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Fall 2018 Courses

M 392C Algebraic Geometry

Prof. Raskin, MWF 9-10 am. This course is an introduction to algebraic geometry, the study of zero sets of systems of polynomial equations. It is a broad subject, with roots in classical Euclidean geometry and deep ties to number theory and differential geometry. In fact, much of the pre-20th century mathematics we now characterize as algebraic geometry was done by groups of researchers who were not in communication, and who would find each other's work incomprehensible and distant.

M 392C Four Manifolds

Prof. Gompf, MWF 11-12 pm. While work of the past four decades has revolutionized our understanding of 4-manifolds, much of the subject now seems more mysterious than ever. Virtually nothing was known about 4-manifolds until 1981, when Freedman's revolutionary work led to a complete classification of simply connected topological 4-manifolds. Shortly thereafter, Donaldson led a counter-revolution that showed that smooth 4-manifolds were much more complicated than their topological counterparts. That is, many topological 4-manifolds admit infinitely many diffeomorphism types of smooth structures, and others cannot be smoothed at all, frequently in defiance of the predictions of high-dimensional smoothing theory. Currently there isn't even a good guess about how to organize the manifolds that can be distinguished by the gauge-theoretic techniques of Donaldson and others. Many smooth 4-manifolds admit complex structures, and a much larger class admits symplectic structures, but there seems to be a sense in which "most" smooth 4-manifolds do not admit symplectic structures. While much of current research is focused on how to organize such examples, other basic problems remain completely open - for example, does the 4-sphere admit exotic smooth structures? In contrast, while Rn admits a unique smooth structure for n not equal 4, R4 admits uncountably many smooth structures.

M 392C Morse Theory

Prof. Freed, W 4-7 pm. The critical points of a smooth function on a smooth manifold encode information about its topology, and conversely the topology constrains the structure of critical points. This idea was exploited by Marston Morse, especially in the infinite dimensional example of the loop space of a smooth manifold. An important variation in infinite dimensions was introduced by Floer, which has led to many striking applications. The course will begin with the basic theory, following the classic text by Milnor. In the second half of the course we will treat some of the modern topics, depending on the interest of students. Some of that material will be from a text by Nicolescu. I will lecture half of the time and students in the course, working in pairs, will lecture the other half. I will provide additional materials and coaching to help students with their lectures. Basic knowledge of manifold theory (at the level of the differential topology prelim class) is necessary, and some Riemannian geometry wouldn't hurt either. I will review some necessary background; summer reading is recommended!

M 393C Gibbs Measures & Random Graphs

Prof. Bowen, MWF 12-1 pm.  

This course is on random processes (such as random walks, percolation, uniform spanning trees, the Ising model) on graphs, usually Cayley graphs of groups. The emphasis is on how geometry (or geometric group theory) plays a role in the qualitative aspects of the process. For example, amenable Cayley graphs admit at most one infinite Bernoulli percolation cluster, but non-amenable Cayley graphs can admit infinitely many.

Part of the course will consist of student lectures. I will provide additional materials and coaching to help students with their lectures.

Some useful (but not required) references include:
1) Probability on Trees and Networks by Lyons with Peres
2) N. Alon and J. Spencer, The probabilistic method, 3rd Edition, Wiley, 2008.
3) Itai Benjamini and Oded Schramm. Percolation beyond Zd, many questions and a few answers [mr1423907]. In Selected works of Oded Schramm. Volume 1, 2, Sel. Works Probab. Stat., pages 679–690. Springer, New York, 2011.

4) Itai Benjamini, Russell Lyons, Yuval Peres, and Oded Schramm. Uniform spanning forests. Ann. Probab., 29(1):1–65, 2001.

5) N. C. Wormald. Models of random regular graphs. In Surveys in combinatorics, 1999 (Canterbury), volume 267 of London Math. Soc. Lecture Note Ser., pages 239–298. Cambridge Univ. Press, Cambridge, 1999.

Basic knowledge of probability theory is necessary. No knowledge of amenability or geometric group theory will be assumed.

 

M 394C Stochastic Processes I

Prof. Zariphopoulou, MW 2-3:30 pm. 

In this class the following topics will be covered:

• Part I
o Ito integral and stochastic calculus
o Stochastic Differential Equations (SDE)
o SDE and linear partial differential equations
o Applications to boundary value problems
o Applications to optimal stopping
o Introduction to filtering
• Part II
o Stochastic control of controlled diffusion processes
o The Hamilton-Jacobi-Bellman equation
o Viscosity solutions
o Introduction to risk sensitive control
o Introduction to singular stochastic control
o Applications in Mathematical Finance and other areas will be also presented.

BACKGROUND: The course will build on material covered in Probability I and Probability II. While these courses are not prerequisites, familiarity with their content is strongly recommended. The students must have taken an advanced course of Real Analysis and/or Probability Theory.

M 398T Supervised Teaching in Math

Prof. Maxwell, TTh 9:30-11 am. The purpose of this course is to give you the ability to have enduring success improving the effectiveness and efficiency of your teaching. Human learning is complex, and any insight we can gain regarding how we and our students acquire knowledge and skills is valuable in our role as lifelong learners and educators. This course is intended to help you acquire a basic understanding of the fundamental principles of learning mathematics, a practical perspective of your own strengths and weaknesses in the classroom, and a compelling interest in meeting the challenges that are before you in your graduate education and your future profession.

M 393C Partial Differential Equations I

Prof. Patrizi, TTh 3:30-5 pm. 

