Fall 2017 Courses

M 393C Topics in Mathematical Physics

Prof. Chen, TTh 2-3:30 pm. The purpose of this graduate course is to provide an introduction to mathematical aspects of Quantum Mechanics and Quantum Field Theory, and to make some fundamental topics in this research area accessible to graduate students with interests in Analysis, Mathematical Physics, PDEs, and Applied Mathematics. No background in physics is required.

M 392C Contact Topology

Prof. Gompf, MWF 11-12 pm. Contact structures on 3-manifolds (or more generally, on odd-dimensional manifolds) are closely related to foliations, but are much more stable under perturbations. While foliations have been instrumental in the development of 3-manifold theory, contact structures have only been extensively studied in more recent years. It is becoming clear that the contact structures on a 3-manifold are delicately related to the underlying topology of the manifold, although the relationship is not yet fully understood. Contact structures on a manifold M can be obtained by various methods: cutting and pasting, perturbing a foliation, or restricting suitable complex or symplectic structures from a manifold bounded by M. We will investigate the current state of knowledge about contact structures (particularly in dimension 3), and about knots and links suitably compatible with a contact structure. While the course will mainly be based on research literature, the text below may be helpful for much of it. Some background material can also be found in Chapter 11 of Gompf and Stipsicz (AMS Grad Studies in Math #20, 1999) and in Chapter 8 of Aebischer et al (Progress in Math 124, Birkhauser 1994).

M 393C Random Graphs & Stochastic Geometry

Prof. Baccelli, TTh 2-3:30 pm. This is an introduction to spin geometry.  Topics include the spin groups, Clifford algebras, spin manifolds, Dirac operators and index theory, and applications.  Students should have a solid background knowledge of smooth manifolds.

M 393C Multiscale Modeling and Computation

Prof. Tsai, TTH 11-12:30 pm. This course will cover a selection of mathematical theory and computational methods commonly used in multiscale computations. The topics include:  survey of physical models used at different length and time scales, classical asymptotic and perturbation analysis, WKG methods and averaging methods for oscillatory problems, and analytical and numerical homogenization methods. We shall also discuss basic ideas of wavelet transform and L1 regularization and sparsity promoting algorithms.

M 394C Stochastic Processes I

Prof. Zariphopoulou, MW 2-3:30 pm. 

• Part I

In this class the following topics will be covered: 

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. Vasseur, MW 2-3:30 pm.

M 393C Applications of Quantum Field Theory to Geometry

Prof. Neitzke, TTh 9:30-11am. Quantum field theory has found numerous applications to mathematics and particularly to geometry over the last few decades. A particularly significant example is the relationship between Donaldson and Seiberg-Witten invariants, which revolutionized 4-manifold topology in the mid-1990's. In this course I will attempt to give an account of what this relationship is and the physical picture underlying it. This will require us to develop a fair amount of intuition about (four-dimensional, supersymmetric) quantum field theory, and in particular about the notion of "effective" field theory, which in one way or another is underlying many of the deepest applications of quantum field theory to mathematics.

M 390C Algebraic Number Theory

Prof. Ciperiani, TTh 12-12:30 pm. This will be an introductory course. We will study the ring of integers of a number field: prove the finiteness of the class group, prove Dirichlet's unit theorem, analyze the decomposition of prime ideals when lifted to a bigger field. We will continue with a brief discussion of local fields and analytic methods in number theory. We will conclude with an introduction to class field theory without proofs. Prerequisites: M 380C & M 380D

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.