Weekly Bulletin

In this talk, I will talk about essential self-adjointness (ESS) of Klein-Gordon operators on spacetimes which are perturbations of Minkowski spacetime. ESS of differential operators has been studied especially for elliptic operators such as Laplace-Beltrami operators and Schrödinger operators. Here we focus on ESS for Klein-Gordon operators which are not elliptic, where previous methods cannot be applied. Moreover, it has an application to a construction of Feynman propagator in Quantum Field Theory. This is a joint work with Shu Nakamura.

In this talk, I'll discuss an enriched version of (mixed) Hodge structures and the existence of such structures on the cohomology of `links', with the goal of understanding the algebraic K-theory of convergent power series rings via these new invariants. This is based on joint work with M. Nori and V. Srinivas.

In Diophantine approximation, it is a classical problem to determine the size of the sets related to \(\psi\) approximable set for a given non-increasing function \(\psi\). Bugeaud determined the Hausdorff dimension of the exact \(\psi\) approximable set answering a question posed by Beresnevich, Dickinson, and Velani. We compute the Hausdorff dimension of the exact set in the general setup of Ahlfors regular spaces. Our result applies to approximation by orbits of fixed points of a wide class of discrete groups of isometries acting on the boundary of hyperbolic metric spaces. This is joint work with Anish Ghosh and Debanjan Nandi.

When numbers are added in the usual way, carries occur. For typical numbers, how many carries are likely (and how do they go; if there was just a carry is it more or less likely that there is a carry in the next place?). It turns out that carries form a Markov chain with an 'Amazing transition matrix' (are any matrices amazing?). This same matrix occurs in the analysis of the usual way of shuffling cards. I will use the correspondence to explain the 'seven shuffles' theorem, try to explain the correspondence (carries are cocycles) and show how the same matrix occurs in taking sections of generating functions. This is joint work with Jason Fulman.

By linking conceptual theories with observed data, generative models can support reasoning in complex situations. They have come to play a central role both within and beyond statistics, providing the basis for power analysis in molecular biology, theory building in particle physics, and resource allocation in epidemiology. This talk will survey some applications of modern generative models and show how they inform experimental design, iterative model refinement, goodness-of-fit evaluation, and agent based simulation. We emphasize a modular view of generative mechanisms and discuss how they can be flexibly recombined in new problem contexts. Current research in generative models is currently split across several islands of activity, and we highlight opportunities lying at disciplinary intersections. This is joint work with Kris Sankaran that was recently published and for which practical illustrations are available at https://github.com/krisrs1128/generative_review.

Let N be a compact Riemannian manifold of dimension 3 or higher, and g a Lipschitz non-negative (or non-positive) function on N. In joint works with Neshan Wickramasekera we prove that there exists a two-sided (well-defined global unit normal) closed hypersurface M whose mean curvature attains the values prescribed by g. Except possibly for a small singular set (of codimension 7 or higher), the hypersurface M is C^2 immersed; more precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections (around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially, as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when g is a constant, in which case the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature. The proof employs an Allen--Cahn approximation scheme, classical minmax, and gradient flow arguments. A key issue in the construction (and, more generally, in related compactness questions) is the possible formation of "hidden boundaries".

The most basic question in enumerative Geometry is: How many lines are there through two distinct points? A natural extension of this question is to count the number Nd of rational curves of degree d passing through 3d-1 points in general position in the complex projective plane. I will explain how to formulate this problem using Gromov-Witten theory and Kontsevich solution to the problem in terms of a recursive formula which reduces the question to the computation of N1 = 1.

Brill-Noether theory is a rich subject dating back to the late 19th century. It governs the geometry of the space of maps from curves to projective space, such as the dimensions of its irreducible components and the dimensions of their images in the moduli space of curves. Logarithmic geometry suggests an alternative geometry: the space of rational maps from smooth pointed curves to an algebraic torus, known in some circles as the interior of the higher double ramification cycle. I will explain how to use logarithmic and tropical techniques to gain access to the geometry of this space, and prove “Brill-Noether existence” results. These specialise in a transparent fashion to classical results of Kempf, Kleiman, Laksov, as well as the recent Hurwitz-Brill-Noether direction pioneered by K. Cook-Powell, E. Larson, H. Larson, D. Jensen, and I. Vogt. The talk is based on work with D. Jensen.

