Polynomial chaos and scaling limits of disordered systems

Nikolaos Zygouras
University of Warwick
December 3, 2013
Inspired by recent work of Alberts, Khanin and Quastel, we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz.

KPZ line ensemble

Ivan Corwin
Clay Mathematics Institute, Columbia University and MIT
December 4, 2013
We construct a \(\mathrm{KPZ}_t\) line ensemble -- a natural number indexed collection of random continuous curves which satisfies a resampling invariance called the H-Brownian Gibbs property (with \(H(x)=e^x\)) and whose lowest indexed curve is distributed as the time \(t\) Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ) stochastic PDE with narrow-wedge initial data. We prove four main applications of this construction:

What's Next?

Nathan Seiberg
Professor, School of Natural Sciences
December 4, 2013
In recent decades, physicists and astronomers have discovered two beautiful Standard Models, one for the quantum world of extremely short distances, and one for the universe as a whole. Both models have had spectacular success, but there are also strong arguments for new physics beyond these models. In this lecture, Nathan Seiberg, Professor in the School of Natural Sciences, reviews these models, their successes, and their shortfalls.

Local eigenvalue statistics at the edge of the spectrum: an extension of a theorem of Soshnikov

Alexander Sodin
Princeton University
December 5, 2013
We discuss two random decreasing sequences of continuous functions in two variables, and how they arise as the scaling limit from corners of a (real / complex) Wigner matrix undergoing stochastic evolution. The restriction of the second one to certain curves in the plane gives the Airy-2 time-dependent point process introduced by Praehofer and Spohn in the context of random growth.

Feynman categories, universal operations and master equations

Ralph Kaufmann
Purdue University; Member, School of Mathematics
December 6, 2013
Feynman categories are a new universal categorical framework for generalizing operads, modular operads and twisted modular operads. The latter two appear prominently in Gromov-Witten theory and in string field theory respectively. Feynman categories can also handle new structures which come from different versions of moduli spaces with different markings or decorations, e.g. open/closed versions or those appearing in homological mirror symmetry. For any such Feynman category there is an associated Feynman category of universal operations.

Eigenvalues and eigenvectors of spiked covariance matrices

Antti Knowles
December 9, 2013
I describe recent results on spiked covariance matrices, which model multivariate data containing nontrivial correlations. In principal components analysis, one extracts the leading contribution to the covariance by analysing the top eigenvalues and associated eigenvectors of the covariance matrix. I give non-asymptotic, sharp, high-probability estimates relating the principal components of the true covariance matrix to those of the sample covariance matrix.

Art History Lecture Series, Orientations in Renaissance Art

Alexander Nagel
New York University
December 9, 2013
In this lecture, Alexander Nagel, Professor of Fine Arts at the Institute of Fine Arts at New York University, illustrates some ways in which art produced during the Renaissance period points ­eastward towards Constantinople, towards the Holy Land, and to places further east, even as far as China. Nagel focuses on the forms this "orientation" took between 1492-1507, years during which new lands were being discovered, to great fanfare, but were still believed to belong to the continent of Asia.

Rigidity phenomena in random point sets and applications

Subhroshekhar Ghosh
Princeton University
December 11, 2013
In several naturally occurring (infinite) point processes, we show that the number (and other statistical properties) of the points inside a finite domain are determined, almost surely, by the point configuration outside the domain. This curious phenomenon we refer to as "rigidity". We will discuss rigidity phenomena in point processes, and their applications to stochastic geometry and to random instances of some classical questions in Fourier analysis.