Veranstaltungen

Kim, Kresch and Oh constructed compactified moduli spaces of unramified maps from nodal curves to an arbitrary smooth projective target. The associated invariants were called unramified Gromov-Witten invariants. In dimension 1, these spaces are familiar moduli spaces of admissible covers introduced by Harris and Mumford. In dimension 3,Pandharipande conjectured that unramified Gromov-Witten invariants are equal to Gopakumar-Vafa invariants (BPS invariants) for Fano and primitive Calabi-Yau classes. After an introduction to unramified Gromov-Witten theory, I will present a work in progress which aims to prove this conjecture. The approach is based on a certain wall-crossing technique, whose wall-crossing invariants are given by Hodge integrals on moduli spaces of stable marked curves.

We introduce new invariants to the existence of Lagrangian cobordisms in R^4. These are obtained by studying holomorphic disks with corners on Lagrangian tangles, which are Lagrangian cobordisms with flat, immersed boundaries. We develop appropriate sign conventions and results to characterize boundary points of 1-dimensional moduli spaces with boundaries on Lagrangian tangles.  We use these to define algebraic structures on a related "linear" case  of Lagrangian cobordisms in surface times complex plane. This talk is based on my thesis work under the supervision of Y. Eliashberg and on work in progress joint with J. Sabloff.

Spectral invariants are quantitative measurements in symplectic topology coming from Floer homology theory. We study their dependence on the choice of coefficients in the context of Hamiltonian Floer homology. We discover phenomena in this setting which hold for $\Z$-coefficients and fail for all field coefficients. We apply this to answer a symplectic version of a question of Hingston.

The friendly partition problem involves determining whether a given graph allows for a partition of its nodes into two nonempty sets, where each node has at least as many neighbors in its own set as in the other. Notably, not all graphs permit such a friendly partition, and even fewer accommodate a partition where a node requires an additional margin of neighbors in its own set compared to the other. We investigate the existence of such partitions and the algorithmic feasibility of finding them. A natural question is: how does a graph evolve when nodes directly adapt their states to meet these local constraints? When this adaptation occurs synchronously, it models scenarios like majority voting or cellular automata. However, it is not a given that a graph, where each node greedily iteratively applies the local rule, converges to a global solution. Our analysis examines the different types of attractors that emerge in locally constrained problems and the role of initialisation in shaping the outcome. Our tool to answer these questions is the cavity method and the backtracking dynamical cavity method from statistical physics for synchronous update processes on regular graphs. They provide the sharp transitions on the existence of solutions, as well as the dynamical phase transitions of local processes in the large system limit.

I will talk about a recent preprint in which we establish an optimal local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations on a connected fluid domain. Some components of this result include: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: A uniqueness result which holds at the level of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove essentially scale invariant energy estimates for solutions, relying on a newly constructed family of refined elliptic estimates; (v) Continuation criterion: We give the first proof of a continuation criterion at the same scale as the classical Beale-Kato-Majda criterion for the Euler equations on the whole space. Roughly speaking, we show that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of the construction of regular solutions. \\ Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is based on joint work with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.

Kim, Kresch and Oh constructed compactified moduli spaces of unramified maps from nodal curves to an arbitrary smooth projective target. The associated invariants were called unramified Gromov-Witten invariants. In dimension 1, these spaces are familiar moduli spaces of admissible covers introduced by Harris and Mumford. In dimension 3, Pandharipande conjectured that unramified Gromov Witten invariants are equal to Gopakumar-Vafa invariants (BPS invariants) for Fano and primitive Calabi-Yau classes. After an introduction to unramified Gromov-Witten theory, I will present a work in progress which aims to prove this conjecture. The approach is based on a certain wall-crossing technique, whose wall-crossing invariants are given by Hodge integrals on moduli spaces of stable marked curves.

Waves of all kinds permeate our world. We are surrounded by light (electromagnetic waves), sound (acoustic waves), and mechanical vibrations. Quantum mechanics revealed that, at the atomic level, all matter has a wavelike character. And classical gravitational waves have been very recently detected. At the cutting edge of today’s science, it has become possible to manipulate individual atoms. This provides us with precise measurements of a world that exhibits myriad irregularities — dimensional, structural, orientational, and geometric— simultaneously. For waves, such disorder changes everything. In complex, irregular, or random media, waves frequently exhibit astonishing and mysterious behavior known as ‘localization’. Instead of propagating over extended regions, they remain confined in small portions of the original domain. The Nobel Prize–winning discovery of the Anderson localization in 1958 is only one famous case of this phenomenon. Yet, 60 years later, despite considerable advances in the subject, we still notoriously lack tools to fully understand localization of waves and its consequences. We will discuss modern understanding of the subject, recent results, and the biggest open questions.

In this talk, we'll explain how Milnor ingeniously (and surprisingly) used modular forms to give an answer to a purely geometrical question, namely ``Is there a pair of isospectral but non-isometric Riemannian manifolds?''. His proof leverages the finite-dimensionality of spaces of modular forms and the connexion between spectra of flat tori and Jacobi theta functions.

In this talk, we explore approximation properties and statistical aspects of Neural Ordinary Differential Equations (Neural ODEs). Neural ODEs are a recently established technique in computational statistics and machine learning, that can be used to characterize complex distributions. Specifically, given a fixed set of independent and identically distributed samples from a target distribution, the goal is either to estimate the target density or to generate new samples. We first investigate the regularity properties of the velocity fields used to push forward a reference distribution to the target. This analysis allows us to deduce approximation rates achievable through neural network representations. We then derive a concentration inequality for the maximum likelihood estimator of general ODE-parametrized transport maps. By merging these findings, we are able to determine convergence rates in terms of both the network size and the number of required samples from the target distribution. Our discussion will particularly focus on target distributions within the class of positive $C^k$ densities on the $d$-dimensional unit cube $[0,1]^d$.

Reinforcement learning (RL) algorithms aim to maximise the accumulated reward for a suitably chosen reward function. However, designing such a reward function often requires task-specific prior knowledge which may be not available in closed quantitative form. To alleviate these issues, preference-based reinforcement learning algorithms have been proposed that can directly learn from an expert’s preferences instead of a hand-designed numeric reward. In this talk I give an overview of preference-based reinforcement learning and illustrate its main principles on examples from mathematical finance. In particular, I discuss what type of human feedback can be assumed and how preferences can be build in the optimization problem via penalisation.

The F-Conjecture is an old problem about the Mori Cone of the moduli space of curves. In this talk I will describe the conjecture, its origins, what is known, and what we would know if it were to be true. I’ll also describe some evidence that vertex operator algebras may play a role in refining our understanding of the problem.