Isogeny-based cryptography is an active research area in post-quantum cryptography, and its security depends on variants of isogeny problems, namely the problem of finding an explicit isogeny between two abelian varieties. Although much of the research focused on isogenies of elliptic curves so far, it is mathematically interesting to understand isogeny graphs in dimension 2 in full generality, and the recent attacks using isogenies in higher dimensions switched the attention to dimension 2 more than ever. Unlike known methods, we use very geometrical and arithmetical objects called Refined Humbert Invariants defined by Kani. We will propose possible promising applications using Refined Humbert Invariants of abelian surfaces. We will investigate how to understand isogeny graphs of abelian varieties, and explain possible applications such as determining minimum walks, finding split abelian surfaces, a security reduction to computing the degree of an isogeny, and multilinear maps. This is a joint work in progress with Harun Kir. <BR> <BR> (**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)

It was conjectured by John Milnor in 1968 that the fundamental group of a complete Riemannian manifold with non negative Ricci Curvature is finitely generated. I will present recent joint work with Elia Bruè and Aaron Naber where we construct a complete 7-dimensional Riemannian manifold with nonnegative Ricci Curvature and infinitely generated fundamental group, thus providing a counterexample to the Milnor conjecture.

Soft tissues and other nearly incompressible media pose a challenge for simulating elastic wave propagation, due to the slow propagation of shear waves compared to pressure waves. To overcome this challenge, a classical Helmholtz-Hodge decomposition is used to split the displacement field into scalar pressure (P -) and shear (S-) waves, allowing for separate treatment of the two dynamics and the construction of discretization spaces suited for each type of wave. This presentation focuses on the simulation of 2D soft scattering elastic wave propagation in isotropic homogeneous media, using the scalar potential decomposition in the time-harmonic regime. For problems defined in bounded domains, a Virtual Element Method (VEM) with varying mesh sizes and degrees of accuracy is proposed to approximate the two scalar potentials. For unbounded domains, a boundary element method is coupled with the VEM. The proposed approach performs better than standard methods that directly use the vector formulation, as it allows for tracking the different wave numbers associated with P - and S-speeds of propagation. This makes it possible to use a high-order method for the approximation of waves with higher wave numbers. We establish the stability of our method and present a convergence error estimate in the L2-norm for the displacement field. Notably, our estimate separates the contributions to the error associated with the P - and S- waves. We provide numerical results to demonstrate the effectiveness of the proposed approach. This presentation is the result of collaborative work with M. Ferrari and L. Scuderi from the Polytechnic University of Turin.

Low-Rank Parity-Check (LRPC) codes are a class of rank metric codes that have many applications specifically in network coding and cryptography. Recently, LRPC codes have been extended to Galois rings which are a specific case of finite rings. In this talk, we first define LRPC codes over finite commutative local rings, which are bricks of finite rings, with an efficient decoder. We improve the theoretical bound of the failure probability of the decoder. Then, we extend the work to arbitrary finite commutative rings. Certain conditions are generally used to ensure the success of the decoder. Over finite fields, one of these conditions is to choose a prime number as the extension degree of the Galois field. We have shown that one can construct LRPC codes without this condition on the degree of Galois extension. <BR> <BR> (**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)

Random planar maps are toy models in random geometry that allow to easily define discrete random surfaces. One hope that when letting the number of faces tend to infinity and their size to zero at the correct speed, one can get a continuum random surface, in the same way Brownian motion arises from large random walks. A decade ago, after a series of works by different authors, the convergence of random quadrangulations and a few other models to the so-called Brownian map was finally obtained simultaneously by Le Gall and Miermont. In this talk we will discuss the more general model of maps sampled uniformly at random given their face degrees and will discuss the convergence depending on these degrees.

