for quick checks in lower-division courses.</p>
</li>
<li>
<p>Learn something like <a href="http://www.sagemath.org/index.html">SageMath</a>, which can do symbolic computations and can be used for quite a bit of number theory, linear algebra, group theory, and more.</p>
</li>
</ul>
<hr />
<h1 id="preparing-for-graduate-school">Preparing for Graduate School</h1>
<ul>
<li><i class="fas fa-tv"></i>
<a href="https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6KzmkUQBQ8TSWVX">Introduction to Higher Mathematics</a>
<ul>
<li>A great survey that highlights many different areas of advanced mathematics.</li>
</ul>
</li>
<li><i class="fas fa-book"></i>
<em>Garrity</em>, <a href="https://www.amazon.com/gp/product/0521797071">All the Mathematics You Missed: But Need to Know for Graduate School</a>
<ul>
<li>If you’re thinking about grad school at all, read this! Even if you’re not, it’s a pretty good collection of mathematics that it’s good to at least be familiar with. You can also use this to get an idea of some of the major theorems and results in a variety of subfields.</li>
</ul>
</li>
</ul>
<h1 id="resources-by-subject">Resources by Subject</h1>
<p>Below are resources for specific courses/subjects; I’ve tried to roughly organize these by increasing complexity with respect to a typical undergraduate Math degree.</p>
<p><strong>Legend</strong></p>
<ul>
<li><i class="fas fa-book"></i> – Textbooks</li>
<li><i class="fas fa-tv"></i> – Videos or online lectures</li>
<li><i class="fas fa-link"></i> – Websites or other online collections of resources</li>
<li><i class="fas fa-star"></i> – Particularly excellent resources that I highly recommend</li>
<li><em>Details</em> – Expand these sections for notes on the mathematical content or subject-specific advice.</li>
</ul>
<h2 id="lower-division">Lower Division</h2>
<h3 id="calculus">Calculus</h3>
<h4 id="single-variable-calculus">Single Variable Calculus</h4>
<ul>
<li><i class="fas fa-book"></i>
<em>Stewart</em>,
<strong>Calculus: Early Transcendentals</strong>
<ul>
<li>What can I say? It’s a Calculus book, and it covers the standard curriculum. This one’s a good choice because there are a few solution manuals floating around for older editions.</li>
</ul>
</li>
<li><i class="fas fa-book"></i>
<em>Spivak</em>,
<strong>Calculus</strong>
<ul>
<li>Exposition is a little more advanced, and closer to introductory real analysis. Good for an honors-level course, or if you want to see a more “rigorous” exposition of the topics from Stewart.</li>
</ul>
</li>
<li><i class="fas fa-book"></i>
<em>Apostol</em>,
<strong>Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra</strong>
<ul>
<li>A good alternative to Spivak.</li>
</ul>
</li>
<li><i class="fas fa-tv"></i>
<a href="https://www.youtube.com/user/patrickJMT/videos">patrickJMT</a>
<ul>
<li>Videos of how to solve many specific calculus and engineering problems</li>
</ul>
</li>
<li><i class="fas fa-link"></i>
<a href="http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx">Paul’s Online Notes</a>
<ul>
<li>Follows the standard curriculum very closely, with many examples and in-depth explanations.</li>
</ul>
</li>
</ul>
<h4 id="multivariable--vector-calculus">Multivariable / Vector Calculus</h4>
<ul>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Schey</em>,
<strong>Div, Grad, Curl, and All That: An Informal Text on Vector Calculus</strong>
<ul>
<li>Most textbooks introduce these operators in a very formal way; this text expands and motivates these definitions greatly.</li>
</ul>
</li>
<li><i class="fas fa-tv"></i> <i class="fas fa-star"></i> <a href="https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/">MIT OCW Denis Auroux</a>
<ul>
<li>A very geometric approach, with lots of great imagery!</li>
</ul>
</li>
<li><i class="fas fa-link"></i> <a href="http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx">Paul’s Online Notes</a>
<ul>
<li>More fantastic notes from Paul!</li>
</ul>
</li>
</ul>
<blockquote>
<p>Many Calculus books cover both single and multivariable, so most of the resources from the single variable section will be applicable here as well. However, these particular resources focus almost entirely on the multivariable setting.</p>
</blockquote>
<h3 id="ordinary-differential-equations">Ordinary Differential Equations</h3>
<ul>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Goode and Annin</em>,
<strong>Differential Equations and Linear Algebra</strong>
<ul>
<li>Good precisely because it sets up the language of linear algebra first, making many concepts in ODEs much easier to explain and understand (e.g. solutions as eigenfunctions of derivative operators).</li>
</ul>
</li>
<li><i class="fas fa-tv"></i>
<a href="https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/index.htm">MIT OCW: Differential Equations</a>
<ul>
<li>Very Physics-motivated approach, for better or worse.</li>
</ul>
</li>
</ul>
<h3 id="linear-algebra">Linear Algebra</h3>
<ul>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Axler</em>,
<strong>Linear Algebra Done Right</strong>
<ul>
<li>This subject is usually taught as a bag of computational tricks and algorithms, which obscures the absolute beauty of the subject – this text motivates the theory nicely and shows how powerful it can be.</li>
</ul>
</li>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Goode and Annin</em>,
<strong>Differential Equations and Linear Algebra</strong>
<ul>
<li>Good balance of rigor vs. brevity and computation vs. theory. Very concise, gives you what you need to start calculating, but also takes time to list vector space axioms, mentions fields, a nice way of viewing the determinant formula, and (best of all) lists of many conditions that are equivalent to a matrix being singular or non-singular.</li>
</ul>
</li>
<li><i class="fas fa-book"></i>
<em>Anton</em>,
<strong>Elementary Linear Algebra</strong>
<ul>
<li>Has an entire chapter on many cool applications of Linear Algebra – things like graph theory, computer graphics, and Google’s Pagerank algorithm. Also has a lot of “historical note” blurbs that are pretty interesting.</li>
</ul>
</li>
<li><i class="fas fa-book"></i>
<em>Strang</em>,
<strong>Introduction to Linear Algebra</strong>
<ul>
<li>Strang is a giant in the world of linear algebra, so it’s worth seeing how he approaches the subject.</li>
</ul>
</li>
<li><i class="fas fa-tv"></i>
<a href="https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/">MIT OCW: Linear Algebra with Gilbert Strang</a>
<ul>
<li>Again, it’s Strang, so worth checking out!</li>
</ul>
</li>
<li><i class="fas fa-tv"></i>
<a href="https://www.youtube.com/watch?v=bqBacABVCeQ">JJ’s Nullspace Trick</a>
<ul>
<li>Many computations in linear algebra boil down to computing the nullspace of a matrix, and this is an excellent shortcut that lets your write the basis of the nullspace of a matrix almost directly from its reduced row-echelon form.</li>
</ul>
</li>
</ul>
<h3 id="discrete-mathematics-and-proofs">Discrete Mathematics and Proofs</h3>
<ul>
<li><i class="fas fa-book"></i>
<em>Rosen</em>,
<strong>Discrete Mathematics and Its Applications</strong>
<ul>
<li>Huge variety of topics, good prep for many Math-related computer science courses, also just a good survey of many topics at an introductory level.</li>
</ul>
</li>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Grimaldi</em>,
<strong>Discrete and Combinatorial Mathematics: An Applied Introduction</strong>
<ul>
<li>Absolutely excellent presentation of recurrence relations, mirroring how solutions of differential equations are found. Also has a good presentation of how to commute quantifiers.</li>
</ul>
</li>
<li><i class="fas fa-book"></i>
<em>Eccles</em>,
<strong>An Introduction to Mathematical Reasoning: Numbers, Sets and Functions</strong>
<ul>
<li>Good for a solid introduction to proofs.</li>
</ul>
</li>
<li><i class="fas fa-link"></i> <i class="fas fa-star"></i>
<a href="http://www.eecs70.org/">UC Berkeley’s EECS 70 Course</a>
<ul>
<li>A wonderful lower-division course for which excellent notes have been posted for a number of years.</li>
</ul>
</li>
<li>
<p><i class="fas fa-book"></i>
<em>Bender and Orszag</em>
<strong>Advanced mathematical methods for scientists and engineers</strong></p>
</li>
<li><i class="fas fa-book"></i>
<em>Graham, Knuth and Patashnik</em>,
<strong>Concrete Mathematics</strong>
<ul>
<li>An encyclopaedic reference for a huge swath of discrete Mathematics.</li>
</ul>
</li>
</ul>
<details>
<summary>
Standard Topics
</summary>
<ul>
<li>Basic logic (truth tables, quantifiers, implications, contrapositive and converse, induction)</li>
<li>Basic set theory (set-builder notation, venn diagrams, cartesian products)</li>
<li>Functions (injectivity/surjectivity, inverse images)</li>
<li>Relations (partial orders, equivalence relations)</li>
<li>Basic combinatorics (permutations and combinations, inclusion/exclusion, pigeonhole principle)</li>
<li>Recurrence relations</li>
<li>Graphs (Königsberg problem, Hamiltonian/Eulerian cycles)</li>
<li>Number theory (divisibility, modular arithmetic, the Euclidean algorithm)</li>
<li>Generating functions</li>
<li>Probability (odds for dice rolls and cards, distributions, Stirling’s approximation)</li>
</ul>
</details>
<hr />
<h2 id="upper-division">Upper Division</h2>
<h3 id="combinatorics">Combinatorics</h3>
<ul>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Wilf</em>,
<a href="https://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>
<ul>
<li>Freely provided by Wilf on his site, exposition is excellent and it provides a comprehensive overview of how to work with generating functions.</li>
</ul>
</li>
<li><i class="fas fa-book"></i>
<em>Bona</em>,
<strong>A Walk Through Combinatorics</strong></li>
</ul>
<h3 id="algebra">Algebra</h3>
<h4 id="abstract-algebra">Abstract Algebra</h4>
<ul>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Dummit and Foote</em>,
<strong>Abstract Algebra</strong>
<ul>
<li>Essentially the de-facto standard, plus it also serves as an encyclopaedic reference.</li>
</ul>
</li>
<li><i class="fas fa-book"></i>
<em>Beachy and Blair</em>,
<strong>Abstract Algebra</strong>
<ul>
<li>A good undergraduate-level reference.</li>
</ul>
</li>
<li><i class="fas fa-tv"></i> <i class="fas fa-star"></i>
<em>Matthew Salomone</em>
<a href="https://www.youtube.com/playlist?list=PLL0ATV5XYF8DTGAPKRPtYa4E8rOLcw88y">Abstract Algebra Series</a>
<ul>
<li>Fantastically well-motivated series, covers the equivalent of an entire year of material that naturally leads into Galois theory.</li>
</ul>
</li>
<li><i class="fas fa-tv"></i> <i class="fas fa-star"></i>
<em>Benedict Gross</em>,
<a href="https://www.youtube.com/watch?v=VdLhQs_y_E8&list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5">Lectures at Harvard</a>
<ul>
<li>Very clear with lots of examples.</li>
</ul>
</li>
</ul>
<h4 id="category-theory">Category Theory</h4>
<ul>
<li><i class="fas fa-tv"></i>
<a href="https://www.youtube.com/watch?v=o6L6XeNdd_k">Talk on Category Theory by Tom LaGatta</a></li>
</ul>
<blockquote>
<p>Expected in some classes, but often assumed. Used in Algebraic Topology and Algebraic Geometry heavily. You can also find many full lectures online by people like Steve Awodey, Bartosz Milewski, and Eugenia Cheng.</p>
