Weekly Bulletin

Ιn this talk, I will address the problem of defining and computing the U(1)^n Chern–Simons (CS) partition function on closed, oriented 3-manifolds in a fully gauge-invariant framework. In contrast to usual approaches, we do not gauge fix. Instead, using the tools of Deligne–Beilinson cohomology, which naturally incorporates both differential and topological data, we work on the gauge classes of connections. By applying the reciprocity formulas of Deloup and Turaev, I will establish the so-called CS duality and demonstrate its relation to a Reshetikhin–Turaev topological invariant. Finally, I define Wilson loop observables within this framework and compute their expectation values, completing a fully consistent treatment of the theory in the closed manifold case.

We prove an intersection result between Lagrangian tori inside the 4 dimensional cylinder and certain codimension 1 hypersurfaces with Lagrangian torus boundary. Intersections between polydisks and the hypersurfaces are also obtained, under weaker conditions. Consequences are computations of various shape type invariants, which result in symplectic embedding obstructions. There is a lot of rigidity, provided the domain is not 'thin' relative to the target. This is joint work with Ely Kerman.

Variational formulations of layer potentials and boundary integral operators generalize their classical construction based on Green's functions. Unlike classical approaches, our method applies even when Green's functions are not explicitly available, such as for Helmholtz problems with rough (e.g., piecewise Lipschitz) coefficients. Wave-number explicit estimates and properties like jump conditions are obtained directly from variational principles. From this, we obtain a generalised Calderón identity and derive nonlocal (integral) formulation of acoustic transmission problems in heterogeneous media. The well-posedness of the resulting boundary integral equations is directly inherited from the underlying partial differential equation. Our analytical framework treats general spatial dimensions and complex wave numbers simultaneously by imposing an artificial boundary and employing recent insights into the associated Dirichlet-to-Neumann map.

The Ramsey number R(k) is the minimum n such that every red-blue colouring of the edges of the complete graph on n vertices contains a monochromatic copy of K_k. It has been known since the work of Erdos and Szekeres in 1935, and Erdos in 1947, that 2^{k/2} < R(k) < 4^k, but until recently the only improvements were by lower order terms. In this talk I will give an introduction to the area, and also sketch the proof of a recent result, which improves the upper bound of Erdos and Szekeres by a (small) exponential factor. Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.

<p>We propose to study actions of countable groups on measure spaces such that every group element individually acts as a conservative transformation, that is, a transformation such that no set of positive measure is disjoint from all its translates. We construct such actions of the free group, using the measurable full group of a hyperfinite equivalence relation. The motivation for our work was to find interesting examples of boomerang subgroups.<br />Based on joint work with Y. Glasner and T. Hartnick.</p>

In this talk, I will show that fixed-domain Gromov–Witten invariants of a positive symplectic manifold (e.g., a smooth Fano variety) are signed counts of J-holomorphic curves in X satisfying prescribed incidence conditions. This provides a symplectic analogue of a conjecture of Lian and Pandharipande, recently disproved in the algebraic setting by Beheshti, Lehmann, Lian, Riedl, Starr, and Tanimoto. The proof relies on constructing the fixed-domain Gromov–Witten pseudocycle without the use of inhomogeneous or domain-dependent perturbations, answering an old question posed by Ruan and Tian.

We will present injective metric spaces, which are metric spaces where any family of pairwise intersecting balls has a non-empty global intersection. We will explain how they can be used to develop a rich theory of nonpositively curved simplicial complexes, somewhat parallel to the theory of CAT(0) cube complexes. We will show that they arise naturally for numerous spaces and groups: hyperbolic groups, buildings, braid groups, Artin groups, Garside groups, some arc complexes on surfaces...

Scientific modeling and computation traditionally rely on structured mathematics and hand-designed algorithms. In this talk, I propose a new perspective: treating both modeling and computation as processes operating within the space of natural language. I will introduce two complementary approaches that realize this vision. The first uses symbolic learning based on tree structures to generate mathematical expressions, where modeling is performed by constructing symbolic trees and computation is governed by operator rules. The Finite Expression Method (FEX) exemplifies this approach by discovering interpretable, high-accuracy solutions to PDEs and physical systems. The second approach employs large language models (LLMs) for automatic code generation and reasoning to translate scientific problem descriptions into formal mathematical models and executable solvers to solve these problems. As an example, the OptimAI framework demonstrates how multi-agent LLM collaboration enables reliable end-to-end optimization problem modeling and solving. Together, these methods point toward a unified paradigm where symbolic and language models form the foundation for interpretable, scalable scientific discovery and computation

