L5

The 12th of Hilbert's 23 problems posed in 1900 asks for an explicit description of abelian extensions of a given base field. Over the rationals, this is given by the exponential function, and over imaginary quadratic fields, by meromorphic functions on the complex upper half plane.  Darmon and Vonk's theory of rigid meromorphic cocycles, or "RM theory", includes conjectures giving a $p$-adic solution over real quadratic fields. These turn out to be closely linked to purely topological questions about intersections of geodesics in the upper half plane, and to $p$-adic deformations of Hilbert modular forms. I will explain an extension of results of Darmon, Pozzi and Vonk proving some of these conjectures, and some ongoing work concerning analogous results on Shimura curves.

We give a purely topological formula for the square class of the central value of the L-function of a symplectic representation on a curve. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds. This is related to the theory of epsilon factors in number theory and Meyer’s signature formula in topology among other topics. We will present some of these ideas and sketch aspects of the proof. This is joint work with Akshay Venkatesh.

The Mixing Conjecture of Michel-Venkatesh has now taken on additional arithmetic significance via Wiles' new approach to modularity. Inspired by this, we present the best currently available method, pioneered by Khayutin's proof for quaternion algebras over the rationals, which we have successfully applied to totally real fields. The talk will overview the method, which brings a suprising combination of ergodic theory, analysis and geometry to bear on this arithmetic problem.

Scaling limits of random developments of a path into a matrix Lie Group have recently been used to construct signature-based kernels on path space, while mitigating some of the dimensionality challenges that come with using signatures directly. General linear group developments have been shown to be connected to the ordinary signature kernel (Muça Cirone et al.), while unitary developments have been used to construct a path characteristic function distance (Lou et al.). By leveraging the tools of random matrix theory and free probability theory, we are able to provide a unified treatment of the limits in both settings under general assumptions on the vector fields. For unitary developments, we show that the limiting kernel is given by the contraction of a signature against the monomials of freely independent semicircular random variables. Using the Schwinger-Dyson equations, we show that this kernel can be obtained by solving a novel quadratic functional equation. 

This is joint work with Thomas Cass.

In this talk, we study concentration properties for laws of non-linear Gaussian functionals on metric spaces. Our focus lies on measures with non-Gaussian tail behaviour which are beyond the reach of Talagrand’s classical Transportation-Cost Inequalities (TCIs). Motivated by solutions of Rough Differential Equations and relying on a suitable contraction principle, we prove generalised TCIs for functionals that arise in the theory of regularity structures and, in particular, in the cases of rough volatility and the two-dimensional Parabolic Anderson Model. Our work also extends existing results on TCIs for diffusions driven by Gaussian processes.

We study the stochastic quantisation for the fractional $\varphi^4$ theory. The model has been studied by Brydges, Mitter and Scopola in 2003 as a natural extension of $\phi^4$ theories to fractional sub-critical dimensions. The stochastic quantisation equation is given by the (formal) SPDE 

\[

(\partial_t + (-\Delta)^{s}) \varphi = - \lambda \varphi^3 + \xi\]

where $\xi$ is a space-time white noise over the three dimensional torus. The equation is sub-critical for $s > \frac{3}{4}$.

We derive a priori estimates in the full sub-critical regime $s>\frac{3}{4}$. These estimates rule out explosion in finite time and they imply the existence of an invariant measure with a standard Krylov-Bogoliubov argument. 

Our proof is based on the strategy developed for the parabolic case $s=1$ in [Chandra, Moinat, Weber, ARMA 2023]. In order to implement this strategy here, a new Schauder estimate for the fractional heat operator is developed. Additionally, several algebraic arguments from [Chandra, Moinat, Weber, ARMA 2023] are streamlined significantly. 

This is joint work with Hendrik Weber (Münster). 

Given a pair of metric tensors g_j ≥ g₀ on a Riemannian manifold, M, it is well known that Vol_j(M)≥Vol₀(M). Furthermore, the volumes are equal if and only if the metric tensors are the same, g_j=g₀. Here we prove that if for a sequence g_j, we have g_j≥g₀, Vol_j(M)→Vol₀(M) and diam(M_j) ≤ D then (M,g_j) converges to (M,g₀) in the volume preserving intrinsic flat sense. The previous result will then be applied to prove stability of a class of tori.
 

