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 2017 courses

M 394C Topics in Stochastic Analysis

Prof. Sirbu, TTh 9:30-11 am. The course will treat a collection of topics in stochastic analysis, assuming a reasonable understanding of  measure theoretic probability and stochastic calculus for Brownian motion (for example, Theory of Probability I and II). Possible choices include: stochastic differential equations, some topics in stochastic control, stochastic calculus for processes with jumps (Levy processes and semi-martingales) and applications.

M 392C Tiling Theory

Prof. Sadun, TTh 2-3:30 pm. Aperiodic tilings give natural models for (a) extensions of symbolic dynamics to higher dimensions,  (b) expanding attractors, (c) physical quasicrystals, (d) Turing machines, (e) matchbox manifolds, (f) C* algebras, and (g) spaces with unusual rotational symmetries.  Although this course will emphasize the topological aspects of tiling theory (such as inverse limit constructions and Cech cohomology), with connections to current research, I'll also go over the historical roots of the field in logic, theoretical computer science, and crystallography.  Much of the material will be drawn from my 2008 book, ``Topology of Tiling Spaces", together with a survey of more recent results.

M 392C Riemannian Geometry

Prof. Freed, TTh 11-12:30 pm. Riemannian Geometry: This is a standard second year graduate course which begins with Riemann's notion of curvature. We will explore connections and curvature more generally, filling in some differential geometry background as we go. The second part of the course will cover some of the following themes: curvature and topology, geodesics, comparison theorems, etc.

M 392C Representation Theory

Dr. Gunningham, MWF 1:00-2:00pm. This will be an introduction to representation theory of Lie groups and Lie algebras. The prerequisites are a familiarity with basic graduate algebra and some differential or algebraic geometry. Here are some topics which may be covered (may be altered based on the background and taste of the audience):
- Basics of Lie groups and Lie algebras.
- Representation theory of compact Lie groups and reductive algebraic groups. Highest weight theory, Weyl character formula. Perhaps the Borel-Weil(-Bott) theorem.
-Representations of reductive Lie algebras. Verma modules, Whittaker modules, Category O.
-Representations of real reductive Lie groups (possibly just SL_2(R)). Harish-Chandra modules, admissible and tempered representations, principal/discrete/complementary series.
-Geometric methods. Beilinson-Bernstein localization, Kazhdan-Lusztig conjectures.

M 393C Topics in PDE

Prof. Pavlovic, 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. We will start by studying existence and regularity of solutions to linear equations of parabolic and hyperbolic type. Then we will discuss introductory topics in the theory of some important nonlinear equations (equations of fluid motion as well as dispersive equations will be considered).

M 383C Applied Harmonic Analysis

Prof. Ward, TTh 2-3:30 pm. This course should serve as an introduction to mathematical building blocks from time-frequency analysis (e.g. Fourier series, wavelets, sampling theorems) that can be used for signal and image processing, numerical analysis, and statistics. The course will emphasize the connection between the analog world and the discrete world, and focus on approximation and compression of functions and data. We will also discuss recent advances in sparse representations and compressive sensing.

M 392C Representation Theory

Dr. Gunningham, MWF 11-12 pm.

M 393C Topics in Statistical and Kinetic Transport in Natural Sciences

Prof. Gamba, TTh 12:30-2 pm. This topics course covers issues on conservative and non-conservative systems in connection to non-equilibrium and transport statistical models with applications with applications to mechanics, physics and and many other areas in natural science. More specifically, it will cover introductory elementary properties of solutions to the Boltzmann and Smoluchowski type equations an connections of entropy, time irreversibility, and energy inequalities. Asymptotics methods to grazing collision limits the Landau Fokker Plank Equation. Model reduction. Applications to kinetic models for plasmas and charge transport at nano-scale. Numerical methods of kinetic particle systems: deterministic vs. Montecarlo transport solvers for linear and non-linear collisional forms. Spectral and FEM methods.