Date/Time : 04 Sep 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Jia Jia, Tsinghua University

We build bridges between moduli theory of sheaf stable pairs on one hand and birational geometry on the other hand. We will in particular treat moduli of sheaf stable pairs on smooth projective curves in detail and present some calculations in low degrees.

This is based on a joint work with Caucher Birkar and Artan Sheshmani.

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Date/Time : 04 Sep 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Fabrizio Bianchi, University of Pisa

Let f be a holomorphic endomorphism of P^k (C) of algebraic degree d. The Julia set J of f is the support of the unique measure mu of maximal entropy k log d. De Thélin and Dinh proved that every measure whose entropy is larger than (k-1)(log d) is also supported on J. Their proof crucially relies on the existence of the Green current. In particular, it follows from a delicate induction which makes use of the successive self-intersections T^j of the Green current T. It is then unclear how to generalize this result to more general non-algebraic settings, where a dynamical Green current does not exist.

I will present a new proof of this theorem which does not rely on the existence of a Green current, but instead on a quantified speed of convergence of preimages of points towards mu, which was established by Dinh and Sibony. In particular, the result can be extended to the more general setting of polynomial-like maps of large topological degree.

This is a joint work with Sardor Bazarbaev and Karim Rakhimov.

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Date/Time : 04 Sep 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Wilderich Tuschmann, Karlsruhe Institute of Technology

Consider a smooth manifold with a Riemannian metric satisfying some sort of geometric constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasakian, of special holonomy, etc. A natural question to ponder is then what the space of all such metrics does look like. Moreover, one can also study this question for the corresponding moduli spaces of metrics,i.e., quotients of the former by the diffeomorphism group of the manifold, acting by pulling back metrics. These spaces are customarily equipped with the topology of smooth convergence on compact subsets and the quotient topology, respectively, and their topological properties then provide the right means to measure ‘how many’ different metrics and geometries the given manifold actually does exhibit.

The history of the subject as a whole indeed goes back more than a century, and since H. Weyl’s early result on the connectedness of the space of positive Gaussian curvature metrics on the two-sphere and the foundings of Teichmüller, infinite- dimensional manifold and Lie group theory, uniformization and geometrization, the study of spaces of metrics and their corresponding moduli has been a topic of interest for differential geometers, global and geometric analysts and topologists alike.

In my talk, I will provide a gentle introduction to the subject with a focus on lower curvature bounds and present recent results and open questions about the global topological properties of moduli spaces of nonnegatively curved Riemannian metrics on manifolds, and as well, if time permits, also corresponding facts and questions for the moduli of nonnegative curvature metrics on RCD spaces.

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Date/Time : 29 Aug 2024 16:00
Venue : S17 #04-06 (SR1)
Speaker : Minsung Kim, Centro di Ricerca Matematica Ennio De Giorgi

The deviation of the Birkhoff integral for area-preserving flow on compact surfaces was first studied by Forni. He proved that the deviation spectrum was determined by the Lyapunov exponents of a renormalization cocycle so-called Kontsevitch-Zorich cocycle. This deviation result was later proved again by Bufetov and Frączek-Ulcigrai for translation flows and locally Hamiltonian flows for non-degenerate types. In this talk, we study the spectrum for deviations of Birkhoff integrals for locally Hamiltonian flows beyond the case of Forni where the observable vanishes at the singularities. Our new developments include a better understanding of the asymptotics at singularities (degenerate type) and the appearance of a new deviation spectrum. This is a joint work with Krzysztof Frączek.

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Date/Time : 14 Aug 2024 16:15
Venue : S17 #04-05 (SR2)
Speaker : Trung Tuyen Truong, University of Oslo

This talk presents a new equivalence and approach to the Riemann hypothesis, via a relatively new root finding method called Backtracking New Q-Newton’s method (BNQN), which is a variant of Newton’s method. Besides having strong global guarantee for finding roots of meromorphic functions, this new method also has connections to Voronoi’s diagrams, which is an important input for the new approach. This is joint work with Thuan Quang Tran (University of Oslo).  

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Date/Time : 08 Aug 2024 15:00
Venue : S17 #05-12 (SR4)
Speaker : Tirasan Khandhawit, Mahidol University

Seiberg-Witten invariants and monopole Floer homology have been important tools in the study of 3-manifolds and 4-manifolds. In this talk, I will give an overview of the stable homotopy refinement of these invariants. In particular, I will describe Bauer-Furuta invariants of 4-manifolds and Seiberg-Witten-Floer spectra of 3-manifolds. I will also discuss some applications of these refinements.

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Date/Time : 08 Aug 2024 13:45
Venue : S17 #06-11 (SR6)
Speaker : Binbin Xu, Nankai University

One generalization of flat structure on torus to oriented closed surface of genus greater than 1 is called translation surface structure. When a surface is equipped with such a structure, we call it a translation surface. Given a translation surface S, like on a flat torus, there is a globally well-defined notion of direction on S, and geodesics on S along a same direction are disjoint from each other, hence induces a flow on S. Such flows and their dynamical properties are important in the study of the symmetry group of S which is called Veech group. In particular, when the flow on S along one direction has only periodic regular orbits, we call this direction a periodic direction of S. In a joint-work with Anja Randecker, using the close relation between interval exchange transformations and flows on surfaces, we obtain a full classification of translation surfaces of genus 2 with one singularity according to the size of their periodic directions sets. In this talk, I will first give some necessary background, then I will explain our result and give the rough idea of the proof.

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Date/Time : 07 Aug 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Yongnam Lee, KAIST, Korea

In this talk, I will discuss the positivity of the tangent bundle of smooth projective surfaces. The first part of the talk will be focused on the classification of non-ruled surfaces with pseudo-effective tangent bundles, which is a joint work with Jia Jia and Goulei Zhong. The second part will be focused on weak del Pezzo surfaces with big tangent bundles. The bigness of the tangent bundle is expected to be determined through the configuration of (−2)-curves. Our approach is to produce irreducible effective divisors on the projectivized tangent bundle, called the total dual VMRT, associated to a conic fibration on weak del Pezzo surfaces.

This part is a joint work with Hosung Kim and Jeong-Seop Kim.

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Date/Time : 24 Jul 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Chen Qile, Boston College

Gauged Linear Sigma Models introduced by physicists in the 90’s is a way of counting curves in critical loci of a holomorphic function called superpotentials. In this talk, I will report on a joint project in progress with Felix Janda and Yongbin Ruan on computing Gauged Linear Sigma Models via logarithmic compactifications. Our method will lead to quantum lefschetz formulas in all genera, relating Gromov-Witten invariants of complete intersections with enumerative geometry of their ambient spaces. Furthermore the correction terms in these formulas are virtual counts of “generalized special linear series”. In case of Fano complete intersections, we show that the vanishing of these correction terms are controlled by invariants from birational geometry.

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