# Seminars & Events for 2010-2011

##### Rearrangement and convection

Rearrangement theory is about reorganizing a given function (or map) in some specific order (monotonicity, cycle monotonicity etc...). This is somewhat similar to the convection phenomenon in fluid mechanics, where fluid parcels are continuously reorganized in a stabler way (heavy fluid at bottom and light fluid at top).

##### Schrödinger equation, deformation theory and $tt^*$ geometry

I will talk about my recent work on the deformation theory of Schrödinger equations. This is an attempt to construct the rigorous mathematical foundation for topological B model, which has tight relation to the deformation theory of complex structure and the singularity theory.

##### From random tilings to representation theory

Lozenge tilings of planar domains provide a simple, yet sophisticated model of random surfaces. Asymptotic behavior of such models has been extensively studied in recent years.

We will start from recent results about q-distributions on tilings of a hexagon or, equivalently, on boxed plane partitions. (This part is based on the joint work with A.Borodin and E.Rains).

##### The size Ramsey number of a directed path

Given a graph $H$, the size Ramsey number $r_e(H,q)$ is the minimal number m for which there is a graph $G$ with $m$ edges such that every $q-coloring$ of $G$ contains a monochromatic copy of $H$. We study the size Ramsey number of the directed path of length $n$ in oriented graphs, where no antiparallel edges are allowed. We give nearly tight bounds for every fixed number of colors.

##### One condition for rational hyperbolicity for moment angle complexes

There is a conjecture in rational homotopy which states "If a finite dimensional space X has two indecomposable classes in the rational cohomology whose cup product is zero, then its rational homotopy has exponential growth , ie, it is hyperbolic." We verify this conjecture for the space X a moment angle complex.

##### On local combinatorial formulae for Pontryagin classes

The talk will be devoted to the problem of combinatorial computation of the rational Pontryagin classes of a triangulated manifold. This problem goes back to the famous work by A. M. Gabrielov, I. M. Gelfand, and M. V. Losik (1975). Since then several different approaches to combinatorial computation of the Pontryagin classes have been suggested.

##### Expanding the scope of Hilbert irreducibility

If $K$ is the rational function field $K=Q(t)$, then a polynomial $f$ in $K[x]$ can be regarded as a one-parameter family of polynomials. If $f$ is irreducible, then a basic form of Hilbert's irreducibility theorem states that there are infinitely many parameters in $Q$ for which the corresponding polynomial is also irreducible.

##### Lower Ricci Curvature, Convexity and Applications

We prove new estimates for tangent cones along minimizing geodesics in GH limits of manifolds with lower Ricci curvature bounds. We use these estimates to show convexity results for the regular set of such limits.

##### Well-posedness theory for compressible Euler equations in a physical vacuum

An interesting problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of vacuum. A particular interest is so called physical vacuum which naturally arises in physical problems. The main difficulty lies in the fact that the physical systems become degenerate along the boundary.

##### Feedback, Lineages and Cancer

A multispecies continuum model is developed to simulate the dynamics of cell lineages in solid tumors. The model accounts for spatiotemporally varying cell proliferation and death mediated by the heterogeneous distribution of oxygen and soluble chemical factors. Together, these regulate the rates of self-renewal and differentiation of the different cells within the lineages.

##### Trace Formulas for Large Random d-Regular Graphs

Trace formulas for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulas depend on a parameter (w) which can be tuned continuously to assign different weights to different periodic orbit contributions.

##### Characterization of Lee-Yang polynomials

I shall give a characterization of the multi-affine polynomials in n variables that occur in the proof of the Lee-Yang Circle Theorem.

##### Unbalanced Allocations

Recently, there has been much research on processes that are mostly random, but also have a small amount of deterministic choice; e.g., Achlioptas processes on graphs. This talk builds on the balanced allocation algorithm first described by Azar, Broder, Karlin and Upfal. Their algorithm (and its relatives) uses randomness and some choice to distribute balls into bins in a balanced way.

##### Algebraic structures from operads and moduli spaces

Some classical algebraic structures like Gerstenhaber's bracket on the Hochschild complex have an operadic origin. We discuss generalizations of these operations coming from different operadic type settings. This includes geometric constructions from moduli spaces and master equations appearing in string topology and in a "pedestrian" version of string field theory.

##### Fox re-embedding and Bing submanifolds

Let $M$ be an orientable closed connected 3-manifold, and $Y$ be a connected compact 3-manifold.

##### Constructing Abelian varieties over Qbar not isogenous to a Jacobian

We discuss the following question of Nick Katz and Frans Oort: Given an Algebraically closed field K, is there an Abelian variety over K of dimension g which is not isogenous to a Jacobian? For K the complex numbers its easy to see that the answer is yes for g>3 using measure theory, but over a countable field like Qbar new methods are required.

##### Diffusive or superdiffusive asymtotics for periodic and non-periodic Lorentz processes

After the first success in establishing the diffusive, Brownian limit of planar, finite-horizon, periodic Lorentz processes, in 1981 Sinai turned the interest toward studying models when periodicity is hurt, in particular, to locally perturbed Lorentz processes.

##### Novel Phenomena and Models of Active Fluids

Fluids with suspended microstructure - complex fluids - are common actors in micro- and biofluidics applications and can have fascinating dynamical behaviors. A new area of complex fluid dynamics concerns "active fluids" which are internally driven by having dynamic microstructure such as swimming bacteria.

##### 4-dimensional symplectic holomorphic contractions

In the talk I will consider birational projective morphisms from smooth holomorphic symplectic fourfolds into affine normal varieties. The ultimate goal will be to classify such maps. For the moment I can present special features of them and discuss some important examples.

##### Bethe-Ansatz for the two species totally asymmetric diffusion model

We study the two species asymmetric diffusion model that describes two species and vacancies diffusing asymmetrically on a one-dimensional lattice. Our method is the algebraic Bethe Ansatz. We will explain this technique which we use to find the finite-size scaling behavior of the lowest lying eigenstates of the quantum Hamiltonian describing the model.