</blockquote>
<h4 id="galois-theory">Galois Theory</h4>
<ul>
<li><i class="fas fa-tv"></i> <i class="fas fa-star"></i>
<em>Matthew Salomone</em>,
<a href="https://www.youtube.com/playlist?list=PLL0ATV5XYF8DTGAPKRPtYa4E8rOLcw88y">Galois Theory Lectures</a>
<ul>
<li>A portion of his Algebra series, the exposition is fantastic because the series follows a cohesive narrative that introduces some of the major results and benefits of Galois theory early on. Highly recommended.</li>
</ul>
</li>
</ul>
<h3 id="analysis">Analysis</h3>
<h4 id="real-analysis">Real Analysis</h4>
<ul>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i> <em>Rudin</em>, <strong>Principles of Mathematical Analysis</strong>
<ul>
<li>
<p>Essentially a standard in undergraduate real analysis, written in a very terse style but covers a great deal of material. Often referred to as “Baby Rudin”.</p>
</li>
<li>
<p>Extra <a href="https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf">notes/commentary/suggested exercises</a> from George Bergman.</p>
</li>
</ul>
</li>
<li><i class="fas fa-tv"></i> <i class="fas fa-star"></i>
<em>Francis Su</em>,
<a href="https://www.youtube.com/playlist?list=PL0E754696F72137EC">Lectures from Harvey Mudd College</a>
<ul>
<li>Recorded lectures, extremely clear and well-motivated!</li>
</ul>
</li>
</ul>
<details>
<summary>
Standard Topics
</summary>
<ul>
<li>The reals as an ordered field, construction</li>
<li>Metric spaces, basic topology</li>
<li>The Riemann-Stieltjes Integral</li>
<li>Sequences and series, Cauchy sequences and completeness</li>
<li>Limits and continuity, pointwise and uniform convergence</li>
<li>The Mean Value Theorem</li>
<li>Measure theory and the Lebesgue integral</li>
</ul>
</details>
<h4 id="complex-analysis">Complex Analysis</h4>
<ul>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Brown and Churchill</em>,
<strong>Complex Variables and Applications</strong>
<ul>
<li>Good overview of computational techniques at an undergrad level.</li>
</ul>
</li>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Needham</em>,
<strong>Visual Complex Analysis</strong>
<ul>
<li>Absolutely phenomenal book! The exposition and imagery is truly excellent, although this is perhaps not the best book for learning computations.</li>
</ul>
</li>
</ul>
<details>
<summary>
Standard Topics
</summary>
<ul>
<li>Arithmetic with complex numbers, roots of unity, the geometry of $\CC$</li>
<li>The complex integral</li>
<li>Residue theorems</li>
<li>Analytic vs. holomorphic vs. complex differentiable</li>
</ul>
</details>
<h4 id="numerical-analysis">Numerical Analysis</h4>
<ul>
<li><i class="fas fa-book"></i> <em>Burden</em>, <strong>Numerical Analysis</strong>
<ul>
<li>Has good info on fixed point theory and root-finding, Newton’s method, least squares.</li>
</ul>
</li>
</ul>
<h3 id="topology">Topology</h3>
<h4 id="point-set-topology">Point-Set Topology</h4>
<ul>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Munkres</em>,
<strong>Topology</strong>
<ul>
<li>A standard - the good stuff starts about 10 chapters in, everything before that is aimed at providing a solid grounding in set theory and proofs.</li>
</ul>
</li>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Lee</em>,
<strong>Introduction to Smooth Manifolds</strong>
<ul>
<li>The appendix has a great über-compressed review of point-set.</li>
</ul>
</li>
</ul>
<h4 id="algebraic-topology">Algebraic Topology</h4>
<ul>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Munkres</em>,
<strong>Topology</strong>
<ul>
<li>Mostly point-set, but introduces things like the fundamental group in the later chapters.</li>
</ul>
</li>
<li><i class="fas fa-book"></i> <i class="fas fa-star"></i>
<em>Hatcher</em>,
<strong>Algebraic Topology</strong>
<ul>
<li>Love it or hate it, this seems to be the standard reference!</li>
</ul>
</li>
<li><i class="fas fa-tv"></i>
<em>NJ Wildberger</em>,
<a href="https://www.youtube.com/watch?v=Ap2c1dPyIVo&list=PL6763F57A61FE6FE8">Introduction to Algebraic Topology</a>
<ul>
<li>A good undergraduate-level series, just be aware that he expresses some extremely non-mainstream views in his other videos!</li>
</ul>
</li>
</ul>
<h4 id="differential-geometry--manifolds">Differential Geometry / Manifolds</h4>
<ul>
<li><i class="fas fa-book"></i>
<em>Spivak</em>,
<strong>Calculus on Manifolds</strong>
<ul>
<li>A good follow-up to Spivak’s Calculus book, the exposition is at an undergraduate level. Worth checking out if you like his style.</li>
</ul>
</li>
<li><i class="fas fa-tv"></i> <i class="fas fa-star"></i>
<em>Frederic Schuller</em>,
<a href="https://www.youtube.com/watch?v=7G4SqIboeig">International Winter School on Gravity and Light 2015</a>
<ul>
<li>This guy is just phenomenal!</li>
</ul>
</li>
</ul>
<h3 id="number-theory">Number Theory</h3>
<ul>
<li><i class="fas fa-book"></i>
<em>LeVeque</em>,
<strong>Fundamentals of Number Theory</strong>
<ul>
<li>Short but good!</li>
</ul>
</li>
</ul>
<details>
<summary>
Standard Topics
</summary>
<ul>
<li>The prime counting function</li>
<li>Modular arithmetic, solving equations in rings, multiplicative functions (like Euler’s totient function)</li>
<li>The Chinese remainder theorem, Euler’s theorem, Fermat’s Little Theorem</li>
<li>Quadratic reciprocity, the Legendre and Jacobi symbol</li>
</ul>
</details>
<h3 id="algebraic-geometry">Algebraic Geometry</h3>
<ul>
<li><i class="fas fa-book"></i>
<em>Reid</em>,
<strong>Undergraduate Algebraic Geometry</strong>
<ul>
<li>Great introduction to the field, weaves in a lot of history and classical results.</li>
</ul>
</li>
<li><i class="fas fa-book"></i>
<em>Cox, Little and O’Shea</em>,
<strong>Ideals, Varieties and Algorithms</strong></li>
<li><i class="fas fa-book"></i>
<em>Cox, Little and O’Shea</em>,
<strong>Using Algebraic Geometry</strong></li>
</ul>
<h3 id="probability-and-statistics">Probability and Statistics</h3>
<ul>
<li>
<p><i class="fas fa-book"></i>
<em>Ross</em>,
<strong>A First Course in Probability</strong></p>
</li>
<li>
<p><i class="fas fa-book"></i>
<em>Wasserman</em>,
<strong>All the Statistics: A Concise Course in Statistical Inference</strong></p>
</li>
</ul>
<blockquote>
<p>It is useful to take (either beforehand or concurrently) introductory classes in both statistics and combinatorics.</p>
</blockquote>
<h3 id="misc--topics">Misc / Topics</h3>
<h4 id="dynamics">Dynamics</h4>
<ul>
<li>
<p><i class="fas fa-book"></i>
<em>Milnor</em>,
<strong>Dynamics in One Complex Variable</strong></p>
</li>
<li>
<p><i class="fas fa-book"></i>
<em>Arnol’d</em>,
<strong>Methods in Classical Mechanics</strong></p>
</li>
<li>
<p><i class="fas fa-book"></i>
<em>Strogatz</em>,
<strong>Nonlinear Dynamics and Chaos</strong></p>
</li>
<li>
<p><i class="fas fa-book"></i>
<em>Hale & Koçak</em>,
<strong>Dynamics and Bifurcations</strong></p>
</li>
</ul>
<h4 id="computer-science">Computer Science</h4>
<ul>
<li><i class="fas fa-book"></i>
<em>Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein</em>,
<strong>Introduction to Algorithms</strong> (CLRS)
<ul>
<li><i class="fas fa-external-link-alt"></i>
<a href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-introduction-to-algorithms-sma-5503-fall-2005/index.htm">MIT OCW: Introduction to Algorithms</a> is a good supplement.</li>
</ul>
</li>
<li><i class="fas fa-book"></i>
<em>Sipser</em>,
<strong>Introduction to the Theory of Computation</strong>
<ul>
<li>The standard resource for an introduction to theoretical computer science. Covers things like computability, automata, and Turing machines.</li>
</ul>
</li>
<li><i class="fas fa-book"></i>
<a href="https://www.amazon.com/gp/product/0984782850">Cracking the Coding Interview</a>
<ul>
<li>Covers a number of extremely typical CS interview questions, absolutely read if you are preparing to apply for internships or jobs.</li>
</ul>
</li>
</ul>D. Zack Garzadzackgarza@gmail.comWhat To Do as an UndergraduateIntro to Derived Algebraic Geometry 1: The Cotangent Complex and Derived de Rham Cohomology2020-12-27T16:37:00-08:002020-12-27T16:37:00-08:00https://dzackgarza.com/intro-to-derived-algebraic-geometry-1-the-cotangent-complex-and-derived-de-rham-cohomology<!-- Courtesy of embedresponsively.com //-->
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</div>
<div class="pandoc">
<iframe src="/pandoc/2020-12-27.html" style="width:100%; min-height:800px;">
</div>
</iframe></div>D. Zack Garzadzackgarza@gmail.comSome introductory notes on derived algebraic geometry from an MSRI workshop series.Introduction to Infinity Categories2020-11-28T22:33:00-08:002020-11-28T22:33:00-08:00https://dzackgarza.com/introductory%20notes/mathematics%20research/introduction-to-infinity-categories<!-- Courtesy of embedresponsively.com //-->
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<div class="pandoc">
<iframe src="/pandoc/2020-11-28.html" style="width:100%; min-height:800px;">
</div>
</iframe></div>D. Zack Garzadzackgarza@gmail.comSome notes on a short introductory video on some foundational aspects of infinity categories.How to do well in (my) Precalculus class2020-11-03T15:50:00-08:002020-11-03T15:50:00-08:00https://dzackgarza.com/advice%20and%20resources/precalculus-tips-and-tricks<h1 id="general-student-tips">General Student Tips</h1>
<ul>
<li>
<p>Reflect on (or measure!) how much time you’re spending on the class <em>outside</em> of class.</p>
<ul>
<li>A successful student may average anywhere from 5-10 hours per week outside of lectures, either studying, working problems, or revising/organizing old material. There’s a <em>huge</em> amount of variation here, of course, but often the students that lose the most points are spending the least amount of time practicing outside of class.</li>
</ul>
</li>
<li>
<p>Take good notes.</p>
<ul>
<li>You don’t have to capture everything in lectures, but there are often many small details in lecture that are hard to remember. Try to record general formulas and derivations, of course, but also listen for things like common mistakes, restrictions on how to use formulas (e.g. radians vs degrees), hints about topics that will be on quizzes/exams, or things the instructor points out are particularly important.</li>
</ul>
</li>
<li>
<p>Do all of the work available to you, and use all class materials.</p>
<ul>
<li>If there are materials given to you for the class (such as extra problems or worksheets) and you’re not completing them, this is a lot of wasted potential. Curated materials <em>from your exact class</em> are by far by the best things to work and study to do well in your class.</li>
<li>Similarly, if there are videos, notes, discussions sections, office hours, discussion boards, group chats, or really any other aspects of the class that don’t happen in a lecture, take full advantage of them!</li>
</ul>
</li>
<li>
<p>Revise.</p>
<ul>
<li>That is, don’t just work a problem and/or turn it in and just forget about it. Check your solutions ASAP, and if you get something wrong, have a systematic way of marking those problems and revisiting them.</li>
</ul>
</li>
<li>
<p>Go to office hours!</p>
<ul>
