# درس‌های کوتاه

Short course title: Some applications of forcing in the model theory of arithmetic and set theory

Time:

Wed. Jan. 7, 2015  15:00-17:00

Thu. Jan. 8, 2015  10:00-12:00

Location: Lecture Hall 2, Niavaran Building, IPM

Short course title: Toric geometry and moduli spaces of sheaves

Torus action can be used to explore some aspects of the geometry of the moduli spaces of coherent sheave on toric varieties, such as Euler characteristic, cell decomposition, Poincare polynomial, etc. We will give combinatorial descriptions of these invariants and show that some of them have modular properties. Our main examples are the moduli spaces of rank 1 and rank 2 torsion free sheaves on smooth toric varieties and smooth toric DM stacks. The talk will include an introduction to Donaldson-Thomas invariants in the toric setting.

Time:

Wed. Dec. 24, 2014  10:30-12:30 and 14:00-16:00

Location: Room 221, Dept. Math. Sci., Sharif University of Technology

Short course title: Topics in the study of the Laplacian eigenvalues

Eigenvalues of the Laplacian are important invariants which appear in different areas of mathematics and physics. In particular, spectral geometry aims at revealing the deep relationships between these eigenvalues and geometry of the underlying manifold. Among various approaches, the focus of this mini-course would be on the study of the multiplicity of eigenvalues. Extensive studies have been devoted in this direction which give deep and beautiful results. We give a survey on main results in this direction starting from the work of Cheng on multiplicity bounds on Riemannian surfaces. Then we study the multiplicity of eigenvalues on manifolds of dimension greater than two, including the work of Li and Colin de Verdiere, illustrating the main difference between two and higher dimensional manifolds.  Finally we discuss recent developments and  interesting open questions in this direction.

Time:

Sat. Dec. 27, 2014  15:30-17:00

Sun. Dec. 28, 2014  15:30-17:00

Location: Room 317, Dept. Math. Sci., Sharif University of Technology

Short course title: An Introduction to Prime Number Theory

I'm going to give three expository talks about recent developements in Prime Number Theory. First lecture will be about Green-Tao Theorem which states prime numbers contain large arithmetic progressions. Second talk will be about recent developments about Goldbach Conjecture. In my third talk, I will talk about recent developements about Twin Prime Conjecture. I'll try make my talks as elementary as possible but it would be useful to know techniques in Analytic Number Theory such Higher Fourier Analysis, Sieve Theory and Hardy–Littlewood circle method.

Time:

Sat. Jan. 3, 2015  10:00-12:00

Sat. Jan. 3, 2015  14:00-16:00

Location: Lecture Hall 1, Niavaran Builiding, IPM

Short course title: Algebraic equations and convex bodies

We develop an intersection theory for subspaces of rational functions on algebraic varieties extending the intersection theory of divisors. We will discuss relations and analogies with theory of convex bodies in Euclidean space. I will assume familiarity with basic notions of algebraic geometry such as variety, regular and rational functions.

Time:

Sat. Dec. 27, 2014  10:30-12:00 and 13:30-15:00

Sun. Dec. 28, 2014  10:30-12:00 and 13:30-15:00

Location: Room 317, Dept. Math. Sci., Sharif University of Technology

Short course title: Values of Quadratic Forms and Oppenheim Conjecture

If $Q=\sum_{i,j=1}^n a_{ij}x_ix_j$ is a quadratic form in $n$ variables with real coefficients, the set of values $\{Q(x)\ |\ x\in\mathbb{Z}^n\}$ is denoted by $Q(\mathbb{Z}^n)$. It is easy to see that if $Q$ is definite or it is a real multiple of an integral quadratic form then $Q(\mathbb{Z}^n)$ is not a dense subset of the real numbers. In 1929, Oppenheim conjectured that if $n$ is at least 5 and the above two conditions are not satisfied then $Q(\mathbb{Z}^n)$ is dense in $\mathbb{R}$.
Ragunathan reformulated a version of this conjecture in 70s in the language of the action of Lie groups, where the condition $n\geq 5$ is replaced by $n\geq 3$. More precisely, Ragunathan proved the equivalence of Oppenheim conjecture with the statement that every orbit of the action of $S=SO(2,1)$ on $SL_3(\mathbb{R})/SL_3(\mathbb{Z})$ is compact. This formulation of the Oppenheim conjecture was subsequently proved by Margulis in 1986.
In this short course, we will first show the equivalence of Oppenheim conjecture with Ragunathan's formulation, and will present a simplified version of Margulis' proof. Finally, we will mention some of the generalizations of this theorem without going into details.

Time:

Tue. Jan. 6, 2015 13:30-15:00 and 15:30-17:00

Wed. Jan. 7, 2015 13:30-15:00

Location: Lecture Hall 1, Niavaran Building, IPM

Short course title: L-functions and principle of functoriality

Time:

Sun. Jan. 4, 2015  14:00-15:30

Mon. Jan. 5, 2015  14:00-17:00

Location: Lecture Hall 1, Niavaran Building, IPM

Short course title: An introduction to estimation and inference in graphical models and networks

Networks are widely used to model interactions among components of biological, social and physical systems. We will discuss statistical methods for estimation and inference in high dimensional networks, with a particular emphasis on graphical models and recent developments in statistical machine learning. Both directed and undirected networks are discussed, and applications and case studies in biology are covered. Codes for selected examples are demonstrated in the R programing language.

Time:

Sat. Dec. 20, 2014  15:00-17:00

Mon. Dec. 22, 2014  15:00-17:00

Wed. Dec. 24, 2014  8:00-12:00

Location: Room 317, Dept. Math. Sci., Sharif University of Technology

Short course title: Disintegration of measures and fine ergodic property of smooth dynamics

1. Preliminaries on measurable partitions, conditional expectation, disintegration of measures along leaves of a foliation

2. Absolute continuity, atomic dichotomy in disintegration of partially hyperbolic dynamics

3. Hierarchy in ergodic properties: Katok's example on non equivalence of Kolmogrov and Bernoulli systems and new results on equivalence of Kolmogrov and Bernoulli in one dimensional central case in partially hyperbolic systems.

The third item may or may not be reached in the minicourses. But in the Frontier talk we try to cover the third part.

Time:

Sun. Dec. 14, 2014  14:00-15:30

Mon. Dec. 15, 2014  9:00-11:30

Tue. Dec. 16, 2014  9:00-11:30

Location: Lecture Hall 1, Niavaran Building, IPM

Short course title:a Conference on Algorithmic Game Theory

Time:

Sun. Dec. 28, 2014  9:00-14:00

Location: Khwarizmi Hall, Computer Engineering Department, Sharif University of Technology