The focus of the course will be consistency proofs.
Discussion of why Gödel's second incompleteness theorem does not preclude a finitary consistency proof for Peano Arithmetic (PA).
A finitary consistency proof for PA.
Critique of finitism.
Construction of Polynomial Time Arithmetic (PTA), weaker than Primitive Recursive Arithmetic.
Some polynomial time consistency proofs.

# Graduate courses and seminars

##### MAT515 Topics in Analytic Number Theory: Spectal theory of automorphic forms

We will discuss the proof of analytic continuation of Eisenstein Series, first in rank 1 cases and then in general. Various applications will be highlighted.

##### MAT516 Topics in Algebraic Number Theory: Galois Representations

This course will cover the basic properties, examples, and deformation theory of l-adic representations. Time permitting, the following will also be covered: the relationship with automorphic forms, particularly, automorphic lifting theorems and potential automorphy.

##### MAT517 Topics in Arithmetic Geometry: Arithmetic Gan-Gross-Prasad Conjecture

I will start with geometry and arithmetic of abelian varieties, including the moduli spaces, Mordell-Weil theorem, the Birch and Swinnerton-Dyer conjecture, and their conjectured high codimension analogues by Beilinson and Bloch. Then, I will introduce Gross-Zagier type formulae for Shimura curves and their triple products, high codimensional analogues by Gan-Gross-Prasad for unitary Shimura varieties, and a relative trace formula approach by Wei Zhang.

##### MAT518 Topics in Automorphic Forms: Philosophy of cusp forms

In this course, we will focus on some aspects of the foundation of automorphic forms. A general organization principle is that cusp forms should be the fundamental building blocks of the theory, which has many implications both to arithmetic and to local representation theory. The first part of the course will be a rigorous introduction to this circle of ideas. The aim is to give precise definitions in arbitrary dimension. The second part will be concerned with the structure of representations of real Lie groups, notably the Langlands classification, cohomological representations, the tempered dual and Vogan theory.

##### MAT520 Functional Analysis

The course is intended as a basic introductory course to the modern methods of Analysis. Specific applications of these methods to problems in other fields, such as Partial Differential Equations, Probability, and Number Theory, will also be presented. Topics will include Lp spaces, tempered distributions and the Fourier transform, the Riesz interpolation theorem, the Hardy–Littlewood maximal function, Calderon–Zygmund theory, the spaces H1 and BMO, oscillatory integrals, almost orthogonality, restriction theorems and applications to dispersive equations, the law of large numbers and ergodic theory. We will also discuss applications of Fourier methods to discrete counting problems, using the Poisson summation formula.

##### MAT522 Introduction to Partial Differential Equations

Introduction to the techniques necessary for the formulation and solution of problems involving partial differential equations in the natural sciences and engineering, with detailed study of the heat and wave equations. Topics include method of eigenfunction expansions, Fourier series, the Fourier transform, inhomogeneous problems, the method of variation of parameters. Prerequisite MAT202 or MAT204 or MAT218.

##### MAT523 Advanced Analysis

The course covers the essentials of the first eleven chapters of the textbook, "Analysis" by Lieb and Loss. Topics include Lebesque integrals, Measure Theory, L^p Spaces, Fourier Transforms, Distributions, Potential Theory, and some illustrative examples of applications of these topics.

##### MAT526 Topics in Geometric Analysis and Relativity: Introduction to general relativity

This is a fast moving introductory course in General Relativity with the goal of presenting some recent important results in the field. The following will be covered: quick introduction to general relativity, break-down criterion, a discussion of bounded L2 curvature conjecture, formation of trapped surfaces, and uniqueness of black holes.

##### MAT527 Topics in Differential Equations: Global solutions of nonlinear evolutions

We will discuss the question of existence of global smooth solutions of certain quasilinear evolution equations. These equations include plasma models and the water wave problem. The main techniques include energy methods and semilinear harmonic analysis.

##### MAT539 Topics in Complex Analysis: Brill-Noether theory for Riemann surfaces

The course will deal primarily with Brill-Noether theory on compact Riemann surfaces, the question of the number of meromorphic functions with at most some specified singularities. That has been discussed extensively on generic (or primitive) Riemann surfaces, as in the book Geometry of Algebraic Curves I by Arbarello, Cornalba, Griffiths and Harris.
I will focus more on special curves, in particular on relations between discrete invariants of the analytic structures of curves such as the maximal sequence and the Luroth sequence, and mostly in terms of line bundles. If there is time and interest perhaps some discussion of the less complete case of vector bundles, or some possibilities in higher dimensions. I will try to make the course fairly self-contained, so initially not assuming much detailed knowledge of function theory on compact Riemann surfaces; but I will not attempt to prove the basic background results.

##### MAT547 Topics in Algebraic Geometry: Arithmetic algebraic geometry (continuation)

*A continuation from Fall 2012*
We will discuss equidistribution questions about L functions, or rather "families" of L functions, in various finite field contexts.

##### MAT555 Topics in Differential Geometry: Kähler geometry

In this course, I will start with basic notions and classical results in Kähler geometry. I will discuss the Kodaira embedding theorem and the almost isometry theorem. Next, I will discuss extremal Kähler metrics and their basic geometric properties. I will then introduce the K-stability and discuss its applications. If time permits, I will also discuss some existence theorems and open problems.

