Modern Pure Mathematics for Modern Applied Mathematicians
Consider the following questions.
If you answered yes to any of these questions, then MoPuMMAM might be the course for you.
The goals of MoPuMMAM are twofold:
Note that MoPuMMAM is really intended as a course for applied mathematicians. (We will use this catch-all phrase to mean really anybody who uses mathematics to solve problems: engineers, physicists, chemists, numerical scientists, etc.) If you are a pure mathematician, this course may whet your appetite for future study, but ultimately may be a poor use of your time; almost certainly you will need to learn each of the four topic areas we cover in the full depth of a semester-long course, not just the three-week treatment we give each area here.
MoPuMMAM will cover four primary subject areas, devoting roughly equal time to each one. (What follows is a brief synopsis; see here for a more detailed overview of the topics we will cover.)
We begin with number theory, the use of sophisticated analytic and algebraic techniques to study patterns and other interesting phenomena in the integers; we will touch briefly on both the analytic and algebraic branches of the field. We start here because the problems to be solved are easily stated and understood by any applied mathematician---for example, which integers are squares modulo 13?---but the techniques used to solve them, such as Gauss' law of quadratic reciprocity, quickly lead us in to some of the more sophisticated pure-math machinery we will encounter again and again in our course. In particular, many algebraic tools (groups, rings, fields, modules, homomorphisms) have an especially natural and intuitive significance in the number-theoretical context.
Next we move on to algebraic topology, the use of algebraic tools---including some of the same tools (groups, rings, modules, homomorphisms) we encountered above---to understand complicated high-dimensional spaces. This subject is important not only because it gives us powerful tools for understanding the properties of Klein bottles, high-dimensional spheres, fiber bundles, and other spaces---spaces that generally lie well beyond the reach of our feeble three-dimensional geometric intuition, and would thus remain entirely mysterious to us without the tools of algebraic geometry---but also because it furnishes a particularly clear-cut example (indeed, historically the original motivating example) of the power of category-theoretical reasoning. This is the observation, first codified into a formal mathematical theory in the 1940s, that many structures in mathematics crop up over and over again in different guises in different branches of mathematics, and that by studying the wigs, costumes and makeup needed to convert one guise into another we can wind up learning new things about the mathematical branches in question. The logic and terminology of category theory is ubiquitous in modern pure mathematics and essentially absent from modern applied mathematics; a key goal of MoPuMMaM is to offer at least a partial bridge over this gap.
Throughout our unit on algebraic topology, we will discuss high-dimensional shapes in a way that will never require us to introduce coordinates or functions. Although it is nice to have such abstract, intrinsic definitions of spaces, eventually it will start to get annoying that we can't integrate or differentiate anything. This will set the stage for our third unit, differential geometry, in which the notions of single-variable and multivariable calculus that you use in your everyday applied-math work are generalized to spaces more complicated than RN. Although this subject can be formulated in a way that requires no exposure to algebraic topology, it is enriched by an understanding of that field; in particular, de Rham cohomology will be easy to understand now that we have already learned simplicial cohomology, while the theory of characteristic classes would hardly make sense at all without having first seen the algebraic structure of fiber bundles. We will begin by discussing the notions of differential forms and the exterior derivative on manifolds, which are easy once you figure out that you already know all about them, just under different names (namely, the gradient, curl, and divergence in R3). Then we will discuss tensors and tensor fields---particularly (1,0)-tensors (vectors), (0,1)-tensors (covectors), and (0,2)-tensors, and we will note the vast world of geometrical richness, including ideas like geodesics and parallel transport, that opens up once we designate a particular (0,2)-tensor field to serve as a metric. As immediate bonus applications we will (a) derive the Schwarzschild black-hole solution to Einstein's general-relativistic gravitational field equation and (b) understand in what sense the vector potential in Maxwell's electromagnetism may be understood as a "connection on a principal U(1) fiber bundle."