M 393C Minimal Surfaces

Prof. Maggi, TTh 11-12:30 pm. We will review the theory of minimal surfaces, starting with the classical examples and the theory developed by Osserman and others in the 50s and 60s, and then coming to more recent developments. The AMS book by Colding and Minicozzi will be used as the main textbook.

M 393C Probability & Stochastic Processes

Prof. Baccelli, TTh 3:30-5 pm. https://www.ma.utexas.edu/users/baccelli/syllabus-ece-18.pdf

M 392C Geometric Foundations of Data Science

Prof. Bajaj, MW 9:30-11 am. Recent topics have included algebraic topology, differential topology, geometric topology, Lie groups.

Spring 2018 courses

M 394C Probabilistic Theory of Mean-Field Interactions

Prof. Zitkovic, MWF 9-10 am. We will take a probabilistic look at the circle of ideas surrounding the notion of mean-field interaction. Tracing its roots to statistical mechanics at the turn of the (19th) century, and the work of Pierre Curie and Pierre Weiss, the mean-field paradigm attempts to deal with the complexity of many-particle systems by replacing the multitude of individual interactions by the interactions of single particles with a well-chosen "statistic" of the whole (the mean field). In recent years, mean-field models, largely in a game-theoretic setting, have found their way into several nontraditional areas of applications, from ``Mexican waves'', over bat-colony dynamics, to jet-lag modeling, stock-market dynamics and principal-agent economics.

The course will start with a review of some basic probabilistic tools and a crash course on stochastic analysis. Other mathematical concepts, such as weak- and Wasserstein-type convergence of measures, will be developed on the go. The bulk of our time will be spent on the McKean-Vlasov model, the ``propagation of chaos'' phenomenon, stochastic differential equations of the McKean-Vlasov type, and related concepts. The last part of the course will deal with mean-field games of Lasry and Lions and some recent developments in that framework.

M 393C Viscosity Solutions and Applications

Prof. Patrizi, TTh 12:30-2 pm. We first cover the basic theory of viscosity solutions: existence, uniqueness, and stability properties. Then, we discuss some applications: homogenization, mean curvature flow, regularity results for second order uniformly elliptic equations. If time permits, we will also study viscosity solutions for nonlocal
operators and the Peierls-Nabarro model.

M 393C Partial Differential Equations II

Prof. Maggi, TTh 11-12:30 pm. This is the second semester of a year long course which serves as an introduction to the modern mathematical treatment of linear and nonlinear partial differential equations. The beginning of the course will be devoted studying existence and some properties of solutions (e.g. regularity) for linear equations of parabolic and hyperbolic type. Then we will discuss introductory topics in the theory of some important nonlinear equations (including but not limited to: nonlinear wave equations and nonlinear dispersive equations).

M 392C Geometry/Topology/Physics

Prof. Freed, W 4-7pm. The Dirac operator is a first-order linear elliptic differential operator, originally introduced in Lorentz signature, but with many incarnations and applications in Riemannian geometry, differential topology, and beyond. On a compact manifold, or family of compact manifolds, a Dirac operator has many topological and geometric invariants. The most basic is the Fredholm index, which only depends on the kernel of the operator. The topological formula for the index is the Atiyah-Singer index theorem. Geometric invariants, such as the Atiyah-Patodi-Singer eta-invariant and determinant line bundle, are constructed from the entire spectrum of the Dirac operator, and there are geometric versions of the index theorem which pertain. These invariants have many contemporary applications in geometry and physics. 

Students in the course will give many of the lectures; the instructor will provide materials and coaching.

 

M 392C Lie Groups

Prof. Allcock, MWF 10-11am. These are groups which are also manifolds, named after Sophus Lie.  Their structure theory and applications are never-endingly rich.  We will go from the beginning of the theory to the classification of simple Lie groups, and cover some basic representation theory and applications.  This course is aimed at graduate students who have already taken both semesters of the graduate topology course (algebraic topology and differential topology), and the first semester of the graduate algebra prelim course (the group theory part).  You will be expected to have this background on day one,  so we can hit the ground running.  There will be homework assigned every week or two. The text will be Rossman's "Lie Groups: An Introduction through Linear Groups", supplemented by additional material.

M 392C Knots/3-Manifolds

Prof. Gordon, TTh 9:30-11am. The course will cover a topic in low-dimensional topology and knot theory. A possible topic is knot concordance, in both the smooth and topological categories.

M 392C Gauge Theory

Prof. Perutz, TTh 11-12 pm.

This will be a course about the Seiberg-Witten equations over 4-dimensional manifolds. We will use these equations to give examples of 4-dimensional homotopy types that admit no smooth manifold structure, and others that admit infinitely many: examples that show that 4-dimensional manifolds do not play by the rules that govern smooth manifolds of any other dimensions. Besides measuring the difference between homotopy theory and smooth topology, we shall also use the Seiberg-Witten equations to detect differences between smooth and symplectic topology in dimension 4.

What rules do in fact govern smooth 4-manifolds? The answer is, to date, the greatest mystery in geometric topology.

Methods used in the course will be from algebraic topology and differential geometry - for which the two topology prelims provide appropriate background - and geometric analysis - for which I will cover background material on elliptic operators. The course will complement the 4-manifolds course taught regularly by Prof. Gompf (which has a quite different flavor), and will provide training for students interested in Heegaard Floer and related theories for 3-manifolds.

Reference: S. K. Donaldson, The Seiberg-Witten equations and 4-manifold topology, Bulletin of the AMS, 1996; http://www.ams.org/journals/bull/1996-33-01/S0273-0979-96-00625-8/ 

M 391C An Introduction to Diffusion Processes

Prof. Caffarelli, MWF 2-3pm.