Events

Following ideas of Symington, Evans calls a Lagrangian in the cotangent bundle of a torus visible if the moment map restricts to a surjective submersion onto a submanifold of the moment space. It follows from the Lagrangian condition that the only possible images are rational affine subspaces (Evans ’23). In this talk, we extend the notion of visibility to smooth Lagrangians in toric symplectic manifolds and ask: given an affine slice of the moment polyhedron, can we find visible Lagrangian(s) whose moment image is the prescribed slice? This is partly based on joint work with Joé Brendel and partly on work in progress.

The Weyl energy on four-manifolds is a quadratic curvature functional related to the Chern-Gauss-Bonnet formula. It is conformally invariant, and its critical points are called Bach-flat metrics and provide an interesting generalization of Einstein metrics or self-dual and anti-self-dual metrics. We will show that, under suitable conditions, it is possible to decrease the total Weyl energy under connected sum between two manifolds. These gluing results might be particularly useful for understanding the compactness properties of minimizing sequences of metrics. This is based on a joint project with Andrea Malchiodi.

<p>While for dimension 3 and lower it is well-known that any compact smooth manifold admits a unique smooth structure, in dimension 4 this is no longer true. Such homeomorphic but not diffeomorphic pairs of manifolds are called exotic, and in particular the smooth 4-dimensional Poincaré conjecture, stipulates there exist no exotic 4-spheres. In this talk I will give an overview of exotic phenomena, relying on the description of 4-manifolds given by handle decomposition, and mention standard techniques allowing to distinguish exotic pairs.</p>

<p><span data-olk-copy-source="MessageBody">I will introduce locally Hamiltonian flows on surfaces and discuss their representation as suspension flows over suitable Poincaré sections. I will review classical and recent results on the ergodic and mixing behavior of these flows, and present recent results on rates of mixing for a subclass of them.</span></p>

Moduli spaces often admit several natural compactifications. The comparison of different compactifications is referred to as wall-crossing. Some compactifications are simpler than others, making wall-crossing one of the most powerful established techniques, for example, in the study of moduli spaces of sheaves. I will discuss wall-crossing phenomena for moduli spaces of maps (Gromov–Witten theory) and other related spaces. The first glimpses of wall-crossing in Gromov–Witten theory go back to its beginnings, namely to Givental’s proof of mirror symmetry for the quintic in 1990s. Later, independently of Givental’s work, Manin and Bayer were the first to explore the idea of wall-crossing in the setting of maps, extending Hassett’s ideas on weighted marked curves. Only with the work of Ciocan-Fontanine, Kim, and Maulik from 2010s, which in turn builds on the work of many others, did it become possible to exploit different compactifications of moduli spaces of maps with considerable flexibility for GIT quotients, thereby revealing Givental’s proof as a wall-crossing calculation. Zhou then made the next major step by proving the Ciocan-Fontanine–Kim wall-crossing formulas for GIT quotients in full generality. The goal is to explain how the ideas of Ciocan-Fontanine, Kim, and Zhou can be applied in three different settings: configuration spaces of points, maps to GIT quotients, and Hurwitz theory. Based on these examples, I will try to formulate some general principles underlying such wall-crossing phenomena. The final goal is to combine all three settings to provide a wall-crossing between spaces of maps and Hilbert schemes associated to threefolds of the form Surface × Curve.

Riemannian manifolds with nonnegative Ricci curvature enjoy several interesting properties. For example, when complete, they have polynomial volume growth and, when compact, their universal covers split isometrically as products of compact manifolds and Euclidean spaces. In the discrete setting of graphs, several analogues of Ricci curvature have been proposed. This talk focuses on Ollivier's notion of Ricci curvature, introduced almost twenty years ago and based on optimal transport. The goal of this talk is to discuss the following rigidity result: if a finitely generated group has a Cayley graph with nonnegative Ollivier–Ricci curvature, then the group is virtually abelian and, in particular, has polynomial growth. This talk is based on joint works with Elia Bruè, Tom Hutchcroft, and Florentin Munch.

