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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Date/Time : 16 Jul 2024 15:30
Venue : LT33 (S17 Level 2)
Speaker : Nishant Chandgotia, Tata Institute of Fundamental Research - Centre for Applicable Mathematics

When can a given subset S be tiled by rectangular tiles in Z^2? This is an old question and for answering it Conway and Lagarias introduced certain groups which are now known as the Conway-Lagarias group (1990). Together with some analytic conditions, Thurston noticed that these questions can be answered completely in certain situations. This later formed the basis for understanding what uniformly random tiling of regions looked like (for instance in the results from Cohn, Kenyon and Propp from 2001). In 1995, Klaus Schmidt realised that this was related to a certain cohomology of dynamical systems and initiated its formal study. We will talk about such questions in the context of graph homomorphisms on the Z^d lattice and see how it relates to the word problem of finitely presented groups.

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Date/Time : 01 Jul 2024 16:00
Venue : S17 #04-04 (SR3)
Speaker : Baohua Fu, The Institute of Mathematics, CAS, Beijing

A smooth projective variety X is called rigid if any deformation of X is isomorphic to itself. The first example is the projective space, but in general, it is a subtle and difficult problem to prove the deformation rigidity. I’ll report some recent progress on this problem. Part of this talk is based on joint works with Yifei Chen(Chinese Academy of Sciences) and Qifeng Li (Shandong Univ).

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Date/Time : 14 Jun 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Greg McShane, Institut Fourier

The modular surface $\mathbb{H}/\Gamma,\, \Gamma= \math rm{SL}(2,\mathbb{Z})$ has many covers -for example the three punctured torus $\mathbb{H}/\Gamma(2)$ and the once punctured torus $\mathbb{H}/\Gamma’$.

We will discuss how classical Diophantine approximation can be interpreted in terms of the behavior of geodesics on the once-punctured torus and a geometric reformulation of the Frobenius uniqueness conjecture. We will then give an account of two theorems of Fermat in terms of the automorphisms of $\mathbb{H}/\Gamma(2)$:
– if $p$ is a prime such that $4|(p-1)$ then can be written as a sum of squares $p = c^2 + d^2$
– if $p$ is a prime such that $3|(p-1)$ then can be written as $ p = c^2 +cd + d^2$

Finally, we will discuss possible extensions to surfaces of the form $\mathbb{H}/\Gamma_0(N)$.

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