Veranstaltungen

A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, an operad, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure. Kaledin classes were introduced as an obstruction theory fully characterizing the formality of associative algebras over a characteristic zero field. In this talk, I will present a generalization of Kaledin classes to any coefficients ring, to other algebraic structures (encoded by operads, possibly colored, or by properads), and to address a more general problem: the existence of homotopy equivalences between algebraic structures. I will prove new formality criteria based on this obstruction theory, presenting applications in several domains such as algebraic geometry, representation theory and mathematical physics.

A contact homeomorphism is defined as a C⁰-limit of a sequence of contact diffeomorphisms. In this talk, we will discuss C⁰-flexibility and C⁰-rigidity properties of various classes of submanifolds in the contact setting. Our primary focus will be on Legendrian submanifolds, which exhibit C⁰-rigidity in the closed case and C⁰-flexibility in the open case. More precisely, we will focus on the following question: if a contact homeomorphism maps a Legendrian submanifold to a smooth submanifold, must the image necessarily be Legendrian?

For a smooth expanding map of the circle, its (unmarked) length spectrum is defined as the set of logarithms of multipliers along all periodic orbits. This set is analogous to the set of lengths of all closed geodesics on negatively curved surfaces -- the classical length spectrum. In the talk, I will present a length spectral rigidity result for expanding circle maps. Namely, I will show that a smooth expanding circle map, under certain assumptions on the sparsity of its length spectrum, cannot be perturbed with an arbitrarily small smooth perturbation (depending on the map) so that the length spectrum stays the same. The proof uses the Whitney extension theorem, a quantitative Livcis-type theorem, and a novel iterative scheme. This is joint work with Vadim Kaloshin.

Irreducible symplectic varieties are one of three building blocks of varieties with Kodaira dimension zero, which are higher-dimensional analogs of K3 surfaces. Despite their rich geometry, there have been only a limited number of approaches to construct irreducible symplectic varieties. In this talk, I will introduce a general criterion for the existence of irreducible symplectic compactifications of non-compact Lagrangian fibrations, based on the minimal model program and the geometry of Lagrangian tori. As an application, I will explain how to get a 42-dimensional irreducible symplectic variety with the second Betti number at least 24. This is a joint work with Yuchen Liu and Chenyang Xu.

After a general introduction to applications of metric graph theory in geometric group theory, focused on quasi-median graphs, I will explain how one can deduce from such a perspective new quasi-isometric invariants for right-angled Artin groups. This is joint work with Carolyn Abbott and Eduardo Martinez-Pedrosa.

In many physical experiments, detectors are only able to measure the intensity of a signal and hence lose access to the phase. The phase retrieval problem deals with the reconstruction of a signal from magnitude measurements, assuming certain physical constraints are satisfied. In this talk, I will briefly discuss the history of this problem and overview some recent progress on proving rigorous stability bounds in infinite dimensions.

I will describe recent work joint with Olga Balkanova and Dmitry Frolenkov on a restricted divisor function and its associated divisor problem. This problem shares properties with the usual Dirichlet divisor problem and the Hardy-Littlewood problem to count lattice points in a right triangle. The latter depends deeply on the arithmentic nature of the slope of the triangle the restricted divisor problem has a similar property. For the H-L problem, Hecke showed how to go much further when the slope comes from a real quadratic field. We apply Hecke’s idea to our problem, which introduces a number of interesting new difficulties,

In the last 10 years colossal cloud infrastructure investments behind the rise of near-ubiquitous global mobile technologies have trickled down to scientific research through innovative infrastructure including cloud compute and storage, I/O tools, data analysis and modeling frameworks, which in turn have generated broad and expanding communities of users and supporters. Arguably, the recent success of Large Language Models were catalyzed by the resulting technological innovations of 1) open and accessible massive data, and 2) re-executable discovery pipelines for model estimation and prediction. These changes are deeply disruptive to the research community since they open new paths to knowledge creation that were previously inaccessible and largely culturally unknown. The scientific community is faced with the challenge of responding to changes in research modalities due to these technological innovations. Research is now conducted as an “Olympics” of benchmarked competitions between Machine Learning models leveraged by the opaque results of Large Language Models, access to massive data, and redeployment of complex scientific discovery workflows. In this seminar I provide a roadmap of challenges and responses by various stakeholders in the research community to ensure that scientific results remain reliable and reproducible, and secure within a position of trust in the broader society.

I will present a formula of the S_n-equivariant Euler characteristics of genus one stable maps to the projective space. It enriches the ordinary Euler characteristics, which are previously unkown, and continues the results of Getzler - Pandharipande on genus zero stable maps. The approach connects torus localization, composition structure on the Grothendieck ring of varieties, and graph enumeration techniques via wreath symmetric functions. Joint work with Siddarth Kannan.