This talk is based on joint works of myself with: Allen and Sormani (https://arxiv.org/abs/2003.01172), and Cabrera Pacheco and Ketterer (https://arxiv.org/abs/1902.03458).

In this talk, we present a graph discretized approximation scheme for diffusions with drift and killing on a complete Riemannian manifold M. More precisely, for a given Schrödinger operator with drift on M having the form A = — Δ — b + V , we introduce a family of discrete time random walks in the  ow generated by the drift b with killing on a sequence of proximity graphs, which are constructed by partitions cutting M into small pieces. As a main result, we prove that the drifted Schrodinger semigroup {e^—tA}_t≥0 is approximated by discrete semigroups generated by the family of random walks with a suitable scale change. This result gives a  nite dimensional summation approximation of a Feynman-Kac type functional integral over M. Furthermore, when M is compact, we also obtain a quantitative error estimate of the convergence.
This talk is based on a joint work with Satoshi Ishiwata (Yamagata University), and the full paper can be found on https://doi.org/10.1007/s00208-024-02809-9.

In their study of GL(N)-GL(m) Howe duality, Cautis-Kamnitzer-Morrison observed that the GL(N) Reshetikhin-Turaev link invariant can be computed in terms of quantum gl(m). This idea inspired Cautis and Lauda-Queffelec-Rose to give a construction of GL(N) link homology in terms of Khovanov-Lauda's categorified quantum gl(m). There is a Spin(2n+1)-Spin(m) Howe duality, and a quantum analogue that was first studied by Wenzl. In the first half of the talk, I will explain how to use this duality to compute the Spin(2n+1) link polynomial, and present calculations which suggest that the Spin(2n+1) link invariant is obtained from the GL(2n) link invariant by folding. In the second part of the talk, I will introduce the parallel categorified constructions and explain how to use them to define Spin(2n+1) link homology.

This is based on joint work in progress with Ben Elias and David Rose.

Dynamical interface motions are important flow patterns and fundamental free boundary problems in fluid mechanics, and have attracted huge attention in the mathematical community. Such waves for purely inviscid fluids are subject to various instabilities such as Kelvin-Helmholtz and Rayleigh-Taylor instabilities unless other stabilizing effects such as surface tension, Taylor-sign conditions or dissipations are imposed. However, in the presence of magnetic fields, it has been known that tangential magnetic fields may have stabilizing effects for free surface waves such as plasma-vacuum or plasma-plasma interfaces (at least locally in time), yet whether transversal magnetic fields (which occurs often for interfacial waves for astrophysical plasmas) can stabilize typical free interfacial waves remain to be some open problems. In this talk, I will show the stabilizing effects of the transversal magnetic fields for some interfacial waves for both compressible and incompressible multi-dimensional magnetohydrodynamics (MHD).

First, I will present the local (in time) well-posedness in Sobolev space of multi- dimensional compressible MHD contact discontinuities, which are the most typical interfacial waves for astrophysical plasma and prototypical fundamental waves for systems of hyperbolic conservations. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversally, which leads to a two-phase free boundary problem that may have nonlinear Rayleigh- Taylor instability and whose front symbols have no ellipticity. We overcome such difficulties by exploiting full the transversality of the magnetic fields and designing a nonlinear approximate problem, which yields the local well-posed without loss of derivatives and without any other conditions such as Rayleigh-Taylor sign conditions or surface tension. Second, I will discuss some results on the global well-posedness of free interface problems for the incompressible inviscid resistive MHD with transversal magnetic fields. Both plasma-vacuum and plasma-plasma interfaces are studied. The global in time well-posedness of both interface problems in a horizontally periodic slab impressed by a uniform non-horizontal magnetic field near an equilibrium are established, which reveals the strong stabilizing effect of the transversal field as the global well- posedness of the free boundary incompressible Euler equations (without the irrotational assumptions) around an equilibrium is unknown. This talk is based on joint work with Professor Yanjin Wang.