Multi-component KPZ equations

Herbert Spohn
Technishe Universitaet Muenchen; Member, School of Mathematics
December 12, 2013
The stochastic Burgers equation (equivalent to the one-dimensional KPZ equation) is a hyperbolic conservation law with random currents. In applications, one often has to deal with several conservation laws, a little explored case. We discuss several examples and the decoupling mechanism.

Complex analytic vanishing cycles for formal schemes

Vladimir Berkovich
Weizmann Institute of Sciences; Member, School of Mathematics
December 12, 2013
Let \(R={\cal O}_{{\bf C},0}\) be the ring of power series convergent in a neighborhood of zero in the complex plane. Every scheme \(\cal X\) of finite type over \(R\) defines a complex analytic space \({\cal X}^h\) over an open disc \(D\) of small radius with center at zero. The preimage of the punctured disc \(D^\ast=D\backslash\{0\}\) is denoted by \({\cal X}^h_\eta\), and the preimage of zero coincides with the analytification \({\cal X}_s^h\) of the closed fiber \({\cal X}_s\) of \(\cal X\).

Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball

Gil Cohen
Weizmann Institute
December 16, 2013
We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even \(n\), there exists an explicit bijection \(f\) from the \(n\)-dimensional Boolean cube to the Hamming ball of equal volume embedded in \((n+1)\)-dimensional Boolean cube, such that for all \(x\) and \(y\) it holds that \(\mathrm{distance}(x,y) / 5 \leq \mathrm{distance}(f(x),f(y)) \leq 4\, \mathrm{distance}(x,y)\) where \(\mathrm{distance}(\cdot,\cdot)\) denotes the Hamming distance.

Deeper Combinatorial Lower Bounds

Siu Man Chan
Princeton University
January 21, 2014
We will discuss space and parallel complexity, ranging from some classical results which motivated the study, to some recent results concerning combinatorial lower bounds in restricted settings. We will highlight some of their connections to boolean complexity, proof complexity, and algebraic complexity.

Exact formulas for random growth off a flat interface

Daniel Remenik
Universidad de Chile
January 22, 2014
We will describe formulas for the asymmetric simple exclusion process (ASEP) starting from half-flat and flat initial data. The formulas are for the exponential moments of the height function associated with ASEP. They lead to explicit formulas for certain generating functions of the model, which in the flat case can be expressed as a Fredholm Pfaffian. We will explain also how these formulas can be used to provide formal derivations of the conjectured limiting fluctuations, and discuss some related models. This is joint work with Janosch Ortmann and Jeremy Quastel.

Unique games, the Lasserre hierarchy and monogamy of entanglement

Aram Harrow
Massachusetts Institute of Technology
January 27, 2014
In this talk, I'll describe connections between the unique games conjecture (or more precisely, the closely relatedly problem of small-set expansion) and the quantum separability problem. Remarkably, not only are the problems related, but the leading candidate algorithms turn out to be essentially equivalent: for unique games, the algorithm is called the Lasserre hierarchy, and for quantum separability, it is called the "k-extendable" relaxation.

Rigidity and Flexibility of Schubert classes

Colleen Robles
Texas A & M University; Member, School of Mathematics
January 27, 2014
Consider a rational homogeneous variety \(X\). (For example, take \(X\) to be the Grassmannian \(\mathrm{Gr}(k,n)\) of \(k\)-planes in complex \(n\)-space.) The Schubert classes of \(X\) form a free additive basis of the integral homology of \(X\). Given a Schubert class \([S]\), represented by a Schubert variety \(S\) in \(X\), Borel and Haefliger asked: aside from the Schubert variety, does \([S]\) admit any other algebraic representatives? I will discuss this, and related questions, in the case that \(X\) is Hermitian symmetric.

Simplicial complexes as expanders

Ori Parzanchevski
Institute for Advanced Study; Member, School of Mathematics
January 28, 2014
Expanders are highly connected sparse graphs. Simplicial complexes are a natural generalization of graphs to higher dimension, and the notions of connectedness and expansion turn out to have interesting analogues, which relate to the homology and the spectral theory of the complexes. I will explain these notions, and discuss results and problems. No prior knowledge is assumed.

Self-avoiding walk in dimension 4

Roland Bauerschmidt
University of British Columbia; Member, School of Mathematics
January 28, 2014
The (weakly) self-avoiding walk is a basic model of paths on the d-dimensional integer lattice that do not intersect (have few intersections), of interest from several different perspectives. I will discuss a proof that, in dimension 4, the susceptibility of the weakly self-avoiding walk diverges with an explicit logarithmic correction as the critical point is approached. The argument is based on a representation of the weakly self-avoiding walk as a supersymmetric field theory which is studied with a renormalization group method.