The course will culminate in a discussion of algebraic geometry, in which simultaneous systems of polynomials are attacked by considering the geometry of the sets of points that solve them. Out of all four pure-math areas we survey, this field, with its clever tricks for solving polynomial systems, seems to me most ripe for exploitation by the applied mathematician; my guess as to why it remains relatively unknown in the engineering and science communities is that, throughout its history, its primary contributors, expositors, and champions have hewed toward a relentlessly abstract, austere, modern presentation that is undoubtedly beautiful and profound to those who understand it, but can come across as somewhat private to those of us who don't. Thankfully, in the 1990s there appeared a new generation of textbooks (especially those by Cox, Little, and O'Shea) that, while not shying away from the rigorous foundations and intrinsic beauty of the subject, also emphasize its practical value in a way that applied-thinking mathematicians can immediately appreciate. Our discussion of algebraic geometry will largely adopt this approach; we will learn the tools of resultants and Gröbner bases, build on the algebraic foundations we have developed earlier in the course to understand why these tools work, and then put them to work to solve real-world science and engineering problems requiring the solution of simultaneous systems of polynomials. Toward the end of our unit we will attempt to go beyond the realm of practical applications to give a glimpse of the more abstract approach to the subject pursued by most practicing algebraic geometers.
|Monday 1/23/2017||2:30-4 PM||E25-117||Invitation: Cohomology for Dummies: Deciphering the world of Pure Math for Applied Mathematicians|
|Wednesday 1/25/2017||2:30-4 PM||E25-111||
|Friday 1/27/2017||2:30-4 PM||E25-111||
|Monday 1/30/2017||2:30-4 PM||E25-111||
|Wednesday 2/1/2017||2:30-4 PM||E25-111||
|Friday 2/3/2017||2:30-4 PM||6-120||Homological algebra in algebraic geometry: The resultant of multiple polynomials in multiple variables|
There is no assigned textbook for this course, but you may find the following references helpful. The links will direct you to online versions (in some cases you will need MIT certificates to access them).
This is my favorite general-purpose textbook on number theory, and one of the first pure-math books I ever read. It is chatty and forgiving enough to be readable already by an engineer with no pure-math experience (as I was), but fairly quickly introduces a host of advanced topics that will take you right up to the frontiers of modern research. Highly recommended.
These are big books of worked problems that guide you through the analytic and algebraic branches of the field. Excellent for self-study.
A more advanced tome for those familiar with the content of the first Ram Murty problem book and ready to tackle the main modern tools in the analytic branch of the field.
A fun book describing all sorts of interesting problems to which the LLL lattice-basis-reduction algorithm and related tools may be profitably applied. Gives a good sense of the experimental and computational flavor of the field.
An impossibly valuable reference on pretty much all branches of modern mathematics, with number theory being no exception. On this site you will find everything ranging from detailed expositions of basic topics to contemporary accounts of recent breakthroughs. Highly recommended.
A big, friendly book with tons of pictures and diagrams. Offers a thorough and well-motivated introduction to all the essential tools in the elementary topologist's kit.
In my opinion, this is not the place to start for algebraic topology or for differential geometry, but is rather a beautiful book to pick up once you have acquired a first exposure to both fields. The long and lovely Introduction to this book alone will immediately advance your understanding of the two fields---and how they fit together---by a significant factor, and the rest of the book will either teach you new ideas or else hammer home old ones more clearly. In particular, ideas you might previously have seen presented somewhat abstractly in an algebraic topology setting are here formulated in the more explicit context of the de Rham complex on a manifold.
These two books by the same trio of authors---which cover similar content at different levels of breadth and sophistication---are, in my view, without question the best place for an applied mathematician to start with algebraic geometry. Already in the title of the first volume we have an indication that the techniques are to be put to some use beyond just standing around gawking at how beautiful they are supposed to be; I think just the first three chapters of UAG should suffice to get anybody interested in and comfortable with the basics of the field. Highly recommended.
Although slightly different in content, I am placing these books together here because they are quite similar in style---in particular, they are both chatty, readable, and informal, adjectives which describe a painfully slim sector of the algebraic geometry textbook market.
In particular, the book by Reid (no relation) contains some invaluable insights into the sociology of the field of algebraic geometry; for example, he acknowledges head on, and bemoans, the extreme levels of abstraction and formalism that render many other textbooks useless for applied-minded readers, and offers valuable historical background on the development of the field to explain how things came to such a pass.
Meanwhile, the book by Schenck is a useful complement to the books by Cox, Little and O'Shea---it starts at roughly the same place (and assumes roughly the same level of background, which is to say not much), but then develops the material in different directions.
These are notes for a course on algebraic geometry pitched at math graduate students who are sure they want to focus on algebraic geometry. That is to say, they are rigorous, comprehensive, and not hand-holding. But they give a glimmer of the abstract side of the field which complements the understanding you will get from the more applied approach of the Cox, Little, and O'Shea books above or the more elementary perspective of the Reid and Schenck books.
|MoPuMMAM IAP 2017||Main course page|