Weekly Bulletin

We will review how one can design convex integration schemes using volume-preserving diffeomorphisms to address problems that are typically not amenable to more traditional schemes. To illustrate this philosophy, we will consider the construction of rough steady Euler flows with certain prescribed topological properties and of Hölder continuous dissipative solutions to ideal MHD. The talk is based on joint work with Daniel Peralta-Salas and Javier Peñafiel-Tomás.

This talk, based on joint work with Ambrosioni and Biran, will discuss some metric properties of some spaces of Lagrangian submanifolds endowed with the spectral metric. In particular, I will show how some natural notions of complexity in triangulated categories can be applied to some triangulated persistence refinements of Fukaya categories to identify examples of such metric spaces that are not precompact.

The Landau equation is a fundamental collisional kinetic model arising in plasma physics. In the spatially homogeneous setting, global well-posedness is known, and solutions exhibit strong regularizing effects. This naturally raises the question: can singularities arise due to spatial inhomogeneity? In this talk, I will present a construction of smooth, strictly positive initial data for the inhomogeneous Landau equation with γ \in (\sqrt{3}, 2] that develops a finite-time implosion singularity. Interestingly, the singularity manifests at the hydrodynamic level: macroscopic quantities such as density and temperature blow up, while the L^\infty-norm of the distribution function remains uniformly bounded. Working in self-similar variables, our blow-up profile is asymptotically self-similar (Type II). The distribution function converges to a local Maxwellian, and the associated macroscopic fields approach to a smooth imploding solution of the compressible Euler equations. To our knowledge, this provides the first example of a collisional kinetic equation that is globally well-posed in the homogeneous setting but exhibits finite-time singularity formation for smooth inhomogeneous data. This is joint work with Jacob Bedrossian, Jiajie Chen, Maria Gualdani, Vlad Vicol, and Jincheng Yang.

<div>The first nontrivial eigenvalue of the Laplacian on a closed Riemannian manifold (called the spectral gap) is a geometrical invariant that encodes many interesting properties of the manifold. Notably, it gives some quantitative information on the connectedness of the manifold, and on the rate of convergence to equilibrium of the heat flow.</div> <div> </div> <div>I will focus on hyperbolic surfaces, which are closed manifolds of dimension 2 with constant curvature -1. After defining these objects, I will review some results about the spectral gap in this setting. Recent developments provide probabilistic constructions of random surfaces that display optimal spectral gaps.</div>

<p><span style="caret-color: #4d4d4d; color: #4d4d4d; font-family: gothambook, sans-serif; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">We discuss how chaos, i.e., sensitivity to initial conditions, arises in the setting of polygonal billiards. In particular, we give a complete classification of the rational polygons whose billiard flow is weak mixing in almost every direction, proving a longstanding conjecture of Gutkin. This is joint work with Jon Chaika and Giovanni Forni. No previous knowledge on the subject will be assumed.</span></p>

Let G be a connected commutative algebraic group over the complex numbers. In this talk, I will introduce a class of multiplicative line bundles with flat connection on G, called character sheaves, and describe the construction and geometry of their moduli space. These objects are inspired by ℓ-adic local systems that play a central role in modern analytic number theory. The construction of the moduli space relies on the so-called de Rham space, which provides a stack-theoretic approach to de Rham cohomology. If time permits, I will explain how the geometry of this moduli space leads to generic vanishing theorems for de Rham cohomology, and how these results give rise to a common framework for differential and difference Galois theory.

Hochschild homology has proved to be an important invariant in algebra and homotopy theory, in particular due to its relevance in algebraic K-theory and fixed point theory, leading to the development of numerous variants of the original construction. Ponto's theory of shadows provides a bicategorical axiomatization of Hochschild homology-type invariants, which captures the essential common properties of all known variants of Hochschild homology, such as Morita invariance. In recent work, Nima Rasekh and I clarified the relationship between shadows and Hochschild homology. After extending the notion of Hochschild homology to bicategories in a natural manner, we proved the existence of a universal shadow on any bicategory B, taking values in the Hochschild homology of B, through which all other shadows on B factor. Shadows are thus co-represented by a bicategorical version of Hochschild homology. Using the universal shadow on the free adjunction bicategory, we established a universal Morita invariance theorem, of which all known cases are immediate corollaries. In this talk I will give an overview of my work with Rasekh and provide relevant examples of shadows, including the free loop space construction, then discuss potential generalization and extensions of our results.

