On the Boltzmann equation without angular cut-off

Robert Strain
University of Pennsylvania
March 18, 2014
In this talk we will explain several results surrounding global stability problem for the Boltzmann equation 1872 with the physically important collision kernels derived by Maxwell 1867 for the full range of inverse power intermolecular potentials, \(r^{-(p-1)}\) with \(p > 2\) and more generally. This is a problem which had remained open for quite a long time. Specifically, we now have global solutions that are perturbations of the Maxwellian equilibrium states, and which decay rapidly in time to equilibrium.

List decodability of randomly punctured codes

Mary Wootters
University of Michigan
March 24, 2014
We consider the problem of the list-decodability of error correcting codes. The well-known Johnson bound implies that any code with good distance has good list-decodability, but we do not know many structural conditions on a code which guarantee list-decodability beyond the Johnson bound. We provide a randomized result of this type, and we show that random puncturings of codes with good distance are near-optimally list-decodable, with high probability.

Gambling, Computational Information, and Encryption Security

Bruce Kapron
University of Victoria; Member, School of Mathematics
March 24, 2014
We revisit the question, originally posed by Yao (1982), of whether encryption security may be characterized using computational information. Yao provided an affirmative answer, using a compression-based notion of computational information to give a characterization equivalent to the standard computational notion of semantic security. We give two other equivalent characterizations.

Circular Encryption in Formal and Computational Cryptography

Bruce Kapron
University of Victoria; Member, School of Mathematics
March 25, 2014
The goal of computationally sound symbolic security is to create formal systems of cryptography which have a sound interpretation with respect to complexity-based notions of security. While there has been much progress in the development of such systems, one big impediment is the treatment of circular encryptions. In many typical symbolic systems, it is secure to encrypt a key by itself, but in the computational setting, standard notions of security break down in this case. There are now approaches to this problem from both sides.

From classical to quantum integrability, and back

Vladimir Kazakov
École normale supérieure
March 25, 2014
Hirota relations in their various incarnations play an important role in both classical and quantum integrable systems, from matrix integrals and PDE's to one-dimensional quantum spin chains and two dimensional quantum field theories (QFT). The Wronskian solutions of discrete Hirota equations (T-systems) are related to the symmetry of these systems. They can be used, when supplied with analyticity properties, to find exact energy spectra of quantum spin chains and QFT's in finite volume.

Some results on history dependent stochastic processes

Margherita Disertori
University of Bonn
March 26, 2014
Edge reinforced random walk (ERRW) and vertex reinforced jump processes are history dependent stochastic process, where the particle tends to come back more often on sites it has already visited in the past. For a particular scheme of reinforcement these processes are mixtures of reversible Markov chains whose mixing measure can be related to a non-linear sigma model introduced in the context of random matrices. I will give an overview on these models and explain some recent results in joint work with F. Merkl and S. Rolles.

Anomalous shock fluctuations in TASEP and last passage percolation models

Patrik Ferrari
University of Bonn
March 26, 2014
We consider the totally asymmetric simple exclusion process with initial conditions and/or jump rates such that shocks are generated. If the initial condition is deterministic, then the shock at time t will have a width of order \(t^{1/3}\). We determine the law of particle positions in the large time limit around the shock in a few models. In particular, we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the \(\mathrm{Airy}_1\) process.

Univalent Foundations: New Foundations of Mathematics

Vladimir Voevodsky
Institute for Advanced Study; Faculty, School of Mathematics
March 26, 2014
In Voevodsky’s experience, the work of a mathematician is 5% creative insight and 95% self-verification. Moreover, the more original the insight, the more one has to pay for it later in self-verification work. The Univalent Foundations project, started at the Institute a few years ago, aims to lower the price by giving mathematicians the ability to verify their constructions with the help of computers.