#### Fall 2016 Courses

## M 393C * Nonlocal Elliptical Equations & Related Nonlinear Problems*

Prof. Ros-Oton, TTh 2-3:30 pm. We will study Nonlocal elliptic equations, which involve integro-differential operators like the fractional Laplacian. We will study existence and regularity of solutions, both for linear and nonlinear problems. / Some of the topics we will study are the following: the Dirichlet problem in a bounded domain; interior and boundary regularity of solutions; semilinear equations; obstacle problems and fully nonlinear equations (if time permits).

## M 392C *Symplectic Topology*

Prof. Gompf, MWF 11-12 pm. We will be studying symplectic structures (closed, nondegenerate 2-forms) on manifolds, with an emphasis on 4-dimensional manifolds and topological (cut and paste) constructions. The course will be similar to the 2013 course described here: https://www.ma.utexas.edu/academics/archives/2013f/descriptions/M392Cgom.php

## M 392C *Spin Geometry*

Prof. Korman, 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*

Dr. J. Arledge, 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 390C * Geometric Langlands*

Prof. Ben-Zvi, TTh 9:30-11am. The course will provide a survey centered around the theme Geometric Representation Theory and Topological Field Theory. Starting from the encoding of basic structures in representation theory of finite groups via the topology of curves and surfaces, we'll work our way through the increasingly rich settings of three- and four-dimensional topological field theory as a unifying framework for increasingly subtle structures in representation theory. The end-goal is to gain a broad understanding of the Geometric Langlands Correspondence and its relation to the Fourier transform and electric-magnetic duality.

## M 380C *Algebra*

Prof. Bowen, MWF 1-2 pm. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Algebra.php

## M 382C *Algebraic Topology*

Prof. Danciger, MWF 10-11 am. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Topology.php

## M 393C *Methods of Applied Mathematics*

Prof. Arbogast, MWF 11-12 pm. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Applied_Math.php

## M 381C *Real Analysis*

Prof. Ren, TTh 12:30-2 pm. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Analysis.php

## M 385C *Theory of Probability I*

Prof. Zitkovic, MWF 2-3 pm. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Probability.php

## M 387C *Numerical Analysis: Algebra & Approximations*

Prof. Engquist, MW 8:30-10 am. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Numerical_Analysis.php

#### 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 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.

## M 380D *Algebra*

Prof. Mohammadi, MWF 9-10 am. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Algebra.php

## M 382D *Differential Topology*

Prof. Gompf, MWF 11-12 pm. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Topology.php

## M 383D *Methods of Applied Mathematics*

Prof. Vasseur, TTh 9:30-11 am. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Applied_Math.php

## M 381D *Complex Analysis*

Prof. Koch, MWF 11-11 am. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Analysis.php

## M 385C *Theory of Probability II*

Prof. Zitkovic, MWF 2-3 pm. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Probability.php

## M 387C *Numerical Analysis: Differential Equations*

Prof. Dawson, MWF 9-10 am. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Numerical_Analysis.php