Latent variable models have been an integral part of probabilistic machine learning, ranging from simple mixture models to variational autoencoders to powerful diffusion probabilistic models at the center of recent media attention. Perhaps less well-appreciated is the intimate connection between latent variable models and data compression, and the potential of these models for advancing natural science. This talk will explore these topics. I will begin by showcasing connections between variational methods and the theory and practice of neural data compression. On the applied side, variational methods lead to machine-learned compressors of data such as images and videos and offer principled techniques for enhancing their compression performance, as well as reducing their decoding complexity. On the theory side, variational methods also provide scalable bounds on the fundamental compressibility of real-world data, such as images and particle physics data. Lastly, I will also delve into climate science projects, where a combination of deep latent variable modeling and vector quantization enables assessing distribution shifts induced by varying climate models and the effects of global warming. Short Bio: Stephan Mandt is an Associate Professor of Computer Science and Statistics at the University of California, Irvine. From 2016 until 2018, he was a Senior Researcher and Head of the statistical machine learning group at Disney Research in Pittsburgh and Los Angeles. He held previous postdoctoral positions at Columbia University and Princeton University. Stephan holds a Ph.D. in Theoretical Physics from the University of Cologne in Germany, where he received the National Merit Scholarship. He received the NSF CAREER Award, a Kavli Fellowship of the U.S. National Academy of Sciences, the German Research Foundation's Mercator Fellowship, and the UCI ICS Mid-Career Excellence in Research Award. He is a member of the ELLIS Society and a former visiting researcher at Google Brain. Stephan will serve as Program Chair of the AISTATS 2024 conference, currently serves as an Action Editor for JMLR and TMLR, and frequently serves as Area Chair for NeurIPS, ICML, AAAI, and ICLR.

In recent decades, the study of many-body systems has been an active area of research in both physics and mathematics. In this talk, we will consider a system of N spin 1/2 interacting fermions confined in a box in the dilute regime, with a particular focus on the correlation energy which is defined as the difference between the ground state energy and that of the free Fermi gas. We will discuss some recent results about a first order asymptotics for the correlation energy in the thermodynamic limit where the number of particles and the size of the box are sent to infinity keeping the density fixed. In particular, we will present a new upper bound for the correlation energy, which is consistent with the well-known Huang-Yang formula from 1957.

In this talk, we first provide a brief introduction to quantum computing from a mathematical perspective. No prior knowledge of quantum computing is necessary. We then introduce a quantum Monte Carlo algorithm to solve high-dimensional Black-Scholes PDEs with correlation for high-dimensional option pricing. The payoff function of the option is of general form and is only required to be continuous and piece-wise affine (CPWA), which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension d of the PDE and the reciprocal of the prescribed accuracy ε and so demonstrate that our quantum Monte Carlo algorithm does not suffer from the curse of dimensionality. This talk is based on a joint work with Yongming Li.

The Brill-Noether classes w_{g,d}^r are virtual fundamental classes associated to the Brill-Noether loci W_{g,d}^r in the universal Jacobian Pic^d(C_{g,n}/M_{g,n}), parametrizing curves with line bundles that have at least r+1 linearly independent sections. Pulling back to M_{g,n} via Abel-Jacobi sections produces tautological classes, which can be calculated by Grothendieck-Riemann-Roch. Extending those classes to the compactification \bar{M}_{g,n} is however not straightforward: the right extensions live naturally on blowups of \bar{M}_{g,n}, or, equivalently, in logCH(\bar{M}_{g,n}). Consequently, their calculation is also subtle. In this talk, I will discuss a general formula that allows to calculate these classes. The formula is close in the spirit of Mumford's GRR calculation, but with combinatorial corrections required. I will then discuss the connection of the calculation with relations in the tautological ring, to the limited extent that I understand them. This is joint work with Alex Abreu and Nicola Pagani.