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February 27
Asif Zaman (U of T)
A model problem for multiplicative chaos in number theory
Abstract. Resolving a conjecture of Helson, Harper recently established that partial
sums of random multiplicative functions typically exhibit more than
square-root cancellation. Harper's work gives an example of a problem in
number theory that is closely link to ideas in probability theory
connected with multiplicative chaos. In this talk, I plan to describe some
of these connections and consider a problem that might be thought of as a
simplified function field version of Helson's conjecture. This includes
joint work with Soundararajan.
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March 6
Benjamin Landon (U of T)
Tail estimates for stationary KPZ models
Abstract. We outline probabilistic proofs of tail estimates in the
moderate deviations regime for several different stationary models in the
KPZ class. This includes the four integrable discrete polymers, the
O'Connell-Yor semi-discrete polymer and a class of interacting diffusions
in equilibrium.
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March 13
Li-Cheng Tsai (Utah)
Some recent development in the weak noise theory for the KPZ equation
Abstract. The variational principle, or the least action principle,
offers a framework for the study of the Large Deviation Principle (LDP)
for a stochastic system. The KPZ equation is a stochastic PDE that is
central to a class of random growth phenomena. In this talk, we will
study the Freidlin-Wentzell LDP for the KPZ equation through the lens of
the variational principle. Such an approach goes under the name of the
weak noise theory in physics. We will explain how to extract various
limits of the most probable shape of the KPZ equation in the setting of
the Freidlin-Wentzell LDP. We will also review the recently discovered
connection of the weak noise theory to integrable PDEs.
This talk is based in part on joint works with Pierre Yves Gaudreau
Lamarre and Yier Lin.
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March 20: Two talks, one at 2:10pm, one at 3:10pm.
Miklos Abert (Renyi)
Eigenfunctions of Riemannian manifolds, Berry’s conjecture and Benjamini-Schramm convergence
Abstract.
We investigate the asymptotic behavior of eigenfunctions of Riemannian manifolds, using local weak sampling. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory.
As a result, we present the first mathematically precise formulation of Berry’s conjecture for a compact negatively curved manifold, using Gaussian random eigenwaves and formulate a Berry-type conjecture for sequences of locally symmetric spaces. For the standard tori the question has been investigated by Bourgain. Already there, the picture gets nontrivial, but for a different reason, as typically one has high multiplicities in that case. We prove some weak versions of these conjectures. Using ergodic theory, we also show that Berry's conjecture implies Quantum Unique Ergodicity. Joint work with Nicolas Bergeron and Etienne le Masson.
Omer Tamuz (Caltech)
On the Origin of the Boltzmann Distribution
Abstract.
The Boltzmann distribution is used in statistical mechanics to describe the distribution of states in systems with a given temperature. We give a novel characterization of this distribution as the unique one satisfying independence for uncoupled systems. The theorem boils down to a statement about multiplicative maps of the set of polynomials over the natural numbers.
Joint with Fedor Sandomirskiy