I’ll present some recent work about the connection of vortex filaments/tubes that move according to Navier Stokes Equation and the binormal curvature flow of curves in 3d. This is a joint work with Marco A. Fontelos and Mikel Ispizua.

Recent works have established sharp Fourier decay for subcritical real Gaussian multiplicative chaos (GMC) on the circle, and in this talk I will discuss the corresponding harmonic picture for imaginary GMC. Gaussian multiplicative chaos is obtained by exponentiating log-correlated Gaussian fields; on the unit circle, one may take the trace of the two-dimensional Gaussian free field with covariance $\log 1/|e^{i\theta}-e^{i\theta'}|$. For purely imaginary parameters $\gamma=i\beta$ with $\beta\in(0,1)$, the resulting object $M_{i\beta}$ exists as a complex-valued random distribution and enjoys strong integrability properties. The Fourier dimension captures the decay of the Fourier coefficients $c_n$ of a distribution. It is defined as the supremum of $s\in(0,1)$ such that $|c_n|^2 = O(|n|^{-s})$. We prove that the Fourier dimension of $M_{i\beta}$ is almost surely $1-\beta^2$ and establish a joint CLT for the rescaled coefficients. The proof uses the method of moments specific to the imaginary regime. The moments of $c_n$ (and mixed moments of nearby modes) are rewritten as Coulomb-gas integrals on the circle, and then analysed via the Selberg inner product and Jack polynomial expansions, which convert the moment integrals into positive partition sums amenable to sharp asymptotic analysis. Joint work with Hermanni Rajamäki and Vincent Vargas.

In this talk, we take an in-depth look at the topic of missing value imputation. Focusing on the 'missing at random' (MAR) case, we discuss the qualities that constitute an effective imputation method and how to evaluate them in practice. Crucially, we review common pitfalls in the literature that can bias subsequent analysis and explain how to avoid them, while also discussing some of the most promising imputation methods currently available. If time permits, we also briefly explore state-of-the-art research on evaluating an imputation method for a given dataset based on imputation scores (I-Scores).

<p>In this talk, I will present a recent joint work with Galina Perelman concerning the derivative nonlinear Schrödinger (DNLS) equation on the torus. The DNLS equation which is a canonical dispersive equation arising in a variety of physical contexts is known to be completely integrable (and then it admits a spectral formulation, an infinite number of conservation laws and explicit families of conservation laws). The main difficulty of this equation is the lack of coercivity of its conservation laws in the regime above the algebraic soliton threshold, and this makes large data global well-posedness a challenging issue. This equation was solved a short time ago on the real line, while  the case of the torus is still less understood. In this work, we prove  global well-posedness for DNLS equation on the torus.</p> <p>The first part of my presentation will be devoted to a general overview of DNLS and completely integrable equations, then I will on the case of the torus  by providing the new arguments that enabled us to achieve our goal. </p>

The rough Heston model is a very popular recent model in mathematical finance; however, the lack of Markov and semimartingale properties poses significant challenges in both theory and practice. A way to resolve this problem is to use Markovian approximations of the model. Several previous works have shown that these approximations can be very accurate even when the number of additional factors is very low. Existing error analysis is largely based on the strong error, corresponding to the L2 distance between the kernels. Extending earlier results by [Abi Jaber and El Euch, SIAM Journal on Financial Mathematics 10(2):309--349, 2019], we show that the weak error of the Markovian approximations can be bounded using the L1-error in the kernel approximation for general classes of payoff functions for European style options. Moreover, we give specific Markovian approximations which converge super-polynomially in the number of dimensions, and illustrate their numerical superiority in option pricing compared to previously existing approximations. The new approximations also work for the hyper-rough case H>−1/2. In addition, we provide explicit characterisations of the state space of the Markovian approximations. Joint work with Eduardo Ani Jaber and Simon Breneis.

I will talk about a surprising connection between the space of multiple polylogarithms on a d-dimensional algebraic torus, and the Steinberg representation of GL_d(Q). I will show how this leads to several new identities for multiple polylogarithms, implying, in paricular, that all polylogarithms can be expressed using polylogarithms of the form Li_{m,1,...,1}. The talk is based on a recent joint work with Steven Charlton and Daniil Rudenko.