# درسهای کوتاه

Short course title: Topics in financial risk management

This short course tries to discuss some research I have been doing during past couple of years in risk management with risk measures. I try to cover topics in pricing, portfolio management as well as empirical assessment of the discussed models, within which we deal with actual data. We will discover how risk measures can add to our knowledge about pricing and hedging. This course tries also to give further insight into the financial market principals, suitable for people who are interested in financial discussions. If time allows (probably the last session out of four) I will start a different discussion on agricultural market risk management. This course is suitable for people in mathematics, finance and economics.

1. 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)

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

Graphs arise naturally in algebraic geometry in the study of degenerations of algebraic curves. In addition, non-archimedean analytic geometry in the sense of Berkovich provides a neat way of realizing metric graphs as skeleton of analytic curves. However, it has been only recently that an algebraic geometry over metric graphs has been developed.
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)

Short course title: Additive Number Theory

Additive Number Theory is a theory of counting additive structures in sets. It is a new and rapidly growing area of mathematics that uses techniques from harmonic analysis, extremal combinatorics, ergodic theory, and analytic number theory. This mini-course will be an introduction to some of those techniques: the core of the course will be around proving one of the central results in the area, namely Szemeredi's theorem on arithmetic progressions, which states that for every δ > 0 there exists a positive integer N such that every subset of {1,2, . . . , N} of size at least δN must contain an arithmetic progression of length.

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)

Short course title: Diffusions with Rough Drifts and Fluid Mechanics

In the Lagrangian description of an inviscid fluid, the velocity of fluid particles satisfies the Euler Equation. Classically, a Lipschitz continuous vector (velocity) field has a Lipschitz continuous flow. However, the velocity field coming from Euler equation is rough and we cannot make sense of its flow in classical sense. According to DiPernaLions theory, velocity fields with weak derivatives in Lp 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)

Short course title: Variational Inequalities

A typical problem in Calculus of variations is to find the minimizer of functionals of the form $\int F(x,u(x),\nabla u(x))dx$ over a subset of a Banach space of functions, consisting of functions satisfying some boundary conditions. There are situations in applications where we need the minimizer to satisfy some extra constraint like $u(x)\leq g(x)$ or $|\nabla u|\leq 1$. In these problems the “Euler-Lagrange equation” is no longer an equation but is an inequality, hence the name variational inequalities. Variational inequalities arose in the study of several mechanical problems, but have found applications in other fields from engineering to economics. They also have a close connection to free boundary problems.
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)

Short course title: Recent Developments in Obfuscating Programs

Obfuscating a program is a process through which a program is taken as input and an equivalent program is generate as output which is "unintelligible". In 2001 Barak et al [1] showed that a "strong" form of obfuscation is impossible. Very recently, Garg et al [2] showed that a "relaxed" form of obfuscation is "plausible" to be possible by presenting a candidate solution. Since the work of [2] many applications of [2] are shown to be possible. In this short course we will go over these notions of obfuscations and their applications, with the hope that further applications of this strong tool could be explored.
[1]On the (im)possibility of obfuscating programs​​.​ ​by Barak, Goldreich, Impagliazzo, Rudich, Sahai, Vadhan, Yang
[2]Candidate Indistinguishability Obfuscationand Functional Encryption for all circuits. by Garg, Gentry, Halevi, Raykova, Sahai, Waters​

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

(7 دی‌ماه، ساعت 13 تا 17، اتاق 317)

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

Lecture 1: Formulation and the Kantorovich duality:
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)