## Math Calendar

The goal of class fieldtheory is to compute the abelianisation of the absolute Galois group of aglobal or local field. Since the 1950s, the proofs are carried out entirely inthe language of Galois cohomology, consisting of a formal part ('abstract classfield theory') and a few (long) computations. I will give an introduction tothe area and explain the formal part of the story, using the formalism of *Mackeyfunctors* from equivariant stable homotopy theory.

Seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/

Talk:Uncertainty Quantification in Large-Scale Inverse Problems — Challenges and Opportunities

Abstract: In an inverse problem, one tries to infer the cause of a measured effect. Such problems are ubiquitous in science and engineering, and well-known examples include medical imaging and non-destructive testing. The basic approach is to fit a parametrised mathematical model of the underlying process to measurements, using (non-linear) optimisation techniques. Mathematical analysis tells us that such problems are often ill-posed, and additional prior information is needed to make the problem well-posed. Casting this in a Bayesian framework then allows us to quantify uncertainty (UQ) in the resulting estimates. Computing these uncertainties is still a challenge for large-scale applications. Research in the past few years aims to exploit advances in machine learning (ML) and the abundance of available (training) data to solve inverse problems more efficiently and more accurately. With such data-driven techniques the line between the model and prior information is blurring, and one of the challenges is to incorporate the known physics of the system in traditional black-box ML models. In this talk, I will give an overview of some of the recent developments in this area and present results on Bayesian UQ in medical imaging with normalising flows.

Seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/

In many control problems there is only limited information about the actions that will be available at future stages. We introduce a framework where the Controller sequentially chooses actions $a_{0}, a_{1}, \ldots$, one at a time. Her goal is maximize the probability that the infinite sequence $(a_{0}, a_{1}, \ldots)$ is an element of a given subset $G$ of $\N^\N$. The set $G$ is assumed to be a Borel tail set. The Controller's choices are restricted: having taken a sequence $h_{t} = (a_{0}, \ldots, a_{t-1})$ of actions prior to stage $t \in \N$, she must choose an action $a_{t}$ at stage $t$ from a non-empty, finite subset $A(h_{t})$ of $\N$. The set $A(h_{t})$ is chosen from a distribution $p_{t}$, independently over all $t \in \N$ and all $h_{t} \in \N^{t}$. We consider several information structures defined by how far ahead into the future the Controller knows what actions will be available.

In the special case where all the action sets are singletons (and thus the Controller is a dummy), Kolmogorov’s 0-1 law says that the probability for the goal to be reached is 0 or 1. We construct a number of counterexamples to show that in general the value of the control problem can be strictly between 0 and 1, and derive several sufficient conditions for the 0-1 ``law" to hold.

JOINT WORK WITH: J\'{a}nos Flesch, William Sudderth, Xavier Venel

__A CRASH COURSE ON MEAN FIELD MODELS: DERRIDA’S GREM AND APPLICATIONS__**Part 5 - On the GREM Approximation of TAP Free Energies. By Giulia Sebastiani.**

The free energy of TAP-solutions for the SK-model of mean field spin glasses can be expressed as a nonlinear functional of local terms:

we exploit this feature in order to contrive abstract GREM-like models which we then solve by a classical large deviations treatment.

This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy

for the SK-model, and formalizes thetrue cavity dynamics which acts on TAP-space, i.e. on the space of TAP-solutions.

*Joint works with Nicola Kistler, and Marius A. Schmidt.*

**Part 6 - From log-correlated models to (un)directed polymers in the mean field limit. By Adrien Schertzer.**

As seen in the previous lectures, Derrida's Random Energy Models have played a key role in the understanding of certain issues in spin glasses.

The mathematical analysis of these models - in particular the multi-scale refinement of the second moment method as devised by Kistler,

is also particularly efficient to analyse the so-called log-correlated class; the latter consists of Gaussian fields with - as the name suggests,

logarithmically decaying correlations. I will introduce/recall some models falling into this class, and the main steps in their analysis

through the paradigmatic Branching Brownian motion / Branching Random Walk. Finally, I will conclude with recent results on models

which are not even Gaussian, but for which the multiscale treatment still goes through swiftly:

the directed and undirected first passage percolation in the limit of large dimensions, a.k.a. the (un)directed polymers in random environment.

*Joint works with Nicola Kistler, and Marius A. Schmidt.*

Seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/