<li>Many students don’t quite understand what office hours are, so let me frame it this way: it’s like having a private tutor working for you, <em>for free</em>, who also happens to be the exact person grading your assignments and writing your exams. Your professors, teaching assistants, or graduate student assistants are there to help you, and moreover they <strong>want</strong> to help you!</li>
</ul>
</li>
<li>
<p>Model your solutions off of posted solutions</p>
<ul>
<li>If any written-up solutions for your class are posted, you are extremely lucky! They are giving you a fantastic model for exactly what they consider to be a good solution. Absolutely use this to your advantage, and structure your own solutions and write-ups to mimic their format.</li>
</ul>
</li>
<li>
<p>Don’t focus on what specific things you <em>might</em> use “in the real world”!</p>
<ul>
<li>
<p>Not every class is meant to be job training!</p>
<p>Do you have similar expectations from your philosophy class?</p>
<p>Your art appreciation class?</p>
<p>Your literature class?</p>
<p>Mathematics is a technical skill, to be sure, but its broader use is to understand <em>methods of solving problems</em>.</p>
</li>
<li>
<p>We use mathematical equations because they’re much simpler than real life: there actually <em>is</em> a single correct solution in many cases, you can explain the algorithms and methods relatively easily, and the problems are accessible without years of extra scientific or industry-specific training.</p>
</li>
<li>
<p>Depending on your career, no one may ask you to solve for $x$ for the rest of your life. But skills like being able to learn and communicate a technical process, abstracting away difficulties, getting yourself “unstuck” in problems, and keeping an organized account of your work, all give you tools to deal with whatever difficult problems you <em>do</em> end up caring about.</p>
</li>
<li>
<p>In any case, many fields are becoming more data-driven and thus mathematical, and you may very well have to solve for $x$ at some point! Mathematics shows up in surprising places. I personally had to use logarithms and cosines/sines multiple times in one of my industry jobs, you just never know.</p>
</li>
</ul>
</li>
</ul>
<h1 id="writing-tips">Writing Tips</h1>
<ul>
<li>
<p>The process is more important than the solution.</p>
<ul>
<li>Teach a man to fish! In other words, having a solution in hand to one specific problem with specific numbers isn’t so valuable. However, having a process, a recipe, an <em>algorithm</em> for solving that problem, is immensely useful, because that’s what can be used, modified, and adapted to unforeseen new problems.</li>
</ul>
</li>
<li>
<p>Rarely write <em>just</em> a solution!</p>
<ul>
<li>
<p>Try to communicate to your reader a recipe, a logical series of steps they could retrace to arrive at the same solution from scratch. <strong>Teach your reader the algorithm!</strong></p>
</li>
<li>
<p>Bonus: this also provides strong evidence to your instructor or grader that you are really and truly internalizing the material.</p>
</li>
</ul>
</li>
<li>
<p>Know your audience.</p>
<ul>
<li>
<p>At this point in your academic career, you aren’t writing up solutions in order to demonstrate the correct answer or show someone that you did the work. Instead, you should transition to thinking about your solution as <em>a piece of mathematical writing</em>, which (importantly) will be read by other human beings.</p>
</li>
<li>
<p>In particular, you should frame your writeup as though you were explaining it to someone in the first week of your course, and include all of the information they’d need to work the problem themselves.</p>
</li>
</ul>
</li>
<li>
<p>Differentiate “scratch work” from what your present.</p>
<ul>
<li>
<p>In previous classes, we’re very used to writing our work out on some paper, possibly erasing things and reworking until we solve the problem, and turning that piece of paper in in.</p>
<p>Instead, it can help to do your intermediate work on scratch paper, setting that paper aside, and cleanly writing up a series of logical steps (usually equations) in a clean and organized way on the paper you’ll actually turn in.</p>
</li>
</ul>
</li>
<li>
<p>Write your solution as a “mathematical sentence”, i.e. a series of logical steps,</p>
<ul>
<li>
<p>For example,</p>
\[\begin{align*}
&&6x + 2 &= 5x + 1 \\
\implies&& 6x + 2 - 5x &= 1 \\
\implies&& x + 2 &= 1\\
\implies&& x &= -1.
\end{align*}\]
</li>
<li>
<p>Notice that each line is a <strong>statement</strong> that two things are equal.</p>
</li>
<li>
<p>Also notice that I’ve put a period at the end! This is the emphasize that you should be able to read your solution out loud. For example:</p>
<ul>
<li>“Six $x$ plus two equals five $x$ plus 1,”</li>
<li>“Which implies that six $x$ plus 2 minus five $x$ equals one,”</li>
<li>“Which implies that $x$ plus two equals one,”</li>
<li>“Which implies that $x$ is equal to negative one.”</li>
</ul>
</li>
</ul>
</li>
<li>
<p>Your solutions should tell a story.</p>
<ul>
<li>
<p>What I mean here is that there should be some kind of “narrative arc” to what you’ve written, as opposed to a collection of mathematical symbols on a page. There should be a beginning, a middle, and an end.</p>
<p>You should generally start with a known equality or equation, take a series of logically justified steps, and eventually arrive at a destination: the solution to the original question being asked!</p>
</li>
</ul>
</li>
<li>
<p>Write the general form of any formula you use <em>before</em> you use it.</p>
<ul>
<li>Again, just imagine that your reader is a student in Week 1, and they haven’t seen the formula yet. Remind the reader of what it is, and importantly, what it <em>means</em>.</li>
<li>For example, you might use the fact that the area of a circle is given by $A(r) = \pi r^2$. Definitely write this general equation before you use it, but don’t stop there! Tell the reader that this is a formula for the area, and that $r$ denotes the radius of a circle. <strong>Indicate what every symbol means!</strong></li>
</ul>
</li>
<li>
<p>Box and/or highlight the <em>actual</em> solution to the problem 100% of the time.</p>
<ul>
<li>Every mathematical problem is asking a question. What’s the answer? At the end of doing all of your work, always go back to the original statement, and ask yourself, “What question was this problem originally asking?” Then write the answer to that question and distinguish it from the rest of the work: box it, highlight it, put a period on it, whatever works!</li>
</ul>
</li>
</ul>
<h1 id="mathematical-tips">Mathematical Tips</h1>
<ul>
<li>
<p>Draw a picture!!</p>
<ul>
<li>
<p>Ask yourself for every problem, “What’s the picture?”</p>
<p>It doesn’t have to be precise every time, sometimes just having a schematic diagram of the situation can be very helpful.</p>
</li>
</ul>
</li>
<li>
<p>If you use a picture to reason precisely, label it precisely 100% of the time.</p>
<p>This often comes up when graphing functions, in which case you should always</p>
<ul>
<li>
<p>Label the dependent and independent variables.</p>
<p>We’re used to these being $x$ and $y$, but in most applications of mathematics there are other variables with other names like $t,r, \theta$, etc.</p>
</li>
<li>
<p>If you plot the graph of one or more functions, label each one by name: $f(x), g(t)$, etc.</p>
</li>
<li>
<p>Label with numbers enough tick marks to communicate the scale of the graph.</p>
<p>For example, do the ticks each represent $1$ unit? $1000$ units? $0.00001$ units? Are the scales on the $x$ and $y$ axis the same or different?</p>
</li>
<li>
<p>Label several “interesting” features, such as coordinates of points or lines. These can include</p>
<ul>
<li>$x$ or $y$ intercepts</li>
<li>Points computed in a table</li>
<li>Points where a piecewise function changes</li>
<li>Intersection points of two graphs</li>
<li>Horizontal or vertical asymptotes</li>
<li>Two or three points on the graph of a function.</li>
</ul>
</li>
</ul>
</li>
<li>
<p>Don’t modify equations “in-place” in write-ups.</p>
<ul>
<li>
<p>For example, in the above problem it might be tempting to look at</p>
\[6x+2 = 5x + 1\]
<p>and write</p>
\[\begin{align*}
6x+2 &&=&& 5x+1 \\
-5x && && -5x \\ \\
x+2 && = && 1.
\end{align*}\]
<p>This is totally fine in scratch work, but tends to allow errors to creep in and generally makes the “bookkeeping” in your solution hard to track as you modify the same thing multiple times.</p>
<p>Instead, an alternative is just <em>not</em> including that $-5x$ step in your writeup at all, or explaining off to the side how you got from one step to the next. For example,</p>
\[\begin{align*}
6x+2 &= 5x+1 \\
\implies x+2 &= 1 && \text{subtracting $5x$ from both sides}
\end{align*}\]
</li>
<li>
<p>Note that this also applies to dividing both sides of an equation by a number, exponentiating both sides, taking a log of both sides, applying $\sin$ or $\cos$, etc. It’s more clear to just write the result of the operation and explain in words what you did.</p>
</li>
</ul>
</li>
<li>
<p>Carefully distinguish between equalities and approximations with $=$ and $\approx$ respectively.</p>
<ul>
<li>
<p>Saying $a=b$ else is a <strong>very strong</strong> assertion, and can introduce issues in many contexts where numerical precision is important. Calculators can only hold so many digits, and some real numbers have infinitely many!</p>
</li>
<li>
<p>It’s important to use the $\approx$ symbol to communicate to the reader when error from approximation has potentially been introduced.</p>
<blockquote>
<p>As a general rule, the second you plug something into your calculator, you’ve traded in an exact answer for an approximation and should be using $\approx$ instead of $=$ in your notation.</p>
</blockquote>
</li>
</ul>
</li>
<li>
<p>Practice not using your calculator!</p>
<ul>
<li>
<p>At least in my class, you will rarely need a calculator, and often plugging things into one actually makes things <em>more</em> difficult.</p>
<p>It is perfectly fine (and often preferable) to leave something in a form like</p>
\[12 + 7 \log_4(5) \over 2\log_5(3^3) - \sqrt 7\]
<p>instead of approximating it with a calculator.</p>
</li>
</ul>
</li>
<li>
<p>Don’t plug anything in until you absolutely have to!</p>
<ul>
<li>
<p>In other words, don’t use your calculator until the very end!</p>
<p>This will save you a ton of writing – a single letter is much easier to write out than a multi-digit number – but also produces the most accurate and exact answer.</p>
<p>(If you take science courses, this becomes a necessity: approximating at intermediate stages can compound errors and significantly throw off important calculations.)</p>
</li>
</ul>
</li>
</ul>D. Zack Garzadzackgarza@gmail.comGeneral Student TipsSome topics to learn for graduate school in Mathematics2020-10-24T17:42:00-07:002020-10-24T17:42:00-07:00https://dzackgarza.com/advice%20and%20resources/some-topics-to-learn-for-graduate-school-in-mathematics<p>While preparing to apply to graduate schools, I searched around a few math department websites to get some idea of topics they expected students to know upon entry or within their first year.</p>
<p>Some of these lists were sourced from the Math department pages at Columbia and Harvard, but unfortunately I’ve forgotten where the rest came from.
