Abstract: According to the Deligne's conception of Shimura varieties and Tannakian philosophy, the moduli of global G-shtukas may appear as a function field analog for Shimura varieties. Here G is a reductive (or even parahoric) group over a smooth projective geometrically irreducible curve C. In this talk, I will discuss about the uniformization theory of these moduli stacks and the relation to the Langlands-Rapoport conjecture.

Abstract: In a previous work, we introduced a refinement of Juhasz's sutured Floer homology, and constructed a minus theory for sutured manifolds, called sutured Floer chain complex. In this talk, we introduce a new description of sutured manifolds as "tangles" and describe a notion of cobordism between them. Using this construction, we define a cobordism map between the corresponding sutured Floer chain complexes. We also discuss some possible applications. This is a joint work with Eaman Eftekhary.

Abstract: It seems that the index theory for non-compact spaces has found its ultimate formulation in realm of coarse spaces and K-theory of related operator algebras. Relative and partitioned index theorems may be mentioned as two important and interesting examples of this program. In this talk we formulate a combination of these two theorems and establish a partitioned-relative index theorem.

Abstract: The study of the asymptotics of quantum invariants of knot objects and their relation to the limiting geometry and topology has been an important research area of Quantum Topology. Over the last few years, several conjectures have been formulated and most of them are still open. The Volume conjecture and the Witten conjecture about the asymptotic behavior of Witten-Reshetikhin-Turaev invariants are two famous examples of these conjectures. In this talk we study these problems within the framework of TQFT and present some new results and questions.

Abstract: We use a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic branched cover $\Sigma_m(K)$ to give a combinatorial proof for the invariance of the associatedknot Floer homology over $\Z$.

Abstract: Suppose that $\mathcal{C}$ is the space of all middle Cantor sets. We characterize all triples $(\alpha,~\beta,~\lambda)\in \mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$ that satisfy $C_\alpha- \lambda C_\beta=[-\lambda,~1]. $ Also all triples (that are dense in $\mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$) has been determined such that $C_\alpha- \lambda C_\beta$ forms the attractor of an iterated function system. Then we found a new family of the pair of middle Cantor sets $\mathcal{P}$ in a way that for each $(C_\alpha,~ C_\beta)\in\mathcal{P}$, there exists a dense subfield $F\subset \mathbb{R}$ such that for each $\mu \in F$, the set $C_\alpha- \mu C_\beta$ contains an interval or has zero Lebesgue measure.
In sequel, conditions on the functions $f, ~g$ and the pair $(C_\alpha,~C_\beta)$ is provided which $f(C_{\alpha})- g(C_{\beta})$ contains an interval. This leads us to denote another type of stability in the intersection of two Cantor sets. We prove the existence of this stability for regular Cantor sets that have stable intersection and its absence for those which the sum of their Hausdorff dimension is less than one.
At the end, special middle Cantor sets $C_\alpha$ and $C_\beta$ are introduced. Then the iterated function system corresponding to the attractor $C_{\alpha}-\frac{2\alpha}{\beta}C_\beta$ is characterized. Some specifications of the attractor has been presented that keep our example as an exception. We also show that $\sqrt{C_{\alpha}}$ - $\sqrt{C_{\beta}}$ contains at least one interval.

Abstract: This is an exposition of some of Mirzakhani's PhD thesis's results. We will review the moduli space and the Teichmuller space of bordered Riemann surfaces and the Weil-Petersson metric on them. We will explain how the volume of these moduli spaces can be found recursively and finally give some applications of these volume computations.

Abstract: In this talk, we consider the problem of minimizing a functional subject to a pointwise gradient constraint. We show that this problem is equivalent to a double obstacle problem with nondifferentiable obstacles. Then, we prove the regularity of the solution to our problem, which is better than the expected regularity of a generic double obstacle problem with nondifferentiable obstacles.

Abstract: I will talk about the dynamics of a surface homeomorphism restricted to the boundary of an invariant domain. In the area-preserving setting, we give a classification of the dynamics which is very similar to Poincar?'s theory for circle homeomorphisms. I will also discuss some of the consequences of this result. The talk is based on a joint work with Andres Koropecki and Patrice Le Calvez.

Abstract: In this talk we discus the difference between the Lebesgue measure and stationary measure arisen by a semi-group action. We talk a bit more about the minimal IFSs , their unique stationary measure and Erdos problem in this context.

Abstract: To study constrained mechanical systems, there are at least two approaches one may take, namely the "classical nonholonomic approach", which is based on the Lagrange-d'Alembert principle and is not variational in nature, and a variational axiomatic one known as the "vakonomic approach".
In fact there are some fascinating differences between these two procedures, e.g., they do not always give the same equations of motion; the distinction between these two procedures has a long and distinguished history going back to Korteweg (1899), and has been discussed in a more modern context by Arnold, Kozlov and Neishtadt since 1983.
In this seminar, we present the classical nonholonomic mechanics and the vakonomic mechanics of systems with constraints, and will compare them in order to see when these two mechanics are equivalent, i.e., when they give the same system of equations. For the class of mechanical systems that they are not so, we determine which one of these approaches is the appropriate one for deriving the equations of (mechanically possible) motions. .