# درسهای کوتاه

مدرس: هیربد آسا

مدرس: امید امینی

**Short course title:**Minicourse:<\/strong> Arithmetic geometric aspects of metric graphs

The aim of this minicourse is to provide an introduction to these recent developments with a view towards applications in arithmetic geometry. Along the way we will also treat some concepts at the heart of modern graph theory. Topics to be discussed include: Algebraic geometry of metric graphs, semistable reduction and structure of Berkovich analytic curves, graph minors-eigenvalue estimates and rational points on curves, potential theory and applications in arithmetic geometry.

1. December 29, 14:00 - 17:30, Room 317.

(8 دیماه، ساعت 14:00 تا 17:30، اتاق 317)

2. December 30, 14:00 - 17:30, Room 317.

(9 دیماه، ساعت 14:00 تا 17:30، اتاق 317)

مدرس: امید حاتمی

**Short course title:**Additive Number Theory

First Lecture:

I give introduction to this field by introducing basic results and sketching recent developments. Then I will use Fourier Analytic techniques to prove Roth's Theorem which states that every subset of integers of positive upper density contains an arithmetic progression of length three.

Second Lecture:

The idea of using Fourier analysis to study equidistribution goes back to Weyl. I'll introduce Weyl's Criterion and use it to prove that sequence

$\left( \{n^k\alpha\}\right)_{n=1}^{\infty}$ is equidistributed.

Third & Fourth Lectures:

Freiman’s theorem is a remarkable result that gives a complete characterization of sets A ⊂ Z such that the sumset A + A is not too much bigger than the set itself. I will talk about Rusza's proof of Freiman's Theorem which uses techniques from Fourier analysis, graph theory and geometry of numbers.

Perquisites for the course: I assume attendants are familiar with Fourier analysis, elementary number theory and graph theory.

1. December 21, 14:00 - 17:30, Room 317.

(30 آذرماه، ساعت 14 تا 17:30، اتاق 317)

2. December 22, 14:00 - 17:30, Room 317.

(1 دیماه، ساعت 14 تا 17:30، اتاق 317)

3. December 24, 14:00 - 16:00, Room 317.

(3 دیماه، ساعت 14 تا 16، اتاق 317)

مدرس: فریدون رضاخانلو

**Short course title:**Diffusions with Rough Drifts and Fluid Mechanics

^{p}spaces possess weakly regular flows. In the Lagrangian description of an viscid fluid, the evolution of fluid particles is given by a stochastic diffeerential equation. More precisely, the fluid velocity is a solution to Navier-Stokes Equation that is perturbed by a white noise. In this case, the corresponding (stochastic) flow is far more regular in spatial variables. In these talks I give an overview of some recent regularity estimates on the flows of diffusions with rough coefficients. I also discuss some of the consequences of these estimates for Navier-Stokes Equation.

1. December 29, 9:00 - 12:30, Room 317.

(8 دیماه، ساعت 9 تا 12:30، اتاق 317)

2. December 30, 9:00 - 12:30, Room 317.

(9 دیماه، ساعت 9 تا 12:30، اتاق 317)

مدرس: محمد صفدری

**Short course title:**Variational Inequalities

In these talks we introduce variational inequalities and consider questions of existence, uniqueness and regularity of solutions. We also introduce some applications.

1. January 11, 14:00 - 15:30, Room 317.

(11 دیماه، ساعت 14 تا 15:30، اتاق 317)

2. January 12, 14:00 - 15:30, Room 317.

(12 دیماه، ساعت 14 تا 15:30، اتاق 317)

مدرس: محمد محمودی

**Short course title:**Recent Developments in Obfuscating Programs

1. December 28, 13:00 - 17:00, Room 317.

(7 دیماه، ساعت 13 تا 17، اتاق 317)

مدرس: عباس مؤمنی

**Short course title:**An introduction to the theory of optimal mass transportation

we introduce the optimal transport problem and its formulations in terms of transport maps and transport plans. Then we introduce basic tools of the theory, namely the duality formula, the c-monotonicity and discuss the problem of existence of optimal maps for twisted costs.

Lecture 2: Geometry of Optimal Transportation:

We will talk about two situations where, one may solve Monge's problem. the discrete case, where transport maps simply are permutations and transport plans bistochastic matrices, as we shall see a celebrated result of Birkhoff says that the extreme points of bistochasticmeasures are permutation matrices, and the case of strictly convex costs where a careful inspection of the optimality conditions obtained via duality theory enables one to prove that optimal plans are actually induced by transport maps.

1. December 17, 13:30 - 15:00, Room 317.

(26 آذرماه، ساعت 13 تا 15، اتاق 317)

2. December 18, 13:30 - 15:00, Room 317.

(27 آذرماه، ساعت 13 تا 15، اتاق 317)

Short course title:Topics in financial risk management1. December 22, 9:00 - 12:30, Room 317.

(1 دیماه، ساعت 9 تا 12:30، اتاق 317)

2. December 24, 9:00 - 12:30, Room 317.

(3 دیماه، ساعت 9 تا 12:30، اتاق 317)