I’ve just decided to toss all of it online for anyone that might find such a thing useful!</p>
<h1 id="undergraduate">Undergraduate</h1>
<h2 id="linear-algebra">Linear Algebra</h2>
<ul>
<li>Finite dimensional vector spaces (over $\RR$)
<ul>
<li>And linear maps between them</li>
</ul>
</li>
<li>Subspaces</li>
<li>Quotient spaces</li>
<li>Dimension</li>
<li>Bases</li>
<li>Matrix representations</li>
<li>Positive definite inner products</li>
<li>Orthonormal bases</li>
<li>Extensions of orthonormal subsets</li>
<li>Eigenvalues and eigenvectors for automorphisms</li>
<li>Characteristic polynomials</li>
</ul>
<blockquote>
<p><strong>References</strong>:</p>
<ul>
<li>M. Artin, ‘Algebra‘ (Prentice Hall, 1991), Chapters 1,3,4</li>
<li>K. Hoffman and R. Kunze, ‘Linear Algebra‘, Chapters 1-6, (Prentice-Hall, 1971)</li>
</ul>
</blockquote>
<h2 id="abstract-algebra">Abstract Algebra</h2>
<ul>
<li>Definitions of groups, rings, fields, and modules over a ring</li>
<li>Homomorphisms of these objects</li>
<li>Subgroups, normal subgroups, quotient groups</li>
<li>Cyclic groups</li>
<li>The structure theorem for finitely generated abelian groups</li>
<li>Ideals, prime and maximal and their quotients
<ul>
<li>Basic examples such as $\ZZ$, $k[x]$, rings of algebraic integers</li>
</ul>
</li>
<li>Field extensions</li>
<li>Splitting fields of polynomials</li>
<li>Normal extensions</li>
</ul>
<blockquote>
<p><strong>References</strong>:</p>
<ul>
<li>M. Artin, ‘Algebra‘, Chapters 2, 10, 11, 12, 13, 14</li>
<li>I. Herstein, ‘Topics in Algebra‘ (Blaisdell Publishers, 1964)</li>
</ul>
</blockquote>
<h2 id="point-set-topology">Point-Set Topology</h2>
<ul>
<li>Open and closed sets</li>
<li>Continuous functions</li>
<li>Connectedness</li>
<li>Compactness</li>
<li>Hausdorff</li>
<li>Normality</li>
<li>Metric spaces, $\RR^n$</li>
<li>Heine-Borel theorem</li>
</ul>
<blockquote>
<p><strong>Reference</strong>:</p>
<ul>
<li>J. Munkres, ‘Topology, A First Course‘, Part I (Prentice-Hall)</li>
</ul>
</blockquote>
<h2 id="calculus">Calculus</h2>
<ul>
<li>Differential of a smooth mapping between open subsets in Euclidean spaces</li>
<li>Matrix of partial derivatives</li>
<li>Inverse and implicit functions</li>
<li>Multivariable Riemann integration</li>
</ul>
<blockquote>
<p><strong>References</strong>:</p>
<ul>
<li>W. Rudin, ‘Principles of Mathematical Analysis‘ (McGraw-Hill, 1964)</li>
<li>A. Browder, ‘Mathematical Analysis: An Introduction‘ (Springer, 1996)</li>
</ul>
</blockquote>
<h2 id="complex-analysis">Complex Analysis:</h2>
<ul>
<li>Definition of holomorphic functions</li>
<li>Cauchy integral formula</li>
<li>Power series representations of holomorphic functions</li>
<li>Radius of convergence</li>
<li>Meromorphic functions</li>
<li>Residues</li>
</ul>
<blockquote>
<p><strong>Reference</strong>:</p>
<ul>
<li>L. Ahlfors, ‘Complex Analysis‘, (McGraw-Hill, 1973), Chapters 1- 5</li>
</ul>
</blockquote>
<h2 id="real-analysis">Real Analysis:</h2>
<ul>
<li>
<p>A thorough working knowledge of advanced calculus, at the level of the books of W. Rudin or A. Browder as listed under Calculus</p>
</li>
<li>Pointwise uniform convergence of functions</li>
<li>Equicontinuity</li>
<li>$\ell^2$, $L^2(S^1)$</li>
<li>Hilbert spaces</li>
<li>Orthonormal bases</li>
</ul>
<h1 id="first-year-graduate">First Year Graduate</h1>
<h2 id="algebra">Algebra</h2>
<ul>
<li>Group theory:
<ul>
<li>Sylow theorems</li>
<li>$p$-groups</li>
<li>Solvable groups</li>
<li>Free groups</li>
</ul>
</li>
<li>Rings and modules:
<ul>
<li>Tensor products</li>
<li>Determinants</li>
<li>Jordan canonical form</li>
<li>PID’s</li>
<li>UFD’s</li>
<li>Polynomials rings</li>
</ul>
</li>
<li>Field theory:
<ul>
<li>Splitting fields</li>
<li>Separable and inseparable extensions</li>
</ul>
</li>
<li>Galois theory:
<ul>
<li>Fundamental theorems of Galois theory</li>
<li>Finite fields</li>
<li>Cyclotomic fields</li>
</ul>
</li>
<li>Representations of Finite Groups:
<ul>
<li>Character theory</li>
<li>Induced representations</li>
<li>Structure of the group ring</li>
</ul>
</li>
<li>Basics of Lie groups and Lie algebras:
<ul>
<li>Exponential map</li>
<li>Nilpotent and semi-simple Lie algebras and Lie groups</li>
</ul>
</li>
</ul>
<blockquote>
<p><strong>References</strong>:</p>
<ul>
<li>Dummit and Foote: Abstract Algebra, 2nd edition, except chapters 15, 16 and 17</li>
<li>Serre: Representations of Finite Groups (Sections 1-6)</li>
<li>Fulton-Harris: Representation Theory: A First Course (Graduate Texts in Mathematics/Readings in Mathematics)</li>
<li>Lie groups and algebras, Chapters 7-10</li>
</ul>
</blockquote>
<h2 id="algebraic-geometry">Algebraic Geometry</h2>
<ul>
<li>Affine and projective varieties;</li>
<li>Regular functions and maps;</li>
<li>Cones and projections</li>
<li>Projective space and Grassmannians</li>
<li>Ideals of varieties;</li>
<li>The Nullstellensatz</li>
<li>Rational functions</li>
<li>Rational maps and blowing up</li>
<li>Dimension and degree of a variety;</li>
<li>The Hilbert function and Hilbert polynomial</li>
<li>Smooth and singular points of varieties;</li>
<li>The Zariski tangent space;</li>
<li>Tangent cones;</li>
<li>Dual varieties
<ul>
<li>Families of varieties (Chow varieties and Hilbert schemes)</li>
</ul>
</li>
<li>Algebraic curves:
<ul>
<li>Genus;</li>
<li>The genus formula for plane curves</li>
<li>The Riemann-Hurwitz formula</li>
</ul>
</li>
<li>Riemann-Roch theorem</li>
</ul>
<blockquote>
<p><strong>References</strong>:</p>
<ul>
<li>Shafarevich: Basic Algebraic Geometry 1, 2nd edition</li>
<li>Harris: Algebraic Geometry: A First Course</li>
</ul>
</blockquote>
<h2 id="complex-analysis-1">Complex Analysis</h2>
<ul>
<li>Holomorphic and meromorphic functions</li>
<li>Conformal maps</li>
<li>Linear fractional transformations</li>
<li>Schwarz’s lemma</li>
<li>Complex integrals:
<ul>
<li>Cauchy’s theorem</li>
<li>Cauchy integral formula</li>
<li>Residues</li>
</ul>
</li>
<li>Harmonic functions:
<ul>
<li>The mean value property;</li>
<li>The reflection principle;</li>
<li>Dirichlet’s problem</li>
</ul>
</li>
<li>Series and product developments:
<ul>
<li>Laurent series</li>
<li>Partial fractions expansions</li>
<li>Canonical products</li>
</ul>
</li>
<li>Special functions:
<ul>
<li>The Gamma function</li>
<li>The zeta functions</li>
<li>Elliptic functions</li>
</ul>
</li>
<li>Basics of Riemann surfaces</li>
<li>Riemann mapping theorem</li>
<li>Picard theorems</li>
</ul>
<blockquote>
<p><strong>References</strong>:</p>
<ul>
<li>Ahlfors: Complex Analysis (3rd edition)</li>
</ul>
</blockquote>
<h2 id="algebraic-topology">Algebraic Topology</h2>
<ul>
<li>Fundamental groups</li>
<li>Covering spaces</li>
<li>Higher homotopy groups</li>
<li>Fibrations and the long exact sequence of a fibration</li>
<li>Singular homology and cohomology</li>
<li>Relative homology</li>
<li>CW complexes and the homology of CW complexes</li>
<li>Mayer-Vietoris</li>
<li>Universal coefficient theorem</li>
<li>Kunneth formula</li>
<li>Poincare duality</li>
<li>Lefschetz fixed point formula</li>
<li>Hopf index theorem</li>
<li>Čech cohomology and de Rham cohomology</li>
<li>Equivalence between singular, Čech and de Rham cohomology</li>
</ul>
<blockquote>
<p><strong>References</strong>:</p>
<ul>
<li>A. Hatcher: Algebraic Topology, W. Fulton: Algebraic Topology</li>
<li>E. Spanier: Algebraic Topology, Greenberg and Harper: Algebraic Topology: A First Course</li>
</ul>
</blockquote>
<h2 id="differential-geometry">Differential Geometry</h2>
<ul>
<li>Basics of smooth manifolds:
<ul>
<li>Inverse function theorem</li>
<li>Implicit function theorem</li>
<li>Submanifolds</li>
<li>Integration on manifolds</li>
</ul>
</li>
<li>Basics of matrix Lie groups over $\RR$ and $\CC$:
<ul>
<li>The definitions of $\operatorname{Gl}(n)$, $\operatorname{SU}(n)$, $\operatorname{SO}(n)$, $\operatorname{U}(n)$</li>
<li>Their manifold structures</li>
<li>Lie algebras</li>
<li>Right and left invariant vector fields</li>
<li>Differential forms</li>
<li>The exponential map</li>
</ul>
</li>
<li>Bundles:
<ul>
<li>Definition of real and complex vector bundles</li>
<li>Tangent and cotangent bundles</li>
<li>Basic operations on bundles such as
<ul>
<li>Dual bundle</li>
<li>Tensor products</li>
<li>Exterior products</li>
<li>Direct sums</li>
<li>Pull-back bundles</li>
</ul>
</li>
</ul>
</li>
<li>Differential forms:
<ul>
<li>Definition of a differential form</li>
<li>Exterior product</li>
<li>Exterior derivative</li>
<li>De Rham cohomology</li>
</ul>
</li>
<li>Behavior under pull-back</li>
<li>Metric Geometry:
<ul>
<li>Metrics on vector bundles
<ul>
<li>Riemannian metrics</li>
<li>Definition of a geodesic</li>
<li>Existence and uniqueness of geodesics</li>
</ul>
</li>
<li>Definition of curvature, flat connections, parallel transport</li>
<li>Definition of Levi-Cevita connection</li>
<li>Properties of the Riemann curvature tensor</li>
</ul>
</li>
<li>Principal Bundles:
<ul>
<li>Definition of a principal Lie group bundle for matrix groups</li>
<li>Associated vector bundles: Relation between principal bundles and vector bundles</li>
<li>Definition of covariant derivative for a vector bundle</li>
<li>Connection on a principal bundle</li>
</ul>
</li>
</ul>
<blockquote>
<p><strong>References</strong>:</p>
<ul>
<li>Taubes: Differential geometry: Bundles, Connections, Metrics and Curvature</li>
<li>Lee: Manifolds and Differential Geometry (Graduate Studies in Math 107, AMS)</li>
<li>S. Kobayashi and K. Nomizu: Foundations of Differential Geometry</li>
</ul>
</blockquote>
<h2 id="real-analysis-1">Real Analysis</h2>
<ul>
<li>Measure Theory
<ul>
<li>Borel measure</li>
<li>Complex measures of bounded variation</li>
<li>Radon-Nikodym theorem</li>
<li>Lebesgue differentiation theorem</li>
</ul>
</li>
<li>Lebesgue Integration
<ul>
<li>Jensen’s inequality</li>
<li>Convergence theorems for integrals</li>
<li>$L^p$ spaces</li>
<li>Duality of $L^p$ space</li>
<li>Fubini theorem</li>
</ul>
</li>
<li>ODE/PDE Theory
<ul>
<li>Fourier series</li>
<li>Fourier transform</li>
<li>Convolution</li>
<li>Heat equation</li>
<li>Dirichlet problem</li>
<li>Fundamental solutions</li>
</ul>
</li>
