Weekly Bulletin

What is a random Hamiltonian diffeomorphism? In this talk, we will try to answer this question by constructing a family probability measure on the group of Hamiltonian diffeomorphisms. We will see that classical invariants such as the Hofer norm and spectral invariants become well-behaved random variables in this setting. Furthermore, these measures have (for suitable choices of parameters) several desirable properties, such as full support on Ham(M), explicit estimates of the measure of Hofer-balls, and well-behaved extensions to the world of C^0-symplectic geometry. Additionally, these measures behave like Gaussian measures in certain ways. After a careful review of the construction, we will review these properties and some open problems. Note: the seminar begins at 15:00 rather than 15:15. The location is HG 19.2 and not the usual room.

We discuss recent developments in the theory of symmetrization for elliptic partial differential equations subject to Robin-type boundary conditions. While classical symmetrization arguments are well understood in the Dirichlet and Neumann settings, the Robin case presents several new challenges. In this talk I will present sharp comparison results for solutions of elliptic problems with Robin boundary conditions. Applications to spectral inequalities will also be discussed.

<p>Error analysis is a integral part of numerical mathematics. The classical way of doing this- called apriori error analysis- only provides an asymptotic result that involves the exact solution, which is generally unknown, and requires some regularity assumptions on the exact solution. Therefore the a-priori analysis only provides a qualitative result. A-posteriori error analysis on the other hand does not require any regularity assumptions nor any knowledge of the exact solution and provides a computable upper bound to the error.</p> <p>In this talk we will be considering the Poisson model problem in 2d and use the finite element method to approximate the exact solution. We then will develop an a-posteriori error estimator from first principles and prove that this estimator truly is an upper bound to the error. We will then use the error estimator to motivate an adaptive mesh refinement strategy for approximating solutions with point singularities in order to recover optimal convergence rates. The talk closes by a short discussion on an open problem in the field of adaptive finite element methods.</p>

<p>in joint work with Pablo Carrasco we push the theory of<br>thermodynamic formalism from hyperbolic systems to partially<br>hyperbolic systems in different forms. In the meantime we find several<br>interesting open problems and dynamical proofs of some (to us)<br>interesting results,, for example we show Burger-Monod theorem on<br>vanishing of second bounded cohomology for higher rank lattices. The goal of the talk is to discuss this development.</p>

Two classical constructions of cusped hyperbolic 3–manifolds of finite volume are (some) link complements in the sphere, and quotients of hyperbolic space by congruence subgroups of Bianchi groups such as PSL₂(<b>Z</b>[i]). A conjecture of Baker and Reid posits that only finitely many manifolds occur as both. I will discuss this conjecture in relation with well-known and conjectured properties of arithmetic groups, and present a proof of the conjecture obtained in joint work with S. Kionke.

In recent years, probability theory has made a big impact on making progress on many long-standing problems in number theory, and many exciting new probabilistic problems have naturally emerged with strong arithmetic motivations. In this talk, I will tell you part of the story of their interactions. Many of the connections can be summarized as a random walk problem, but with influences from primes. You may find the story interesting if any of the following keywords catch your attention: the Ballot problem, Log-correlated fields, Gaussian multiplicative chaos, Fydorov-Hiary-Keating conjecture, Polya's conjecture, random polynomials, distribution of primes.

Spectral methods are a simple yet effective approach to extract information from data which is high-dimensional, i.e., where sample size and signal dimension grow proportionally. As a prelude, we will consider the prototypical problem of inference from a generalized linear model with an i.i.d. Gaussian design. Here, the spectral estimator is the principal eigenvector of a data-dependent matrix. We will discuss the emergence of a (BBP-like) phase transition in the spectrum of this random matrix and how such phase transition is related to signal recovery. The core of the talk will then deal with two models that capture the heterogeneous and structured nature of practical data. First, we will consider a multi-index model where the output depends on the inner product between the feature vector and a fixed number $p$ of signals, and the focus is on recovering the subspace spanned by the signals via spectral estimators. By using tools from random matrix theory, we will locate the top-$p$ eigenvalues of the spectral matrix and establish the overlaps between the corresponding eigenvectors (which give the spectral estimators) and a basis of the signal subspace. Second, we will consider a generalized linear model with a correlated design matrix. Here, the analysis of the spectral estimator relies on tools based on approximate message passing, and we will present a methodology which is broadly applicable to the study of spiked matrices. In all these settings, the precise asymptotic characterization we put forward enables the optimization of the data preprocessing, thus allowing to identify the spectral estimator that requires the minimal sample size for signal recovery.

