# Seminars & Events for 2011-2012

##### Grothendieck's inequality and the propeller conjecture

##### Ultrametric subsets with large Hausdorff dimensions

##### Domination when the stars are out

We algorithmize the recent structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that several domination problems are fixed-parameter tractable, and even possess polynomial-sized kernels, on claw-free graphs.

##### Ambient metrics and exceptional holonomy

Construction of pseudo-Riemannian metrics whose holonomy is an exceptional Lie group has been of interest in recent years. This talk will outline a construction of an infinite-dimensional family of metrics in dimension 7 whose holonomy is contained in the split real form of the exceptional group $G_2$. An open dense subset of the family has holonomy equal to $G_2$. The datum for the construc

##### Improved Moser-Trudinger inequalities and Liouville equations on compact surfaces

We consider a class of equations with exponential nonlinearities and possibly singular sources motivated from the study of abelian Chern Simons models or from the problem of prescribing a metric with conical singularities through conformal transformations.

##### Cohen-Lenstra Heuristics

The class group of a number field, in colloquial terms, measures the failure of unique factorization in its ring of integers. Despite its central role in number theory, information about its structure, and even its size, mostly remain mysteries.

##### Erdos-Ko-Rado-like theorems for rainbow matchings

Let $f(n,k,r)$ be the smallest number such that every set of more than $f(n,k,r)$ r-sets in $[n]$ contains a matching of size $k$. The Erdos-Ko-Rado theorem states that $f(n,2,r)=(n-1)(r-1)$. A natural conjecture is that if $F_1, F_2, ...F_k \subseteq {[n]}{r}$ are all of size larger than $f(n,k,r)$ then they possess a rainbow matching, that is, a choice of disjoint edges, one from each $F_i$.

##### Transverse invariants in Heegaard Floer homology

Using the language of Heegaard Floer knot homology recently two invariants were defined for Legendrian knots. One in the standard contact 3-sphere defined by Ozsvath, Szabo and Thurston in the combinatorial settings of knot Floer homology, one by Lisca, Ozsvath, Stipsicz and Szabo in knot Floer homology for a general contact 3-manifold. Both of them naturally generalizes to transverse knots.

##### Hessian estimates for special Lagrangian equations with critical and supercritical phases

We talk about a priori Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. The "gradient" graphs of solutions are minimal Lagrangian submanifolds. Our unified approach leads to sharper estimates even for the previously known three dimensional or convex solution cases.

##### Long-time strong instability and unbounded orbits for some nonlinear Schrödinger equations

We establish a relation between long-time strong instability and the existence (in a certain generic sense) of unbounded orbits for dynamical systems on a Banach space. We then discuss some consequences of this relation for nonlinear Schrödinger equations.

##### Understanding 3D Shapes Jointly

The use of 3D models in our economy and life is becoming more prevalent, in applications ranging from design and custom manufacturing, to prosthetics and rehabilitation, to games and entertainment.

##### The Polynomial Method in Harmonic Analysis

We'll discuss incidence problems in harmonic analysis. I'll focus on recent progress which interestingly comes from algebra and topology. In particular, we'll examine Dvir's proof of the Kakeya conjecture over finite fields, Guth's endpoint estimate for the multilinear Kakeya function which used tools from algebraic topology and the joints problem.

##### Recent advances in connecting and contrasting test ideals and multiplier ideals

This talk will focus on two distinct measures of singularities: test ideals (in positive characteristic) and multiplier ideals (characteristic zero). Though known for over a decade to be related via reduction to characteristic p > 0, recent advances have provided a uniform description of these invariants using regular alterations.

##### Some new results on the ground state of the strong coupling (solid)limit of the Bose-Hubbard Model; Possible applicability to solid He-4

The Mott insulator is accepted as the appropriate ground manifold for the strongly interacting Fermion Hubbard model, with solid He-3 as the simplest exemplar. It is a manifold because of the spin degrees of freedom, which order antiferromagneticallly due to atom exchange, below a critical temperature. No corresponding effect of exchange has been known for the Bose solid.

##### The black hole stability problem

##### The size of a hypergraph and its matching number

More than 40 years ago, Erdos asked to determine the maximum possible number of edges in a $k$-uniform hypergraph on n vertices with no matching of size $t$ (i.e., with no $t$ disjoint edges). Although this is one of the most basic problem on hypergraphs, progress on Erdos' question remained elusive.

##### v1-periodic homotopy groups of SU(n)

I will survey the various results that have been obtained during the past 22 years on the v1-periodic homotopy groups of $SU(n)$. The most recent work has been combinatorial fine tuning to make the statements more explicit. I will discuss conceptual differences between 2-primary and odd-primary groups and implications for actual homotopy groups

##### A Bernstein type theorem for entire Willmore graphs

We show that every two-dimensional entire graphical solution to the Willmore equation with square integrable second fundamental form is a plane. This is joint work with Tobias Lamm.

##### On Singularities With Rational Homology Disk Smoothings

##### On the uniqueness of solutions to the 3D periodic Gross-Pitaevskii hierarchy

In this talk, we present a uniqueness result for solutions to the Gross-Pitaevskii hierarchy on the three-dimensional torus, under the assumption of an a priori spacetime bound.