L6

The vacant set of the random walk on the torus undergoes a percolation phase transition at Poissonian timescales in dimensions 3 and higher. The talk will review this phenomenon and discuss recent progress regarding the nature of the transition, both for this model and its infinite-volume limit, the vacant set of random interlacements, introduced by Sznitman in Ann. Math., 171 (2010), 2039–2087. The discussion will lead up to recent progress regarding the long purported equality of several critical parameters naturally associated to the transition. 

It was proved by Selberg's in the 1940's that the typical values of the logarithm of the Riemann zeta function on the critical line is distributed like a complex Gaussian random variable. In this talk, I will present recent work with Emma Bailey that extends the Gaussian behavior for the real part to the large deviation regime. This gives a new proof of unconditional upper bounds of the $2k$-moments of zeta for $0\leq k\leq 2$, and lower bounds for $k>0$. I will also discuss the connections with random matrix theory and with the Moments Conjecture of Keating & Snaith. 

Optimization problems constrained to a smooth manifold can be solved via the framework of Riemannian optimization. To that end, a geometrical description of the constraining manifold, e.g., tangent spaces, retractions, and cost function gradients, is required. In this talk, we present a novel approach that allows performing approximate Riemannian optimization based on a manifold learning technique, in cases where only a noiseless sample set of the cost function and the manifold’s intrinsic dimension are available.

The Steinberg scheme and the equivariant coherent sheaves on it play a very important role in Geometric Representation theory. In this talk we will discuss various t-structures on the equivariant derived category of the Steinberg of importance for Representation theory in positive characteristics. Based on arXiv:2302.05782.

Schauder estimates are a classical tool in linear and nonlinear elliptic and parabolic PDEs. They describe how regularity of coefficients reflects in regularity of solutions. They basically have a perturbative nature. This means that they can be obtained by perturbing the estimates available for problems without coefficients. This paradigm works as long as one deals with uniformly elliptic equations. The nonuniformly elliptic case is a different story and Schauder's theory turns out to be not perturbative any longer, as shown by counterexamples. In my talk, I will present a method allowing to bypass the perburbative schemes and leading to Schauder estimates in the nonuniformly elliptic regime. For this I will concentrate on the case of nonuniformly elliptic functionals with nearly linear growth, also covering a borderline case of so-called double phase energies. From recent, joint work with Cristiana De Filippis (Parma). 

Given a non-orientable Lagrangian surface L in a symplectic 4-manifold, how far
can the cohomology class of the symplectic form be deformed before there is no
longer a Lagrangian isotopic to L? I will properly introduce this and a
related question, both of which are less interesting for orientable
Lagrangians due to topological conditions. The majority of this talk will be
an exposition on Evans' 2020 work in which he solves this deformation
question for a particular Klein bottle. The proof employs the heavy machinery
of symplectic field theory and more classical pseudoholomorphic
curve theory to severely constrain the topology and intersection properties of
the limits of certain pseudoholomorphic curves under a process called
neck-stretching. The treatment of SFT-related material will be light and focus
mainly on how one can use the compactness theorem to prove interesting things.

Spectral properties of Schrödinger operators are studied a lot in mathematical physics. They can give the description of trapped fermionic particles. This presentation will focus on the non-interacting case. I will explain why it is relevant to estimate L^p bounds of orthonormal families of eigenfuntions at the semiclassical regime and then, give the main ideas of the proof.

In this talk, I will firstly give asymptotic formulas for the moments of the n-th derivative of the characteristic polynomials from the CUE. Secondly, I will talk about the connections between them and a solution of certain Painleve differential equation. This is joint work with Jonathan P. Keating.
 

Female PhD students and Postdocs will be giving short talks about their research. This will be followed by a Q&A and a chance to mingle with the speakers over lunch.

Speakers:

• Rhiannon Savage, DPhil Student in Algebra and Geometry
• Shanshan Hua, DPhil Student in Functional Analysis
• Silvia Butti, Postdoc in Theoretical Computer Science
• Anna Berryman, DPhil Student in OCIAM (Oxford Centre for Industrial and Applied Mathematics)
• Carmen Jorge Diaz, DPhil Student in Mathematical Physics