Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare. Relying on a recognition theorem of McDuff for closed symplectic manifolds, we can understand this classification for certain Seifert fibered spaces with their canonical contact structures. We will discuss these classifications and applications in topology. This is a thesis defense.

This is an overview of the ideas and the proof of the Geometric Langland’s Correspondence for GL(1). This is also called Geometric Class Field theory. Namely, given the moduli space of bundles on a curve C, we will describe sheaves on this moduli space. A “basis” for this category of sheaves can be given in terms of connections on the curve C. If there is extra time, we will attempt to motivate why we are forced to consider the language of derived stacks to make the Langland’s correspondence an equivalence.

The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental queueing model with dynamic rates for large scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers gives us fluid and diffusion limits. The diffusion limit suggests a Gaussian approximation to the stochastic behavior. The fluid mean and diffusion variance can form a two-dimensional dynamical system that approximates the actual transient mean and variance for the queueing process. Recent work showed that a better approximation for mean and variance can be computed from a related two-dimensional dynamical system. In this spirit, we introduce a new three-dimensional dynamical system that estimates the mean, variance, and third cumulant moment. This improves on the previous two approaches by fitting the queuing distribution to one of a quadratic function for a Gaussian random variable. This is joint work with Jamol Pender of Cornell University.

In this talk I'll review hep-th/1308.4896. They found that the elliptic genera of 2d N=2 gauge theories are given by the sum of Jeffrey-Kirwan residues of a meromorphic form on the moduli space of flat connections on T^2. After going through the definitions I'll present several examples in this paper, with both Abelian and non-Abelian gauge groups. We'll also see that the elliptic genus provides a powerful tool to check 2d dualities.

In the eighties, Zhikov introduced some integral functionals to model non-homogeneous composites made by different basic materials. A model type is given by a variational integral whose integrand switches between two different types of elliptic phases according to a continuous coefficient, which depends on the position. If the elliptic phases are far from each other in a sharply quantified way, minimizers can be discontinuous. If this is not the case, we prove sharp regularity for minimizers, namely the Holder continuity of the gradient of the deformation.

This is an introductory talk focused on the Bing shrinking criterion and some applications to geometric topology such as Brown's proof of the generalized (topological) Schoenflies theorem and Bing's construction of an "exotic" involution of the the 3-sphere. This talk should be accessible to all graduate students.

If a photon polarized at a 45 degree angle hits a filter at a 90 degree angle, it has a 50-50 chance of passing through (as opposed to bouncing back). Einstein thought there had to be some hidden mechanism that determined which happened, and gave a strong argument supporting that belief. John S Bell gave a stronger argument, that convinved most physicists it was really just up to pure chance.

Stochastic geometry is a branch of probability theory focusing on random phenomena in the Euclidean space. Kolmogorov is credited for having built the foundations of the field – the Boolean model and the Poisson-Voronoi tessellation – for analyzing the growth of crystals. Nowadays stochastic geometry is widely used in computer science (image analysis), communication sciences (wireless networks, information theory), cosmology (analysis of the organization of galaxies), biology and life sciences. The modern theory is a very active branch of probability theory with a rich interplay with topology (topology of closed sets), algebraic topology (e.g. for the Boolean model), ergodic theory (Palm probabilities), convex analysis (Euler’s characteristics, intrinsic volumes), and computational probability. In the first part of the seminar, I will give a survey of the theory of point processes in the Euclidean space and of three fundamental objects of stochastic geometry built from point processes: the Boolean model, random tessellations and shot-noise fields. In the second part, I will give a survey of the connection between stochastic geometry in high dimension and information theory which I currently investigate with V. Anantharam.

In recent years, there has been a growing interest in obstructions to the existence of integral points on affine varieties. For example, given an extension of number fields of degree , one might ask when can values of a polynomial over be represented by norms of elements of ? In 2012, Colliot-Thélène and Harari asked for the integral Hasse principle and strong approximation for the variety defined by , where is separable of degree at least 3. They note that at the time, the answer was out of reach of the current techniques. In this talk, we consider a particular family of examples of this form, for which we construct explicit representatives of the classes in the Brauer group of X that can be used to compute the integral Brauer-Manin obstruction.

We show that for an immersed two-sided minimal surface in R^3, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in R^3 of index 2, as conjectured by Choe. Moreover, we show that the index of an immersed two-sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface. (This is joint work with Otis Chodosh)

I will study BV regularity for solutions of variational problems in Optimal Transportation and present some applications. In particular I will focus on bounds for the Wasserstein projection on the set of measure with density bounded by a prescribed function . I will also show how to recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. This is a joint work with A. Meszaros, F. Santambrogio and B. Velichkov.

The equations of Newton and Boltzmann give two seemingly unrelated descriptions of the dynamics of diluted gas. In this talk we show how one can derive Boltzmann?s equation from Newton?s, and that the solution of Newton?s equation converges to the solution of Boltzmann?s when the number of particles becomes large. This talk is based on the recent work of Isabelle Gallagher, Laure Saint-Raymond and Benjamin Texier.