##### MAT560 Algebraic Topology

Singular homology, cellular complexes, Poincare duality, Lefschetz fixed point theorem. Prerequisite: MAT323 in the old numbering system or MAT345 in the new system; in other words, abstract algebra.
(Replaces MAT326 beginning AY 2012-13)

##### MAT566 Topics in Differential Topology: Topical Invariants for Knots and Three-Dimensional Manifolds

##### MAT568 Topics in Knot Theory: Knot Floer homology

Knot theory involves the study of smoothly embedded circles in
three-dimensional manifolds. There are lots of different techniques to study knots: combinatorial invariants, algebraic topology, hyperbolic geometry, Khovanov homology and gauge theory. This course will cover some of the modern techniques and recent developments in the field.

##### MAT572 Introduction to Combinatorial Optimization

This course will survey the theory of combinatorial optimization.
We will cover:
-The elementary min-max theorems of graph theory, such as Konig's theorems and Tutte's matching theorem
-Network flows (Menger's theorem, the max-flow min-cut theorem, multicommodity flows)
-Linear programming and polyhedral optimization
-Hypergraph packing and covering problems
-Perfect graphs
-Polyhedral methods to prove min-max theorems
-Packing directed cuts, the Lucchesi-Younger theorem
-Packing T-cuts, T-joins and circuits
-Edmonds' matching polytope theorem
-Relations with the four-colour theorem
-Lehman's results on ideal clutters
-Various further topics (the ellipsoid method, the matching lattice, the theta-function) as time permits.

##### MAT576 Advanced Topics in Computer Science: Arithmetic circuits

One the fundamental goals in computational complexity is to understand which functions are easy and which are hard to compute. This course will focus on this question from an algebraic perspective by studying the complexity of computing polynomials over a field using basic arithmetic operations. The basic model of computation is an arithmetic circuit, which is a circuit that takes variables and field constants as inputs and, at each gate, computes a sum or product of previously computed polynomials. Understanding the complexity of even the simplest polynomials in this model is a challenging (and mostly open) problem. However, a rich theory has evolved around this question and there are many beautiful and highly non trivial results known. A partial list of topics includes a sample of some nontrivial arithmetic circuits for problems such as matrix multiplication, Fourier transform, polynomial evaluation etc. Structural results for circuits (homogenization, depth reduction, elimination of division gates). Lower bounds for circuits, formulas and for restricted models. Valiant's complexity classes VP and VNP and the algebraic version of the P vs. NP problem. Approaches for proving lower bounds (matrix rigidity, elusive mappings, tensor rank). Polynomial identity testing.

##### MAT579 Topics in Discrete Mathematics: Coloring and induced subgraphs

There are several well-known open questions about the coloring properties of graphs with certain induced subgraphs forbidden, for instance:
1. Erdos-Hajnal conjecture - for every graph H, there exists c>0 such that every n-vertex graph not containing H as an induced subgraph has a clique or stable set with at least O(n^c) vertices.
2. Gjarfas conjecture - for every integer k>0, there exists c>0 such that every graph with no clique of size k and no odd cycle of length >3 as an induced subgraph has chromatic number at most c.
3. Gyarfas-Sumner conjecture - for every integer k>0 and every forest F, there exists c such that every graph with no clique of size k and not cointaining F as an induced subgraph has chromatic number at most c.
The aim of this course is to survey what is known about these conjectures, and discuss some analogous results and conjectures for tournaments.

##### MAT582 Dynamical Systems

This course will concentrate on chaos in dynamical systems. We will discuss Anosov (globally hyperbolic) diffeomorphisms and flows. I intend to prove their topological stability and then discuss their stable and unstable foliations. Anosov flows and diffeomorphisms are purely chaotic dynamical systems. Ergodic theory provides a means to study chaos, and I intend to discuss the beginning of this study as it pertains to Anosov systems—Markov partitions and Gibbs measures. Time permitting, I will discuss a bit about chaos in systems where random and stable motions exist side-by-side.

##### MAT585/APC520 Mathematical Analysis of Massive Data Sets

This course focuses on spectral methods useful in the analysis of big data sets. Spectral methods involve the construction of matrices (or linear operators) directly from the data and the computation of a few leading eigenvectors and eigenvalues for information extraction. Examples include the singular value decomposition and the closely related principal component analysis; the PageRank algorithm of Google for ranking web sites; and spectral clustering methods that use eigenvectors of the graph Laplacian.

##### MAT596 Mathematical Methods in Physics: Intro to the Calculus of Variations and Spectral Theory

The credo ‘every effect in nature follows a maximum or minimum rule’ is what Euler wrote in 1744 and which since then has driven the development of the mathematical field of Calculus of Variations. We will give an introduction into this area focusing both on problems from pure mathematics and from applications in atomic physics. Symmetries will play a major role in this analysis. Topics include the isoperimetric inequality, the concentration compactness principle, rearrangements and the moving plane method, Sobolev and Lieb-Thirring inequalities and the semi-classical limit of the Schrödinger equation.

##### MATxxx Geometric Measure Theory: Regularity theory for area-minimizing currents

After a brief introduction to the theory of integral currents, I will focus on the regularity theory for area-minimizing ones. The aim is to prove the partial regularity theorem of Almgren, which shows that such objects are regular submanifolds except for a closed singular set of codimension at most two. In the course, I will follow my recent work with Emanuele Spadaro where we give a simpler approach to Almgren's result.