Univariate extreme value analysis often focus on observations that are large in the sense that they exceed a large threshold, above which observations are approximately generalized Pareto distributed under mild assumptions. The choice of threshold has a large impact on inference and its uncertainty is often ignored in subsequent analysis. Starting with the statistical properties underlying the various proposals, this talk provides an extensive review of threshold selection mechanisms, including semiparametric methods based on Hill’s estimator, visual diagnostics, goodness-of-fit tests, extended generalized Pareto models, among others. We perform an extensive simulation study under various tail regimes, with serial dependence and varying sample sizes, to identify the most promising methodologies. Methods are showcased on the Padova rainfall series and we provide critical assessment of methods strengths and weaknesses. This presentation is based on joint work with Sonia Alouini (MeteoSwiss) and Anthony Davison (EPFL).

<div class="elementToProof" style="font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; text-align: left; background-color: white; margin: 0px; font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: black;">Gibbs measures for nonlinear dispersive PDEs have been used as a fundamental tool in the study of low-regularity almost sure global well-posedness of the associated Cauchy problem following the pioneering work of Bourgain in the 1990s. In the first part of the talk, we will discuss the connection of Gibbs measures with the classical Kubo-Martin-Schwinger (KMS) condition. The latter is a property characterizing equilibrium measures of the Liouville equation.  In particular, we show that Gibbs measures are the unique KMS equilibrium states for a wide class of Hamiltonian PDEs, including nonlinear Schrödinger equations with defocusing interactions. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. This is joint work with Zied Ammari (University of Besançon, Bourgogne-Franche-Comté). </div> <div class="elementToProof" style="font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; text-align: left; background-color: white; margin: 0px; font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: black;"> </div> <div class="elementToProof" style="font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; text-align: left; background-color: white; margin: 0px; font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: black;">In the second part of the talk, we study (local) Gibbs measures for focusing nonlinear Schrödinger equations. These measures have to be localized by a truncation in the mass in one dimension and in the Wick-ordered (renormalized) mass in dimensions two and three. We show that local Gibbs measures correspond to suitably localized KMS states. This is joint work with Andrew Rout (Politecnico di Milano) and Zied Ammari (University of Besançon, Bourgogne-Franche-Comté).</div>

With the global shift from defined benefit to defined contribution pension systems, retirement planning is now fully borne on individuals elevating their exposure to longevity, health, and market risks. This transition has prompted more precautionary saving behaviour, as retirees become more conservative in fully consuming their wealth. This research proposes a decumulation strategy which combines long-term care insurance (LTCI) and guaranteed minimum death benefit (GMDB) purchased at retirement with a withdrawal-then-rebalance investment approach. Within this framework, the retirement fund is modelled using a regime-switching structure, while a target volatility strategy detects asset allocation to smooth wealth dynamics and reduces likelihood of extreme losses. The LTCI covers late-life healthcare costs, whereas the GMDB secures a minimum bequest, thereby supporting both consumption confidence and legacy objectives. Numerical experiments compare consumption patterns under this strategy with default account-based pension drawdown strategies. Results reveal that the proposed strategy provides smoother long-term consumption and better resilience to adverse financial shocks. Sensitivity analyses exploring variations in insurance allocation ratios, health state transitions, and target volatility levels are performed. Preliminary results suggest that moderate volatility targets strike an effective balance between risk and sustainability, and that the strategy remains robust across different health scenarios. Joint work with Jennifer Alonso-Garcia, Mengdie Hu and Yuxin Zhou.

In the early 2000s Giovanni Felder and his collaborators began the group-theoretic study of the elliptic gamma function, a remarkable multivariable meromorphic q-series arising from mathematical physics. In particular they obtained a collection of modular functional equations under the group SL3(Z) which make it a higher-dimensional analogue of the Jacobi theta function. In this work, we unveil the significance that this function and its variants have in number theory. Our main thesis is that these functions play the role of modular units in extending the theory of complex multiplication to complex cubic fields. In other words we propose a conjectural solution to Hilbert’s 12th problem for complex cubic fields, following a line of research actually initiated by G. Eisenstein. We give substantial numerical evidence that support this conjecture, and relate it to the Stark conjecture by proving an analogue of the Kronecker limit formula in this cubic setting. If times permits, I will also mention ongoing generalizations to SLn(Z), n>3, as part of Pierre Morain's PhD work at Sorbonne Université. This is joint work with Nicolas Bergeron and Luis Garcia.

It has been conjectured that the Gromov-Witten invariants of a K3 or abelian surface for arbitrary curve classes are determined by those for primitive curve classes by a very simple multiple cover rule. The primitive invariants in turn are known, so the conjecture would determine the entire Gromov-Witten theory of K3 and abelian surfaces. In this talk I will report on joint work in progress with Rahul Pandharipande in which we aim to prove this conjecture.