16:00
Random growth models with half space geometry
Abstract
Selberg’s CLT informs us that the logarithm of the Riemann zeta function evaluated on the critical line behaves as a complex Gaussian. It is natural, therefore, to study how far this Gaussianity persists. This talk will present conditional and unconditional results on atypically large values, and concerns work joint with Louis-Pierre Arguin and Asher Roberts.
Smooth generic representations of $GL_n$ over a $p$-adic field $F$, i.e. representations admitting a nondegenerate Whittaker model, are an important class of representations, for example in the setting of Rankin-Selberg integrals. However, in recent years there has been an increased interest in non-generic representations and their degenerate Whittaker models. By the theory of Bernstein-Zelevinsky derivatives we can associate to each smooth irreducible representation of $GL_n(F)$ an integer partition of $n$, which encodes the "degeneracy" of the representation. By using these "highest derivative partitions" we can define a stratification of the category of smooth complex representations and prove the surprising fact that all of the strata categories are equivalent to module categories over commutative rings. This is joint work with David Helm.
Motivated by a recent experiment on a superconducting quantum
information processor, I will discuss the Floquet quantum Ising model in
the presence of integrability- and symmetry-breaking random fields. The
talk will focus on the relation between boundary spin correlations,
spectral pairings, and effects of the random fields. If time permits, I
will also touch upon self-similarity in the dynamic phase diagram of
Fibonacci-driven quantum Ising models.
I will present joint work with Alessandro Fazzari in which we prove precise conditional estimates for the third (non-absolute) moment of the logarithm of the Riemann zeta function, beyond the Selberg central limit theorem, both for the real and imaginary part. These estimates match predictions made in work of Keating and Snaith. We require the Riemann Hypothesis, a conjecture for the triple correlation of Riemann zeros and another ``twisted'' pair correlation conjecture which captures the interaction of a prime power with Montgomery's pair correlation function. This conjecture can be proved on a certain subrange unconditionally, and on a larger range under the assumption of a variant of the Hardy-Littlewood conjecture with good uniformity.
We define a version of algebraic K-theory for bornological algebras, using the recently developed continuous K-theory by Efimov. In the commutative setting, we prove that this invariant satisfies descent for various topologies that arise in analytic geometry, generalising the results of Thomason-Trobaugh for schemes. Finally, we prove a version of the Grothendieck-Riemann-Roch Theorem for analytic spaces. Joint work with Jack Kelly and Federico Bambozzi.
Let G be a reductive group over a finite field F of characteristic p. I will present work with Tzu-Jan Li in which we determine the endomorphism algebra of the Gelfand-Graev representation of the finite group G(F) where the coefficients are taken to be l-adic integers, for l a good prime of G distinct from p. Our result can be viewed as a finite-field analogue of the local Langlands correspondence in families.
Let X --> S be a family of algebraic varieties parametrized by an infinitesimal disk S, possibly of mixed characteristic. The Bloch conductor conjecture expresses the difference of the Euler characteristics of the special and generic fibers in algebraic and arithmetic terms. I'll describe a proof of some new cases of this conjecture, including the case of isolated singularities. The latter was a conjecture of Deligne generalizing Milnor's formula on vanishing cycles.
This is joint work with Massimo Pippi; our methods use derived and non-commutative algebraic geometry.
Plethysms lie at the intersection of representation theory and algebraic combinatorics. We give a recursive formula for a family of plethysm coefficients encompassing those involved in Foulkes' Conjecture. We also describe some applications, such as to the stability of plethysm coefficients and Sylow branching coefficients for symmetric groups. This is joint work with Y. Okitani.