<li>Probability Theory
<ul>
<li>Central limit theorem</li>
<li>Law of large numbers</li>
<li>Conditional probability</li>
<li>Conditional expectation</li>
</ul>
</li>
<li>Functional Analysis
<ul>
<li>Distributions</li>
<li>Sobolev embedding theorem</li>
<li>Hilbert spaces</li>
<li>Riesz representation theorem</li>
</ul>
</li>
<li>Maximum principle</li>
</ul>
<blockquote>
<p><strong>References</strong>:</p>
<ul>
<li>Rudin: Real and complex analysis</li>
<li>Stein and Shakarchi: Real analysis</li>
<li>Lieb-Loss: Analysis, Chapter 2. Fourier series:</li>
<li>Stein and Shakarchi: Fourier Analysis</li>
<li>Evans: Partial Differential Equations. Chapter 5</li>
<li>Shiryayev: Probability</li>
<li>Feller: An Introduction To Probability Theory And Its Applications</li>
<li>Durrett: Probability: Theory And Examples</li>
</ul>
</blockquote>D. Zack Garzadzackgarza@gmail.comSome notes on topics to learn to help prepare for grad school.Some notes on Benson Farb’s talk on surface bundles, mapping class groups, moduli spaces, and cohomology2020-10-24T12:00:00-07:002020-10-24T12:00:00-07:00https://dzackgarza.com/mathematics%20research/some-notes-on-benson-farb-s-talk-on-surface-bundles-mapping-class-groups-moduli-spaces-and-cohomology<p>Below is a partial transcription and some notes I took while watching the following talk from Benson Farb:</p>
<!-- Courtesy of embedresponsively.com //-->
<div class="responsive-video-container">
<iframe src="https://www.youtube-nocookie.com/embed/E_Ly2NWX1g8" frameborder="0" webkitallowfullscreen="" mozallowfullscreen="" allowfullscreen=""></iframe>
</div>
<iframe src="https://dzackgarza.com/assets/blogposts/some-notes-on-benson-farb-s-talk-on-surface-bundles-mapping-class-groups-moduli-spaces-and-cohomology.html" style="width:100%; min-height: 800px;"></iframe>D. Zack Garzadzackgarza@gmail.comBelow is a partial transcription and some notes I took while watching the following talk from Benson Farb:Advice for Undergraduates in Mathematics2020-10-16T12:27:00-07:002020-10-16T12:27:00-07:00https://dzackgarza.com/advice-for-undergraduates-in-mathematics<h1 id="other-pages-with-great-advice">Other Pages with Great Advice</h1>
<ul>
<li>Rob Candler, <a href="http://poplab.stanford.edu/pdfs/Candler-StuffMostGradStudentsDontAsk-ieee05.pdf">Stuff Most Students Never Ask About Grad School</a></li>
</ul>
<h1 id="on-learning-and-being-a-student">On Learning and Being a Student</h1>
<ul>
<li>
<p><strong>Stay organized</strong>: Find and settle on a good way to track all of your due dates, and in particular have a good idea of when each upcoming major exam is for all of your classes.</p>
<ul>
<li>I use things like <a href="https://calendar.google.com/calendar/r">Google Calendar</a> and <a href="https://todoist.com/app?lang=en">Todoist</a> and map out my entire semester within the first week or two, but there are many other great apps. A physical planner is also an excellent choice.</li>
</ul>
</li>
<li>
<p><strong>Start everything early</strong>: Starting early gives you time to mull over questions and potentially visit professors/TAs for help.</p>
<ul>
<li>Fitting in a lot of work at the last-minute isn’t usually an optimal strategy, so planning when you’ll work on things can be helpful too. Be sure to slot in enough time to avoid last-minute stress, since this tends to hurt how much you can absorb/retain the material.</li>
</ul>
</li>
<li>
<p><strong>Use a wide variety of resources</strong>: Books, lectures, notes, PDFs – use them all! The search modifiers <code class="language-plaintext highlighter-rouge">inurl:edu filetype:pdf</code> on Google are incredibly useful.</p>
</li>
<li>
<p><strong>Extensively use good resources</strong>: There is also some value in working a single, particularly well-regarded book, front to back.</p>
</li>
<li>
<p><strong>Spaced repetition</strong>: <a href="https://en.wikipedia.org/wiki/Spaced_repetition">Read up on it</a> (there is some pretty good science on its effectiveness in learning) and work it into your study habits. Cramming can have diminishing returns in proof-based classes, so it’s worth visiting and revisiting material often enough to be familiar with it.</p>
<ul>
<li>Apps like <a href="https://apps.ankiweb.net/">Anki</a> are great for this.</li>
</ul>
</li>
<li>
<p><strong>Sleep like it is your hobby:</strong> Two hours of extra sleep is almost always a more efficient use of your time than two hours of tired studying.</p>
</li>
</ul>
<h1 id="on-learning-mathematics-specifically">On Learning Mathematics Specifically</h1>
<ul>
<li>
<p><strong>Learn the vocabulary:</strong> Mathematics is (partially) a humanities discipline, so it can help to (partially) study it like one. I.e., the techniques you use to study in an English, Art History, or language class can also be put to good use here.</p>
<ul>
<li>One useful trick is keeping an ongoing list of new words you don’t know and their definitions, and continually reviewing and adding to it.</li>
</ul>
</li>
<li>
<p><strong>Spaced Repetition</strong>: When you run into things you don’t know, memorizing can be a first step (although this will also come naturally with practice). I usually recommend keeping a list of words/terms you’re less comfortable with or don’t know, or longer equations or formulas, and then continually reviewing all of them over the semester.</p>
</li>
<li>
<p><strong>Practice a lot!</strong> The wonderful violinist Jascha Heifetz had this to say about practicing music:</p>
<blockquote>
<p>If I don’t practice one day, I know it; two days, the critics know it; three days, the public knows it.</p>
</blockquote>
<p>In some ways, Maths is very similar.</p>
<ul>
<li>Moreover, doing tiny bits of math every day truly adds up: a nice mathematical analogy is that</li>
</ul>
\[(1.00)^{365} = 1.00 \quad\text{but}\quad (1.01)^{365} \approx 37.7.\]
<ul>
<li>I.e., a tiny bit of extra work/studying/review every day (in this equation, an extra 1% over a year) multiplies your skill level many times over.</li>
</ul>
</li>
<li><strong>Explain concepts and teach them to others</strong>: A great way to learn (anything) is to try teaching it to someone else. This is especially true in Math, since communicating your solutions and process is a huge component. If you stumble a little in your explanation of some particular concept, that is a great way to diagnose and pick out areas you can work on (e.g. re-studying the section, working on more problems, etc).
<ul>
<li>If none of your friends jump at the opportunity to hear about wonderful Maths all of the time, it can also be useful to try recording yourself explaining/summarizing a concept!</li>
</ul>
</li>
<li>
<p><strong>Use multiple books and notes.</strong> For most undergraduate topics, there are many “standard” references. Use them all! Cross-reference liberally as well.</p>
</li>
<li>
<p><strong>Get the definitions down cold.</strong> Record every single definition you come across and just do whatever you have to do to memorize them! Some people like flashcards, I make a “dictionary” document for each subject. Meaning and understanding is often a gradual process, but knowing the full statement of a definition should just be automatic.</p>
<ul>
<li>I like to think of definitions as the analog “multiplication tables” within higher Mathematics – is it essential to memorize? Well no, but you also wouldn’t want to waste time stopping to multiply out when you’re trying to solve an integral.</li>
</ul>
</li>
<li>
<p><strong>Synthesize often</strong>. This includes things like revising class notes, making your own “cheat sheets”, writing up explanations of particular theorems or motivations for certain concepts, etc. Even better, find ways to give talks or teach other people. Present things as if you were trying to teach yourself the topic 6-12 months in the past.</p>
</li>
<li>
<p><strong>Google for notes.</strong> For just about any topic at any level of specificity, someone has <em>probably</em> written up notes or expository articles and posted them online. So it can be extremely beneficial to look through the first page or two of Google results using filetype:pdf for any concept or proof you happen to be studying and see how other mathematicians think about and present them.</p>
</li>
<li>
<p><strong>Work and rework many problems and proofs.</strong> Do more than you are assigned, but be judicious with your time. If you stare at a problem for more than an hour or two without making progress, pivot. Sometimes the best way to make progress on a problem is to take a break, work on something different, or sleep on it.</p>
</li>
<li>
<p><strong>Work the exams.</strong> If possible, do them in a timed setting.</p>
</li>
<li>
<p><strong>Maths is cumulative</strong>: Previous classes tend to be broken up: you’ll use a concept from a unit, then move on to something else. At the university level, we start bringing together all kinds of tools from previous sections/courses and using them to build more complex things. The more tools you have in your tool belt (that you remember how to use!), the easier things will be.</p>
<ul>
<li>On the other hand, something complicated can be much more complicated if you’re rusty with a concept from an earlier section. So continually “honing your old tools” with practice becomes very important (reviewing notes, looking back over old homework, reworking old problems, etc)</li>
</ul>
</li>
</ul>
<h1 id="for-incoming-first-year-undergraduate-math-majors">For Incoming (First Year Undergraduate) Math Majors</h1>
<ul>
<li>
<p>Start thinking about whether or not graduate school is an option. You don’t have to <em>decide</em> within your first year, but you’ll want to pick courses accordingly.</p>
</li>
<li>
<p>Get an idea of what Math is all about - there are great things beyond Calculus!</p>
<ul>
<li>To get an idea of what you’re in for as a Math major, take a look at the GRE Mathematics Subject Exam. It covers a broad array of pretty standardized topics.</li>
