## Fall 2020

**Math 5010: Yang-Mills Theory**

Instructor: Masha Gordina

Description: We will discuss mathematical foundations of the standard model of elementary particle theory.

We will begin with classical physics equations of Newtonian mechanics, then will move to the Lagrangian mechanics, Hamiltonian mechanics, quantum mechanics (including Heisenberg versus Schroedinger picture), and finally arrive to the quantum field theory. In parallel we will need to understand two classical theories, namely, Maxwell’s equations and Yang-Mills equations. The needed differential geometrical notions will be covered as well.

**Math 5020: Enumerative Combinatorics**

Instructor: Tom Roby

Description: The course will give an introduction to enumerative combinatorics at the graduate level, focusing on techniques useful for those specializing in other research areas as well as combinatorics. Topics will include: basic enumeration, multiset permutations and statistics, generating functions, bijective proofs, q-analogues, sieve methods, exponential formula, Lagrange inversion.

Text: Enumerative Combinatorics 1 (2nd edition) and 2, by Richard Stanley (selections from chapters 1,2,4,5, and 6).

**Math 5031: Einstein Manifolds**

Instructor: Fabrice Baudoin

Description: In this course we will study some topics in Riemannian and pseudo- Riemannian geometry. We will mostly focus on Ricci curvature and its applications. The course will start with basics about Riemannian and pseudo-Riemannian geometry. We will assume familiarity with differential manifolds and basic calculus on them.

We will cover the following topics:

• Linear connections on vector bundles: Torsion, Curvature, Bianchi identities

• Riemannian and pseudo-Riemannian manifolds

• Get the feel of Ricci curvature: Volume comparison theorems, Bonnet-Myers theorem

• Ricci curvature as a PDE

• Einstein manifolds and topology

• Homogeneous Riemannian manifolds

• Kahler and Calabi-Yau manifolds

• Quaternion-Kahler manifolds

Text: Einstein manifolds, by A.L. Besse, Springer, 1987.

**Math 5040: Topics in PDE**

Instructor: Xiadong Yan

Description: This topic class will cover two different topics in PDE. In 1970s, De Giorgi made a conjecture regarding level sets for solutions of semilinear PDE’s. Over the years, many mathematicians have made contributions to the problem. For the first topic, we will discuss results and technique for classical De Giorgi conjecture and its extension to fractional PDEs and (possibly) models related to thin film equations.

First introduced by Skyrme in 1960s, skyrmions are topological solitons that emerge in many physical contexts such as superfluid, Bos-Einstein condensates with ferromagnetic order, liquid crystals and magnetism. Magnetic skyrmions have attracted a lot of attentions in recent years due to their topological structure which hold promises for future information technologies. In the second part of the class, we will discuss some recent results on magnetic skyrmions.

**Math 5121: Conformal Dynamics: from the complex plane to Carnot groups**

Instructor: Vasilis Chousionis

Description: The course will cover a variety of topics whose common underlying theme is the iteration of conformal maps. Such as:

- Basics of complex dynamics including iteration of meromorphic functions
- Conformal maps in Euclidean spaces and conformal fractals
- Rigidity of conformal maps in Carnot groups
- Conformal graph directed Markov systems on Carnot groups
- Dynamics of real, complex and Iwasawa continued fractions

On the way we are going to introduce and employ ideas from geometric measure theory, ergodic theory, geometric function theory and sub-Riemannian geometry.