#### 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 *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 380C *Algebra*

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

## M 382C *Algebraic Topology*

Prof. Gordon, TTh 11-12:30 pm. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Topology.php

## M 383C *Methods of Applied Mathematics*

Prof. Gamba, TTh 2-3:30 pm. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Applied_Math.php

## M 381C *Real Analysis*

Prof. Caffarelli, 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 11-12 pm. https://www.ma.utexas.edu/academics/graduate/prelims/exam_syllabi/Probability.php

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

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

#### Spring 2019 courses

## M 394C *Mathematical neuroscience*

Prof. Taillefumier, TTh 9:30-11 am. This course is intended for mathematicians interested in neuroscience and mathematically-inclined computational neuroscientists. The emphasis will be primarily on the analytical treatment of neuroscienceinspired models and algorithms. The objectives of the course is to equip students with a solid technical and conceptual background to tackle research questions in mathematical neuroscience.

The course will be structured in three blocks: neural dynamics, information theory, and machine

learning.