<li>Also take a look at Garrity’s book (<a href="/resources/#resources-to-prepare-for-graduate-school">linked in the resources section</a>) for a short, condensed survey of some of the major topics and theorems you’ll encounter.</li>
<li>One thing that was helpful to me was to make <a href="/resources/world-of-math/">graphs and diagrams</a> of different areas of Math I encountered. Take a look, knowing the landscape you’re traversing is valuable.</li>
</ul>
</li>
<li>
<p>Learn $\LaTeX$ and Mathjax early, and then make efforts to regularly typeset your written notes.</p>
</li>
<li>
<p>It may be intimidating, but try to find ways to <strong>talk to your professors outside of class</strong>. Go to office hours, and don’t hesitate to ask for advice related to Math or your academic career. In my experience, they are usually happy to talk about these kinds of things.</p>
</li>
<li>
<p>For those aiming for grad school, particularly in pure Math: here are what I would consider some of the most essential, core classes to take:</p>
<ul>
<li>Calculus (of course)</li>
<li>Linear Algebra</li>
<li>Ordinary Differential Equations</li>
<li>Abstract Algebra</li>
<li>Real Analysis</li>
<li>Point-Set Topology</li>
</ul>
</li>
<li>
<p>Here are some “nice-to-haves” – still important, still beautiful, and great electives, but missing out on them in undergrad isn’t the end of the world:</p>
<ul>
<li>Probability</li>
<li>Combinatorics</li>
<li>Complex Analysis</li>
<li>Number Theory</li>
<li>Discrete Math (e.g. Graph Theory)</li>
<li>More Linear Algebra (e.g. Numerical Analysis)</li>
</ul>
</li>
<li>
<p>Of course, I recommend taking as many Math courses as possible – explore the subject and discover what you like! But the ideas and concepts from these particular courses are relevant to just about any area you might go into, so having some exposure to them makes life much easier (regardless of what you pursue).</p>
</li>
</ul>
<h1 id="the-math-subject-gre">The Math Subject GRE</h1>
<p>If you are thinking about applying to graduate school in Mathematics (pure or applied), I would recommend that you start looking at the material for this exam within your first few years, and take it as soon as your Junior year.</p>
<p>Note that the Math Subject GRE is vastly different than the Math portion of the General GRE. The latter covers primarily high-school level mathematics and requires no courses beyond Calculus (although the questions can still be tricky). The former is over 50% Calculus, and often includes a wide variety of upper-division topics as well. Many sources confuse these two exams!</p>
<p><strong>Topics</strong></p>
<ul>
<li>
<p>Single Variable Calculus</p>
</li>
<li>
<p>Multivariable Calculus</p>
</li>
<li>
<p>Ordinary Differential Equations</p>
</li>
<li>
<p>Linear Algebra</p>
</li>
<li>
<p>Complex Analysis</p>
</li>
<li>
<p>Abstract Algebra</p>
<ul>
<li>Primarily groups and rings, virtually no module or Galois theory.</li>
</ul>
</li>
<li>
<p>Probability/Statistics</p>
</li>
<li>
<p>Real Analysis</p>
<ul>
<li>Convergence of sums/sequences, topological properties of .</li>
</ul>
</li>
<li>
<p>Combinatorics</p>
<ul>
<li>Graph theory, counting problems</li>
</ul>
</li>
<li>
<p>Point-Set Topology</p>
</li>
<li>
<p>Numerical Analysis</p>
</li>
<li>
<p>Set Theory and Logic</p>
</li>
</ul>
<p><strong>Resources</strong></p>
<ul>
<li><i class="fas fa-external-link-alt"></i>
You can find five sample exams <a href="http://www.math.ucla.edu/~iacoley/greprep.html">here.</a></li>
<li>
<p><i class="fas fa-book"></i>
Princeton Review, <a href="https://www.amazon.com/gp/product/0375429727">Cracking the Math Subject GRE</a></p>
</li>
<li>
<p>Bill Shillito’s Page:
<a href="https://www.mathsub.com/resources/">https://www.mathsub.com/resources/</a></p>
</li>
<li>A “Mathematics Subject GRE Workshop” that I ran at UCSD
<ul>
<li><a href="/assets/talks/02-22-2019-GRE-Workshop/General%20Notes.html">Notes</a>, <a href="/assets/talks/02-22-2019-GRE-Workshop/Slides.html">Slides (HTML)</a>, <a href="https://dzackgarza.com/assets/talks/02-22-2019-GRE-Workshop/Slides.pdf">Slides (PDF)</a>.</li>
</ul>
</li>
<li>
<p>A UCSB Math Subject GRE Seminar:
<a href="http://web.math.ucsb.edu/~padraic/ucsb_2014_15/math_gre_w2015/math_gre_w2015.html">http://web.math.ucsb.edu/~padraic/ucsb_2014_15/math_gre_w2015/math_gre_w2015.html</a></p>
</li>
<li>Christian Parkinson’s GRE seminar page: <a href="https://www.math.ucla.edu/~chparkin/gre.html">https://www.math.ucla.edu/~chparkin/gre.html</a></li>
</ul>
<p><strong>Advice</strong></p>
<ul>
<li>Most problems don’t require extensive computations; there is usually a trick that solves it very quickly.</li>
<li>Time is the most difficult factor, be sure to take timed practice tests. You have right around two minutes per problem.</li>
<li>Sign up early, as these exams are often only held a few times per year.</li>
</ul>D. Zack Garzadzackgarza@gmail.comOther Pages with Great AdviceRecommendations: Graduate Level Texts and Notes2020-10-11T12:27:00-07:002020-10-11T12:27:00-07:00https://dzackgarza.com/recommendations-graduate-level-texts-and-notes<style>
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<p>Inspired by the following Twitter thread:</p>
<blockquote class="twitter-tweet"><p lang="en" dir="ltr">Yo math tweeps, what is an absolutely standard textbook in your field that'd be accessible to advanced undergrads and/or newbie grad students?</p>— D. Zack Garza (∂² ⋘ 0) (@dzackgarza) <a href="https://twitter.com/dzackgarza/status/1315352700779933699?ref_src=twsrc%5Etfw">October 11, 2020</a></blockquote>
<script async="" src="https://platform.twitter.com/widgets.js" charset="utf-8"></script>
<h1 id="multiple-areas">Multiple Areas</h1>
<ul>
<li>
<p><a href="https://www.ocf.berkeley.edu/~abhishek/chicmath.htm">The Chicago Undergraduate Mathematics Bibliography</a></p>
</li>
<li>
<p><a href="http://www.math.hkbu.edu.hk/~homanho/comments.html">Comments on a number of mathematics books</a></p>
</li>
<li>
<p><a href="https://www.jmilne.org/math/CourseNotes/">Milne’s Collection of Course Notes</a></p>
</li>
<li>
<p><a href="http://alpha.math.uga.edu/~pete/expositions2012.html">Pete Clark’s Expositions</a></p>
</li>
<li>
<p><a href="http://dec41.user.srcf.net/notes/">PDF Notes for a variety of Cambridge courses</a></p>
</li>
<li>
<p>Qiaochu Yuan has some <a href="https://qchu.wordpress.com/reading-recommendations/">reading recommendations</a></p>
</li>
<li>
<p>This blog also has a <a href="https://mathblog.com/mathematics-books/">large list of recommendations, sorted by topic</a></p>
</li>
<li>
<p>As does <a href="http://hbpms.blogspot.com/">this one</a></p>
</li>
<li>
<p>There is a similar <a href="https://math.stackexchange.com/questions/302023/best-sets-of-lecture-notes-and-articles?lq=1">community wiki on MSE</a></p>
</li>
<li>
<p>You can find ranked recommendations <a href="https://mathoverflow.net/questions/761/undergraduate-level-math-books">on this question on MO</a></p>
</li>
<li>
<p>UC Berkeley has <a href="https://math.berkeley.edu/courses/archives/announcements/fall-2011-textbooks">a bibliography</a> of books used by class.</p>
</li>
</ul>
<hr />
<h1 id="analysis">Analysis</h1>
<h2 id="real-analysis">Real Analysis</h2>
<h3 id="gerald-b-folland-real-analysis-modern-techniques-and-applications">Gerald B. Folland, Real Analysis: Modern Techniques and Applications</h3>
<ul>
<li>Links to some <a href="http://www.math.ucsd.edu/~bli/teaching/math240Cs20/">homeworks and solutions at UCSD</a></li>
</ul>
<h3 id="walter-rudin-real-and-complex-analysis">Walter Rudin, Real and Complex Analysis</h3>
<ul>
<li>Useful as a general reference, but there are more useful techniques in other books.</li>
</ul>
<h3 id="stein-and-shakarchi-real-analysis-measure-theory-integration-and-hilbert-spaces">Stein and Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces</h3>
<ul>
<li>Has more useful techniques than Rudin.</li>
<li>Doesn’t include $L^p$ spaces or convexity.</li>
</ul>
<h3 id="lieb-loss-real-analysis">Lieb-Loss, Real Analysis</h3>
<ul>
<li>Good source for $L^p$ spaces, convexity, and Fourier analysis.</li>
</ul>
<h3 id="stein-and-shakarchi-fourier-analysis">Stein and Shakarchi, Fourier Analysis</h3>
<ul>
<li>Very elementary.</li>
</ul>
<h3 id="schilling-measures-integrals-and-martingales">Schilling, Measures, Integrals, and Martingales</h3>
<h3 id="royden-real-analysis">Royden, Real Analysis</h3>
<h2 id="complex-analysis">Complex Analysis</h2>
<h3 id="taylor-complex-analysis">Taylor, Complex Analysis</h3>
<h3 id="simon-complex-analysis">Simon, Complex Analysis</h3>
<h3 id="stein-and-shakarchi-complex-analysis">Stein and Shakarchi, Complex Analysis</h3>
<h3 id="lars-ahlfors-complex-analysis">Lars Ahlfors, Complex Analysis</h3>
<h3 id="conway-functions-of-one-complex-variable-i">Conway, Functions of one complex variable I</h3>
<h2 id="functional-analysis">Functional Analysis</h2>
<h3 id="conway-a-course-in-functional-analysis">Conway, A Course in Functional Analysis</h3>
<h2 id="differential-equations">Differential Equations</h2>
<h3 id="evans-partial-differential-equations">Evans, Partial Differential Equations</h3>
<ul>
<li>Good source for Sobolev spaces.</li>
</ul>
<h3 id="v-arnold-ordinary-differential-equations">V. Arnold, Ordinary Differential Equations</h3>
<h2 id="probability">Probability</h2>
<h3 id="s-ross-a-first-course-in-probability-prentice-hall">S. Ross, A First Course in Probability (Prentice-Hall)</h3>
<h3 id="shiryayev-probability">Shiryayev, Probability.</h3>
<h3 id="feller-an-introduction-to-probability-theory-and-its-applications">Feller, An Introduction To Probability Theory And Its Applications</h3>
<h3 id="durrett-probability-theory-and-examples">Durrett, Probability: Theory And Examples</h3>
<hr />
<h1 id="algebra">Algebra</h1>
<h2 id="generalintroductory">General/Introductory</h2>
<h3 id="dummit-and-foote-abstract-algebra">Dummit and Foote, Abstract Algebra</h3>
<ul>
<li>Standard reference, encyclopaedic!</li>
</ul>
<h3 id="hungerford-algebra">Hungerford, Algebra</h3>
<h3 id="isaacs-algebra">Isaacs, Algebra</h3>
<h3 id="m-artin-algebra">M. Artin, Algebra</h3>
<h2 id="commutative-algebra">Commutative Algebra</h2>
