CohomologyForDummies

Modern Pure Mathematics for Modern Applied Mathematicians

(MoPuMMAM)

Fall / Spring 20xx IAP 2017 Not-For-Credit Short Course

Instructor: Homer Reid

I will be offering a short (not-for-credit) version of this course during weeks 3 and 4 of IAP in January 2017. If you might be interested in attending, please email me so I can gauge interest. If you have specific subject areas in mind, or specific reasons for wanting to learn this material, let me know (if you don't, that's fine too!) Also, check the (corrected) preliminary IAP schedule and let me know if you have any conflicts.

Bicycle Tours around Lake Cyclotomic: Algebraic Number Theory



Mission Statement

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:

  1. To survey four of the central pillars of modern pure mathematics---number theory, algebraic topology, differential geometry, and algebraic geometry---and to develop, in each case, at least a feel for the big picture: the most important solved and unsolved problems in the field, the basic logical and computational techniques, and a first rough appreciation of why anyone cares.
  2. Along the way, to introduce and demystify the terminology of modern pure mathematics---including notions such as short exact sequences, functoriality (category theory), and cohomology---so that pure-math papers are rendered less scary and more readable to the applied-math audience. Indeed, our basic conceit will be that many of the ubiquitous and seemingly inscrutable notions of modern pure mathematics---very much including the notions of short exact sequences, functoriality, and cohomology---are in fact just fancy ways of describing simple concepts that are already familiar to any practicing applied mathematician, just under different guises.

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.


Course Outline

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.


Schedule for IAP 2017 Short Course (Not for credit)

Note: classroom locations have changed! See below.

Date Time Location Topic
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
  • Algebraic structures: groups, rings, fields, modules, algebras.
  • More on cohomology: The basic idea and some metaphors to help you picture it.
  • Algebraic topology:
    • Homology: Programming computers to distinguish donuts from beach balls
    • Homotopy: High-dimensional cowboys and lassos and the wild-west frontier of modern pure mathematics
    • Betti numbers: The algebraic-topology version
    • A glimpse at abstract algebraic geometry
Friday 1/27/2017 2:30-4 PM E25-111 Algebraic geometry:
  • Polynomial rings, ideals, modules, Koszul complex, free resolutions, Hilbert function
  • Betti numbers: The algebraic-geometry version
  • The resultant of several polynomials in several variables
Monday 1/30/2017 2:30-4 PM E25-111 Differential Geometry:
Wednesday 2/1/2017 2:30-4 PM E25-111 Number Theory:
Friday 2/3/2017 2:30-4 PM 6-120 Homological algebra in algebraic geometry: The resultant of multiple polynomials in multiple variables


Lecture Notes

General Surveys / Overviews / Invitations

Number Theory

Algebraic Topology

Differential Geometry

Algebraic Geometry

Miscellaneous


Other References

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).

Number Theory References

Algebraic Topology References

Differential Geometry References

Algebraic Geometry References

The Chicago undergraduate mathematics bibliography

A very useful and extensive resource in the sector of pure mathematics for pure mathematicians is the
Chicago undergraduate mathematics bibliography. This is a great place to read about what pure-math undergraduates think about their own textbooks, and which ones they do or don't recommend to each other, for which reasons. (It's particularly affirming when one of them says they found a book difficult---kinda makes you feel like less of a simpleminded applied-math bumpkin for struggling with the pure-math literature yourself.) This resource is a little out of date (last updated probably 10+ years ago), but the pure-math textbook landscape doesn't change very rapidly, so many of the books they discuss are still widely used today.
MoPuMMAM IAP 2017 Main course page