Discrete Exterior Calculus has become a standard tool for computations on meshes, yet it remains mostly limited to scalar-valued forms. Many geometric and physical systems, however, involve bundle-valued forms together with a connection, for which no widely adopted discrete framework exists.This talk will introduce a discrete analogue of the exterior covariant derivative that extends DEC to forms with values in a vector bundle. We will explore the geometric ideas behind the construction, the role of local frame fields in numerical evaluation, and how the resulting operator preserves key structural properties such as Bianchi identities while converging —in contrast to previous approaches— to its smooth counterpart under refinement.

q-Gaussian variables form a family of noncommutative random variables interpolating between free and classical Gaussian behavior. A natural question is whether they admit finite-dimensional random matrix models that converge not only in moments, but also strongly, meaning that the operator norms of all noncommutative polynomials converge. I will explain our construction of strongly convergent random matrix models for finite families of q-Gaussian variables. The construction proceeds in two stages. First, normalized sums of graph-product (Γ-independent) semicircular variables converge strongly to finite q-Gaussian families. Second, the corresponding graph-product semicircular systems are realized as strong limits of finite-dimensional tensor-GUE random matrices. Both steps admit matrix-coefficient versions, and the resulting random-matrix convergence is uniform for growing coefficient dimensions that may even exceed the dimension of the random matrices themselves. Our current arXiv version treats the range |q| < √2 − 1, and we have recently extended the result to the full range |q|<1. I will conclude with a brief discussion of recent connections between our construction and q-Gaussian limits arising from Kikuchi matrices.

Complex engineered systems can no longer be designed by optimizing components in isolation. In domains such as autonomy, mobility, and other society-critical infrastructures, system-level performance emerges from tightly coupled choices across hardware, software, control, operations, infrastructure, and policy. Moreover, these choices are distributed across abstraction levels and stakeholders, and the same subsystem is often both a system in its own right and a component of a larger one. This calls for a compositional approach to design: one that can represent interdependence, capture trade-offs across scales and disciplines, and remain computationally tractable. In this talk, I will present a framework for the compositional co-design of complex systems, drawing on optimization, control, and applied category theory. The central idea is to model components and their interconnections in a modular way, so that large design problems can be assembled, queried, and solved systematically from smaller ones. This perspective gives rise to a new class of compositional optimization problems, in which design trade-offs are expressed structurally and solved through algorithms that exploit interconnection and feedback, rather than collapsing everything into a monolithic formulation. I will illustrate the framework through examples spanning autonomous systems and future mobility, showing how it supports task-driven design across levels: from the choice of algorithms and components in autonomous vehicles to the co-design of services, operations, infrastructure, and policies in intermodal mobility systems. These examples highlight how compositional design can unify heterogeneous models and objectives, enable collaboration across disciplines, and provide quantitative tools to reason about the societal impact of emerging technologies. I will conclude by outlining ongoing efforts that extend this framework toward uncertainty-aware and adaptive design, including distributional formulations of co-design problems, online learning with embedded black-box components, and broader questions at the intersection of compositional optimization, strategic interaction, and multi-stakeholder system design.

Markov categories are a new mathematical framework designed to deal with problems in probability and information theory, using a higher abstraction layer. For an analogy, high-level programming languages (such as Python) allow the user to write code without worrying about lower-level issues such as memory allocation. Similarly, Markov categories allow the user to write rigorous mathematics, possibly involving continuous sample spaces and singularities, without having to know any measure theory. Moreover, they come equipped with a graphical formalism which faithfully reflects the information flow and the stochastic dependencies, and so they give a very general theory of graphical models. In this talk we will also see that several quantitative aspects of information theory can be captured by an enriched version of Markov categories, where the spaces of morphisms are equipped with a divergence or even a metric. For instance, we can define entropy exactly by measuring how far a source or channel is from being deterministic in the Markov category sense. This recovers the Shannon and Rényi entropies, as well as the Gini-Simpson coefficient used in ecology to quantify diversity in an ecosystem, and it can be used to give a conceptual definition of generalized entropy. Relevant paper: Paolo Perrone, "Markov Categories and Entropy", IEEE Transactions on Information Theory 70(3), 2024. (arXiv:2212.11719)

$C^0$ symplectic topology investigates the behaviour of symplectic objects under $C^0$ limits. In this talk, I will give a brief introduction to symplectic topology, and then present some ideas of a proof of the Eliashberg-Gromov rigidity theorem, which states that the set of symplectic diffeomorphisms is closed within the space of diffeomorphisms for the $C^0$ topology. This result led to a natural definition of symplectic homeomorphisms, and opened the way towards the study of $C^0$ symplectic topology. If time permits, I will talk about some other topics in $C^0$ symplectic topology, such as $C^0$ Lagrangian rigidity and/or the structure of the group of Hamiltonian homeomorphisms.