<h3 id="altman-kleiman-a-term-of-commutative-algebra">Altman-Kleiman, A Term of Commutative Algebra</h3>
<p><a href="https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/exercises/Altman_-Kleiman---A-term-of-commutative-algebra-_2017_.pdf">https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/exercises/Altman_-Kleiman---A-term-of-commutative-algebra-_2017_.pdf</a></p>
<p><a href="https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/exercises/Altman_-Kleiman---A-term-of-commutative-algebra-_2017_.pdf">https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/exercises/Altman_-Kleiman---A-term-of-commutative-algebra-_2017_.pdf</a></p>
<h3 id="atiyah-and-macdonald-introduction-to-commutative-algebra">Atiyah and MacDonald, Introduction to Commutative Algebra</h3>
<h2 id="representation-theory">Representation Theory</h2>
<ul>
<li>Gaitsgory, <a href="http://people.math.harvard.edu/~gaitsgde/267y/index.html">Course Notes on Geometric Representation Theory</a></li>
<li>Mcgerty, <a href="https://courses.maths.ox.ac.uk/node/view_material/42444">Notes on Lie Groups/Algebras</a></li>
<li>Shoshany, <a href="http://baraksh.com/BarakShoshanyLieGroupReview.pdf">Notes on Lie Groups</a>
<ul>
<li>From a physicist’s perspective</li>
</ul>
</li>
</ul>
<h3 id="j-p-serre-linear-representations-of-finite-groups">J-P. Serre, Linear Representations of Finite Groups</h3>
<h3 id="humphreys-introduction-to-lie-algebras-and-representation-theory">Humphreys, Introduction to Lie algebras and Representation Theory</h3>
<h3 id="pramod-achar-unreleased-geometric-representation-theory-text">Pramod Achar, Unreleased Geometric Representation Theory Text</h3>
<h3 id="nicolas-libedinsky-gentle-introduction-to-soergel-bimodules">Nicolas Libedinsky, Gentle Introduction to Soergel Bimodules</h3>
<ul>
<li>Notes: <a href="https://arxiv.org/abs/1702.00039">Gentle introduction to Soergel bimodules I: The basics</a></li>
</ul>
<h3 id="hall-lie-groups-lie-algebras-and-representations">Hall, Lie Groups, Lie Algebras, and Representations</h3>
<h3 id="kirillov-introduction-to-lie-groups-and-lie-algebras">Kirillov, Introduction to Lie Groups and Lie Algebras</h3>
<ul>
<li>Recommended by Daniel Litt, <a href="https://www.math.stonybrook.edu/~kirillov/mat552/liegroups.pdf">link to notes</a></li>
</ul>
<h2 id="homological">Homological</h2>
<h3 id="peter-j-hilton-and-urs-stammbach-a-course-in-homological-algebra">Peter J. Hilton and Urs Stammbach, A Course in Homological Algebra</h3>
<ul>
<li>Recommended by Dan Nakano</li>
</ul>
<h3 id="kirillov-lie-groups-and-lie-algebras">Kirillov, Lie Groups and Lie Algebras</h3>
<ul>
<li>Recommended by Daniel Litt</li>
</ul>
<hr />
<h1 id="algebraic-geometry">Algebraic Geometry</h1>
<ul>
<li>Gathmann, <a href="https://www.mathematik.uni-kl.de/~gathmann/de/alggeom.php">Notes</a></li>
<li>Vakil, <a href="http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf">Rising Sea Notes</a></li>
</ul>
<h2 id="robin-hartshorne-algebraic-geometry">Robin Hartshorne, Algebraic Geometry</h2>
<h2 id="eisenbud-and-harris-the-geometry-of-schemes">Eisenbud and Harris, The Geometry of Schemes</h2>
<ul>
<li>Standard! Notes here: <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/eisenbudharris.pdf">The Geometry of Schemes</a></li>
</ul>
<h2 id="mumford-the-red-book-of-varieties-and-schemes">Mumford, The Red Book of Varieties and Schemes</h2>
<hr />
<h1 id="number-theory">Number Theory</h1>
<h3 id="algebraic-number-theory">Algebraic Number Theory</h3>
<h4 id="j-neukirch-algebraic-number-theory">J. Neukirch, Algebraic Number Theory</h4>
<ul>
<li>Large number of exercises <a href="http://www.math.ucsd.edu/~csorense/teaching/math204B_W20/204B_W20_EXC.pdf">here</a></li>
</ul>
<h4 id="cassels-and-fröhlich-algebraic-number-theory">Cassels and Fröhlich, Algebraic Number Theory</h4>
<h4 id="j-milne-algebraic-number-theory">J. Milne, Algebraic Number Theory</h4>
<ul>
<li>Not a textbook: actually <a href="http://www.math.ucsd.edu/~csorense/teaching/math204B_W19/Milne_ANT.pdf">notes</a></li>
</ul>
<h3 id="uncategorized">Uncategorized</h3>
<h4 id="j-p-serre-a-course-in-arithmetic">J.-P. Serre, A Course in Arithmetic</h4>
<h4 id="silverman-the-arithmetic-of-elliptic-curves">Silverman, The Arithmetic of Elliptic Curves</h4>
<h4 id="marcus-number-fields">Marcus, Number Fields</h4>
<ul>
<li>Covers quadratic fields</li>
</ul>
<h4 id="j-p-serre-local-fields">J.-P. Serre, Local fields</h4>
<h4 id="f-lorenz-algebra-volume-ii-fields-with-structure-algebras-and-advanced-topics">F. Lorenz, Algebra Volume II: Fields with structure, algebras and advanced topics</h4>
<h4 id="j-milne-class-field-theory">J. Milne, Class Field Theory</h4>
<ul>
<li>Not a textbook: actually <a href="http://www.math.ucsd.edu/~csorense/teaching/math204B_W19/Milne_CFT.pdf">notes</a></li>
</ul>
<h4 id="weil-basic-number-theory">Weil, Basic Number Theory</h4>
<h4 id="saban-alaca-and-kenneth-williams-introductory-algebraic-number-theory">Saban Alaca and Kenneth Williams, Introductory Algebraic Number Theory</h4>
<ul>
<li>Covers quadratic fields</li>
</ul>
<h4 id="valenza-fourier-analysis-on-number-fields">Valenza, Fourier Analysis on Number Fields</h4>
<hr />
<h1 id="topology">Topology</h1>
<ul>
<li><a href="http://math.uchicago.edu/~shmuel/2nd_steps.html">Second Steps in Algebraic Topology</a>
<ul>
<li>Slightly out-of-date</li>
</ul>
</li>
</ul>
<h2 id="algebraic-topology">Algebraic Topology</h2>
<h3 id="hatcher-algebraic-topology">Hatcher, Algebraic Topology</h3>
<ul>
<li>Standard reference.</li>
</ul>
<h3 id="peter-may-a-concise-course-in-algebraic-topology">Peter May, A Concise Course in Algebraic Topology</h3>
<h3 id="dodson-and-parker-a-users-guide-to-algebraic-topology">Dodson and Parker, A User’s Guide to Algebraic Topology</h3>
<ul>
<li>Covers more advanced topics than a usual course: some sheaf theory, bundles, characteristic classes, obstruction theory</li>
<li>Appendices on algebra, topology, manifolds/bundles, and tables of homotopy groups</li>
</ul>
<h3 id="glen-bredon-topology-and-geometry">Glen Bredon, Topology and Geometry</h3>
<ul>
<li>Blends differential and algebraic topology, can be disorienting as a first pass</li>
</ul>
<h3 id="milnor-topology-from-the-differentiable-viewpoint-princeton">Milnor, Topology from the Differentiable Viewpoint (Princeton)</h3>
<ul>
<li>Classic reference.</li>
</ul>
<h3 id="bott-and-tu-differential-forms-in-algebraic-topology-springer">Bott and Tu, Differential Forms in Algebraic Topology (Springer)</h3>
<ul>
<li>Classic reference.</li>
</ul>
<h3 id="massey-a-basic-course-in-algebraic-topology">Massey, A Basic Course in Algebraic Topology</h3>
<h2 id="homotopy-theory">Homotopy Theory</h2>
<ul>
<li><a href="http://www.math.jhu.edu/~eriehl/616/DwyerSpalinski.pdf">Dwyer-Spalinski, Homotopy Theories and Model Categories</a></li>
<li><a href="https://arxiv.org/abs/math/9801077">Hovey-Shipley-Smith, Symmetric Spectra</a></li>
<li><a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/hovey-symm.pdf">Hovey, Spectra and symmetric spectra in general model categories</a></li>
</ul>
<h3 id="bott-tu-differential-forms-in-algebraic-topology">Bott-Tu, Differential Forms in Algebraic Topology</h3>
<h3 id="griffiths-morgan-rational-homotopy-theory-and-differential-forms">Griffiths-Morgan, Rational Homotopy Theory and Differential Forms</h3>
<h3 id="mosher-tangora-cohomology-operations-and-applications-in-homotopy-theory">Mosher-Tangora, Cohomology Operations and Applications in Homotopy Theory</h3>
<h2 id="manifolds">Manifolds</h2>
<h3 id="milnor-topology-from-the-differentiable-viewpoint">Milnor, Topology from the differentiable viewpoint.</h3>
<h2 id="differential-geometry-and-topology">Differential Geometry and Topology</h2>
<h3 id="manfredo-p-do-carmo-riemannian-geometry">Manfredo P. Do Carmo, Riemannian Geometry</h3>
<h3 id="manfredo-p-do-carmo-differential-geometry-of-curves-and-surfaces">Manfredo P. Do Carmo, Differential Geometry of Curves and Surfaces</h3>
<h3 id="guillemin-and-pollack-differential-topology">Guillemin and Pollack, Differential Topology</h3>
<h3 id="john-m-lee-introduction-to-smooth-manifolds">John M. Lee, Introduction to Smooth Manifolds</h3>
<h3 id="milnor-morse-theory">Milnor, Morse Theory</h3>
<h3 id="pollack-differential-topology">Pollack, Differential Topology</h3>
<h3 id="milnor-lectures-on-h-cobordism">Milnor, Lectures on h-Cobordism</h3>
<h3 id="frank-warner-foundations-of-differentiable-manifolds-and-lie-groups-">Frank Warner, Foundations of Differentiable Manifolds and Lie Groups (?)</h3>
<h3 id="guillemin-stable-mappings-and-their-singularities">Guillemin, Stable Mappings and Their Singularities</h3>
<h2 id="symplectic-geometrytopology">Symplectic Geometry/Topology</h2>
<h3 id="dusa-mcduff-introduction-to-symplectic-topology">Dusa McDuff, Introduction to Symplectic Topology</h3>
<h3 id="eliashberg-from-stein-to-weinstein-and-back">Eliashberg, From Stein to Weinstein and Back</h3>
<h3 id="cannas-da-silva-lectures-on-symplectic-geometry">Cannas da Silva, Lectures on Symplectic Geometry</h3>
<ul>
<li>Notes: <a href="https://people.math.ethz.ch/~acannas/Papers/lsg.pdf">Lectures on Symplectic Geometry</a>
<ul>
<li>Skip chapters 4, 5, 25, 26, 30</li>
</ul>
</li>
</ul>
<h2 id="complex-geometry">Complex Geometry</h2>
<h3 id="claire-voisin-hodge-theory-and-complex-algebraic-geometry-volumes-i-and-ii">Claire Voisin, Hodge Theory and Complex Algebraic Geometry, Volumes I and II</h3>
<h3 id="daniel-huybrechts-complex-geometry-an-introduction">Daniel Huybrechts, Complex Geometry An Introduction</h3>
<h3 id="griffiths-harris-principles-of-algebraic-geometry">Griffiths-Harris, Principles of Algebraic Geometry</h3>
<h3 id="carlson-period-mappings-and-period-domains">Carlson, Period Mappings and Period Domains</h3>
<h3 id="rick-miranda-algebraic-curves-and-riemann-surfaces">Rick Miranda, Algebraic Curves and Riemann Surfaces</h3>
<h3 id="f-kirwan-complex-algebraic-curves">F. Kirwan, Complex Algebraic Curves</h3>
<h3 id="voison-hodge-theory-and-complex-algebraic-geometry-i">Voison, Hodge Theory and Complex Algebraic Geometry I</h3>
<ul>