Drilling a closed hyperbolic 3-manifold along an embedded geodesic is a crucial technique in low-dimensional topology, transforming the fundamental group of the manifold into a relatively hyperbolic group. In this talk, we extend this concept by proving that, under appropriate conditions, a similar "drilling" operation can be applied to a (Gromov) hyperbolic group with the 2-sphere boundary. Our primary motivations and applications revolve around the Cannon Conjecture, which states that if the Gromov boundary of a hyperbolic group is homeomorphic to the 2-sphere, then the group is virtually (i.e., up to a finite-index subgroup) the fundamental group of a closed 3-manifold of constant negative curvature. We also explore the relatively hyperbolic counterpart—the Toral Relative Cannon Conjecture. Using drilling, we show that if the Toral Relative Cannon Conjecture holds, then the Cannon Conjecture is valid for all residually finite hyperbolic groups. The Toral Relative Cannon Conjecture appears more accessible, owing to the presence of additional structure—abelian parabolic subgroups. This is joint work with Daniel Groves, Peter Haïssinsky, Jason Manning, Alessandro Sisto, and Genevieve Walsh.

Min-norm interpolators emerge naturally arise as implicit regularized limits of modern machine learning algorithms. Recently, their out-of-distribution risk was studied when test samples are unavailable during training. However, in many applications, a limited amount of test data is typically available during training. Properties of min-norm interpolation in this setting are not well understood. In this talk, I will present a characterization of the risk of pooled min-L2-norm interpolation under covariate and concept shifts. I will show that the pooled interpolator captures both early fusion and a form of intermediate fusion. Our results have several implications. For example, under concept shift, adding data always hurts prediction when the signal-to-noise ratio is low. However, for higher signal-to-noise ratios, transfer learning helps as long as the shift-to-signal ratio lies below a threshold that I will define. Our results also show that under covariate shift, if the source sample size is small relative to the dimension, heterogeneity between domains improves the risk. Time permitting, I will introduce a novel anisotropic local law that allows to achieve some of these characterizations and is of independent interest in random matrix theory. This is based on joint work with Kenny Gu, Yanke Song and Sohom Bhattacharya.

<p>We will derive a class of nonlinear diffusion equations (covering both porous media and fast diffusion equations) as the hydrodynamic limit of a class of nonlinear, spatially inhomogeneous kinetic equations of BGK-type. These equations share the same superlinearity as the Boltzmann equation and fall into the class of run and tumble equations appearing in mathematical biology. We will look at the Cauchy problem under the parabolic scaling and also provide a quantitative nonlinear hypocoercivity result. The derivation is based on both the hypocoercivity and relative entropy methods. The talk is based on a joint work with Daniel Morris (TU Delft) and Josephine Evans (University of Warwick).</p>

We present structured additive regression models to model probability density functions given scalar covariates. To preserve nonnegativity and integration to one, we formulate our models for densities in a Bayes Hilbert space with respect to an arbitrary finite measure. This enables us to not only consider continuous densities, but also, e.g., discrete densities (compositional data) or mixed densities. Mixed densities occur in our application motivated by research on gender identity norms and the distribution of the woman’s share in a couple’s total labor income, as the woman’s income share is a continuous variable having discrete point masses at zero and one for single-earner couples. We show how to handle the challenging case of mixed densities using an orthogonal decomposition. We discuss interpretation of effect functions in our model via odds-ratios. We consider two cases: First, where densities are observed and are directly used as responses. Second, when only individual scalar realizations of the conditional distributions are observed, but not the whole conditional densities, we use our additive regression approach to model the conditional density given covariates. We show approximate equivalence of the resulting Bayes space penalized likelihood to a certain penalized Poisson likelihood, facilitating estimation. We apply our framework to a motivating gender economic data set from the German Socio-Economic Panel Study (SOEP) to analyze the distribution of the woman’s share in a couple’s total labor income, given year, place of residence and age of the youngest child. Results show a more symmetric distribution in East German than in West German couples after German reunification and a smaller child penalty comparing couples with and without minor children. These West-East differences become smaller, but are persistent over time.

<p><span style="caret-color: #000000; color: #000000; font-family: -webkit-standard; font-size: medium; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">In the radiative Vlasov-Maxwell equations, the Lorentz force is modified by the addition of radiation reaction forces. The radiation forces produce damping of particle energy but the forces are no longer divergence-free in momentum space, which has an effect of concentration to zero momentum. We prove unconditional global regularity of solutions for a class of radiative Vlasov-Maxwell equations with large initial data. This is joint work with Peter Constantin.</span></p>

In this talk, I will discuss my previous work on constructing explicit models of logarithmic Hilbert schemes. This relates to work or Li-Wu on expanded degenerations, Gulbrandsen-Halle-Hulek on degenerations of Hilbert schemes of points and Maulik-Ranganathan on logarithmic Hilbert schemes. The constructions I consider are local. I will then explain how we globalise these in joint work with Shafi and apply them to construct minimal type III degenerations of hyperkähler varieties, namely Hilbert schemes of points on K3 surfaces.