We study a principal-agent problem with adverse selection, where the principal does not know the agent's true cost but must design a contract to optimize a specific criterion. Unlike standard screening frameworks that allow for self-selection, we assume the principal can only offer a unique contract. We show that the agent's optimization problem can be reformulated as a stochastic target problem. After characterizing the viability domain of this target problem, we show that the principal's objective can be solved as a stochastic optimal control problem with partial information and state constraints. The description of the viability domain also allows us to obtain the value of screening contracts. This is a joint work with Ibrahim Ekren and Liwei Huang.

Einige geometrische Sätze, insbesondere der Satz des Pythagoras, werden unter dem Aspekt der invarianten Flächensumme untersucht. Diese neue Sichtweise ermöglicht ein ganzes Feld von Verallgemeinerungen und zugehörigen Illustrationen.

Modern electricity markets are organized around an optimization problem. Generators submit bids, and an independent system operator (ISO) dispatches production by solving a convex program over a transmission network with losses. Prices are its Lagrange multipliers. This talk explains how such markets work and presents a line of results obtained the last years. In discrete time, the generators’ bids make the market a Nash equilibrium problem with the ISO as a common follower. Quadratic transmission losses make the dispatch strictly convex. This yields structural market power, with an explicit lower bound on markups obtained from second-order growth and Lipschitz stability of the dispatch. With epi-convergence and the Simon–Zame theorem, we prove that equilibria exist when generators bid general cost functions. Adding emission caps or carbon costs, we prove the uniqueness and differentiability of the ISO’s primal–dual response and compare the two instruments numerically. In continuous time, the ISO becomes the leader. Pollution regulation under moral hazard reduces to a multidimensional BSDE and a smooth one-dimensional HJB equation, which gives an optimal contract. Investment in random renewable capacity leads to a state-constrained HJB equation on a non-smooth domain, characterized under a pointing-inward condition. Both models are illustrated on a three-node network calibrated to the Chilean grid. Joint work with N. Hernández-Santibáñez, D. Possamaï, L. Brotcorne, G. Guillot, B. Vera, and M. Vera.

I will review the method of Chen, Tropp, van Handel, and myself for proving strong convergence of random matrices. This, roughly speaking, can be thought of as a method for obtaining a sharp control on the norm of random matrices whose randomness possess certain symmetries. I will then state some results and open problems.

In a beautiful paper Duke, Imamoglu and Toth constructed for each (fundamental) closed geodesic on the modular curve a thin subgroup with boundary mapping to this geodesic. The construction relied on minus continued fractions. In this talk I will provide an alternative and more geometric construction and explain how it extends to more general cusped hyperbolic surfaces. This will show connections to more geometric topics such as Scott’s lifting theorem. This is joint work with Ser Peow Tan.

The Maulik-Nekrasov-Okounkov-Pandharipande correspondence predicts an equivalence between the partition functions of (numerical) Gromov-Witten and Donaldson-Thomas invariants of smooth projective threefolds. It was recently proposed by Pardon a solution of this conjectural correspondence by reducing to the simpler case of local curves, which are more amenable for computations by means of TQFT methods. Even more recently, inspired by the seminal work of Nekrasov-Okounkov on the index in M-theory, Brini-Schuler proposed a refined GW/DT correspondence. In this talk, I will present a full solution for the Donaldson-Thomas side of the refined GW/DT correspondence in the case of local curves. In particular, I will explain how to derive the refined DT partition function without relying on degeneration techniques, relative invariants and TQFT methods, and how our formulas recover string-theoretic prediction of Aganagic-Schaeffer.