# Clay Lectures on Mathematics

**Lectures will be hosted by:
School of Mathemtics
Tata Institute of Fundamental Research
Homi Bhabha Rd.
Mumbai, India
**

# Public Lectures

### Integrals, the Change of Variable Formula, and Singularities

*Mircea Mustata, Monday, December 10 at 4:00*

If f is a polynomial with integer coefficients and p is a prime number, then one can encode the number of solutions of f modulo the powers of p in Igusa's zeta function. There are many analogies between this zeta function and the complex power of a polynomial with real or complex coefficients. We will see how the study of these functions brings in the picture the singularities of f, as well as other subtle invariants of singularities.

### The Geometry and Dynamics of Numbers

*Elon Lindenstrauss, Tuesday, December 11 at 4:00*

In the late 19'th century, Minkowski made a wonderful discovery: proof of certain deep facts about number fields can be best explained in geometric terms via the study of the space of lattices in n-dimensional Euclidean space, thus launching the subject of geometry of numbers.

The space of lattices can be studied from several angles, a very fruitful one being the dynamic perspective: employing techniques originally conceived in the context of studying physical systems to study these fascinating spaces brewing with number theoretic data.

# Singularities in Algebraic Geometry

*Mircea Mustata, University of Michigan*

**Singularities in the Minimal Model Program**

Tuesday, December 11, 11:30 to 12:30

**Arc Spaces and Motivic Integration**

Wednesday, December 12, 11:30 to 12:30

**Singularities in Positive Characteristic**

Thursday, December 13, 11:30 to 12:30

Singularities play an important role in the program of classifying higher-dimensional algebraic varieties. Furthermore, it was recently realized that several invariants that naturally appear in this context, show up in various other settings. The goal of these lectures is to discuss some of these points of view, and highlight recent results and the main open problems in the area.

I will start with a non-technical introduction to the role of singularities in birational geometry, and to the main conjectures in the field. We will then discuss an approach via the space of arcs on a given variety, and applications of Kontsevich's idea of motivic integration. In the last lecture I will switch to positive characteristic, and I will describe invariants of singularities of a different flavor. There are intriguing connections between the invariants in characteristic zero, and those in positive characteristic, and understanding this precise connection hints subtle arithmetic properties of the varieties involved.

# Aspects of Homogeneous Dynamics

*Elon Lindenstrauss, Princeton University*

**Flows from Unipotent to Diagonalizable (and what is in between)**

Wednesday, December 12, 4:00 to 5:00

**Values of some Integral and Non-integral Forms**

Thursday, December 13, 4:00 to 5:00

**Equidistribution and Stationary Measures on the Torus**

Friday, December 14, 4:00 to 5:00

We discuss natural group actions on homogeneous spaces G/Gamma and their rigidity properties. The study of such actions has a long and distinguished tradition, with numerous applications, notably in number theory.

Emphasis will be given to two current frontiers in the study: actions of diagonalizable groups, such as the action of the full diagonal group on the space of lattices SL(n, R) / SL(n, Z), and actions of Zariski dense but infinite covolume discrete subgroups.