<li>Notes: <a href="https://www-fourier.ujf-grenoble.fr/~peters/Books/PeriodBook.f/SecondEdition/PerBook.pdf">Period Mappings and Period Domains</a></li>
</ul>
<h2 id="knot-theory">Knot Theory</h2>
<ul>
<li>Justin Roberts, <a href="http://math.ucsd.edu/~justin/Roberts-Knotes-Jan2015.pdf">Knots Knotes</a></li>
</ul>
<h3 id="rolfsen-knots-and-links">Rolfsen, Knots and Links</h3>
<h3 id="livingston-knot-theory">Livingston, Knot Theory</h3>
<h3 id="colin-adams-the-knot-book">Colin Adams, The Knot Book</h3>
<h3 id="turner-five-lectures-of-khovanov-homology">Turner, Five Lectures of Khovanov Homology</h3>
<ul>
<li>Arxiv, <a href="https://arxiv.org/abs/math/0606464">link</a></li>
</ul>
<h3 id="bar-natan-on-khovanovs-categorification-of-the-jones-polynomial">Bar-Natan, On Khovanov’s categorification of the Jones polynomial</h3>
<ul>
<li>Arxiv, <a href="https://arxiv.org/abs/math/0201043">link</a></li>
</ul>
<h3 id="j-kock-frobenius-algebras-and-2d-tqfts">J. Kock, Frobenius algebras and 2D TQFTs</h3>
<h3 id="osvath-and-szabo-grid-homology-for-knots-and-links">Osvath and Szabo, <a href="https://web.math.princeton.edu/~petero/GridHomologyBook.pdf">Grid Homology for Knots and Links</a></h3>
<h2 id="geometric-group-theory">Geometric Group Theory</h2>
<blockquote class="twitter-tweet"><p lang="en" dir="ltr">Here are 4 books that all my grad students read:<br />1. Brown's "Cohomology of groups"<br />2. Serre's "Trees" (followed up w/ Scott's article "Topological methods in group theory")<br />3. Witte-Morris's "Introduction to arithmetic groups"<br />4. Farb-Margalit's "Primer on mapping class groups"</p>— Andrew Putman (@AndyPutmanMath) <a href="https://twitter.com/AndyPutmanMath/status/1315390718492512256?ref_src=twsrc%5Etfw">October 11, 2020</a></blockquote>
<script async="" src="https://platform.twitter.com/widgets.js" charset="utf-8"></script>
<h3 id="mosher-tangora-cohomology-operations-and-applications-in-homotopy-theory-1">Mosher-Tangora, Cohomology Operations and Applications in Homotopy Theory</h3>
<h3 id="serre-trees">Serre, Trees</h3>
<ul>
<li>Recommendation from Andrew Putman: follow up with Scott’s <a href="http://math.hunter.cuny.edu/olgak/scott_wall.pdf">Topological Methods in Group Theory</a></li>
</ul>
<h3 id="witte-morris-introduction-to-arithmetic-groups">Witte-Morris, Introduction to Arithmetic Groups</h3>
<h3 id="farb-margalit-primer-on-mapping-class-groups">Farb-Margalit, Primer on mapping class groups</h3>
<hr />
<h1 id="physics">Physics</h1>
<h2 id="uncategorized-1">Uncategorized</h2>
<h3 id="mark-srednicki-quantum-field-theory">Mark Srednicki, Quantum Field Theory</h3>
<h3 id="pierre-deligne-quantum-fields-and-strings-a-course-for-mathematicians">Pierre Deligne, Quantum Fields and Strings: A Course for Mathematicians</h3>
<h3 id="howard-georgi-lie-algebras-in-particle-physics">Howard Georgi, Lie Algebras in Particle Physics</h3>
<h3 id="kusse-mathematical-physics">Kusse, Mathematical Physics</h3>
<hr />
<h1 id="combinatorics">Combinatorics</h1>
<h2 id="uncategorized-2">Uncategorized</h2>
<h3 id="stanley-enumerative-combinatorics-vol-1">Stanley, Enumerative Combinatorics Vol 1</h3>
<h3 id="bruce-sagan---springer-the-symmetric-group">Bruce Sagan - Springer, The Symmetric Group</h3>
<h3 id="doug-west-introduction-to-graph-theory">Doug West, Introduction to Graph Theory</h3>
<hr />
<h1 id="logic">Logic</h1>
<h2 id="set-theory">Set Theory</h2>
<h3 id="kunen-set-theory-an-introduction-to-independence-proofs">Kunen, Set Theory: An introduction to independence proofs</h3>
<h2 id="model-theory">Model Theory</h2>
<h3 id="tent-and-ziegler-a-course-in-model-theory">Tent and Ziegler, A Course in Model Theory</h3>
<hr />
<h1 id="unsorted-recommendations">Unsorted Recommendations</h1>
<h3 id="sipser-introduction-to-the-theory-of-computation">Sipser, Introduction to the Theory of Computation</h3>
<h3 id="murray-mathematical-biology">Murray, Mathematical Biology</h3>
<h3 id="mac-lane-moerdijk-sheaves-in-geometry-and-logic-a-first-introduction-to-topos-theory">Mac Lane-Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory</h3>D. Zack Garzadzackgarza@gmail.comLaTeX Handwriting Practice Worksheets2020-08-03T18:27:00-07:002020-08-03T18:27:00-07:00https://dzackgarza.com/advice%20and%20resources/latex-handwriting-practice-worksheets<p>Having never quite been able to write $\zeta$ or $\xi$ correctly, I cobbled together some code to generate worksheets to help practice my handwriting.
Since a few people have expressed interest in them, I thought I’d share the documents and code here!</p>
<p>The idea is fairly simple: you toss in a bunch of symbols you want to practice writing, and it generates a grade-school-style worksheet with the symbols sketched out in dotted lines:</p>
<p><img src="/assets/images/image-20200803211425525.png" alt="image-20200803211425525" /></p>
<p>You can then either print this out and write over the symbols with any real-life writing implement, or just import them into your favorite PDF annotation software and practice using a stylus.</p>
<p>If you just want to take a direct look, <a href="https://dzackgarza.com/assets/pdfs/handwriting.pdf">here is a sample PDF</a> and <a href="https://dzackgarza.com/assets/pdfs/handwriting.tex">here is raw tex code</a>.</p>
<p>The code is fairly simple:</p>
<figure class="highlight"><pre><code class="language-latex" data-lang="latex"><span class="k">\documentclass</span><span class="p">{</span>article<span class="p">}</span>
<span class="k">\usepackage</span><span class="p">{</span>graphicx<span class="p">}</span>
<span class="k">\usepackage</span><span class="p">{</span>dashrule<span class="p">}</span>
<span class="k">\usepackage</span><span class="p">{</span>pgf, pgffor<span class="p">}</span>
<span class="k">\usepackage</span><span class="p">{</span>amsfonts<span class="p">}</span>
<span class="k">\newsavebox\myboxX</span>
<span class="k">\newsavebox\myboxx</span>
<span class="k">\newdimen\heightX</span>
<span class="k">\newdimen\heightx</span>
<span class="k">\newcommand</span><span class="p">{</span><span class="k">\setline</span><span class="p">}</span>[2]<span class="p">{</span><span class="c">%</span>
<span class="k">\savebox\myboxX</span><span class="p">{</span><span class="k">\scalebox</span><span class="p">{</span>#1<span class="p">}{</span>X<span class="p">}}</span><span class="c">%</span>
<span class="k">\savebox\myboxx</span><span class="p">{</span><span class="k">\scalebox</span><span class="p">{</span>#1<span class="p">}{</span>x<span class="p">}}</span><span class="c">%</span>
<span class="k">\heightX</span>=<span class="k">\ht\myboxX</span>
<span class="k">\heightx</span>=<span class="k">\ht\myboxx</span>
<span class="k">\noindent\ooalign</span><span class="p">{</span><span class="k">\rule</span><span class="na">[\heightX]</span><span class="p">{</span><span class="k">\textwidth</span><span class="p">}{</span>.1pt<span class="p">}</span><span class="k">\cr</span>
<span class="k">\noindent\hdashrule</span><span class="na">[\heightx]</span><span class="p">{</span><span class="k">\textwidth</span><span class="p">}{</span>.1pt<span class="p">}{</span>1mm<span class="p">}</span><span class="k">\cr</span>
<span class="k">\noindent\rule</span><span class="p">{</span><span class="k">\textwidth</span><span class="p">}{</span>.1pt<span class="p">}</span><span class="k">\cr</span>
<span class="k">\noindent\scalebox</span><span class="p">{</span>#1<span class="p">}{</span><span class="k">\pdfliteral</span><span class="p">{</span>q 1 Tr [.1 .2]0 d .1 w<span class="p">}</span>#2<span class="k">\pdfliteral</span><span class="p">{</span>Q<span class="p">}}}</span><span class="c">%</span>
<span class="p">}</span>
<span class="nt">\begin{document}</span>
<span class="k">\foreach</span> <span class="k">\n</span> in <span class="p">{</span>
<span class="k">\zeta</span>, <span class="k">\eta</span>,
<span class="k">\kappa</span>, <span class="k">\nu</span>, <span class="k">\xi</span>, <span class="k">\rho</span>, <span class="k">\sigma</span>,
<span class="k">\tau</span>, <span class="k">\phi</span>, <span class="k">\varphi</span>, <span class="k">\chi</span>, <span class="k">\psi</span>, <span class="k">\omega</span>,
<span class="k">\mathfrak</span><span class="p">{</span>g<span class="p">}</span>, <span class="k">\mathfrak</span><span class="p">{</span>h<span class="p">}</span>, <span class="k">\mathfrak</span><span class="p">{</span>n<span class="p">}</span>,
<span class="k">\mathfrak</span><span class="p">{</span>b<span class="p">}</span>, <span class="k">\mathfrak</span><span class="p">{</span>z<span class="p">}</span>, <span class="k">\mathfrak</span><span class="p">{</span>s<span class="p">}</span>,
<span class="k">\mathfrak</span><span class="p">{</span>l<span class="p">}</span>, <span class="k">\mathfrak</span><span class="p">{</span>p<span class="p">}</span>,
<span class="k">\Phi</span>
<span class="p">}{</span>
<span class="k">\foreach</span> <span class="k">\j</span> in <span class="p">{</span>1, 2, ..., 10<span class="p">}{</span>
<span class="k">\setline</span><span class="p">{</span>5<span class="p">}{$</span><span class="nv">\n\n\n\n\n\n\n\n\n\n</span><span class="p">$}</span><span class="k">\vspace</span><span class="p">{</span>0.25cm<span class="p">}</span><span class="k">\\</span>
<span class="p">}</span>
<span class="p">}</span>
<span class="nt">\end{document}</span></code></pre></figure>
<p>You can swap out anything appearing in the <code class="language-plaintext highlighter-rouge">foreach</code> section for whatever symbols you’d like to practice. So far, it has seemed to work with just about everything I’ve thrown at it, so give it a try!</p>
<p>If you have any ideas for improvements or generate cool worksheets, let me know and I’d be more than happy to link and share them here!</p>
<blockquote>
<p>Note that there is a slight caveat: each symbol is printed with a wide interior region, which can make tracing with fine-tipped pens difficult.
I don’t currently know a way around this.</p>
</blockquote>D. Zack Garzadzackgarza@gmail.comEver had trouble writing ξ? Yeah, me too, so I made these practice worksheets. Enjoy!