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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Date/Time : 13 May 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Sheng Meng, East China Normal University

A critical point x of a differentiable self-map f of X=R^n or manifolds means the determinant of the Jacobian |J_f(x)|=0. We collect these points and call it the critical set C_f. This set C_f also carries algebraic data, denoted as R_f which is nothing but the zeros (counting multiplicities) of the equation |J_f(x)|=0, called the ramification.

In the algebraic geometric setting, R_f turns out to be an effective divisor, but not necessarily smooth anymore. In this talk, I shall show you the behaviour of the coefficients of R_f (multiplicity in codimension 1), and also the much more complicated, the singularities of R_f (multiplicity in higher codimension).
Some applications will be discussed if time permits.

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Date/Time : 23 Apr 2024 16:00
Venue : S8 #03-14
Speaker : Philipp Kunde, Jagiellonian University

One of the oldest and most influential problems in ergodic theory is the classification of invertible measure-preserving transformations (MPT’s) up to isomorphism or Kakutani equivalence. Starting from the late 1990s, so-called anti-classification results use concepts from descriptive set theory to show precisely that classification for ergodic MPT’s is impossible in terms of computable invariants, even with a very inclusive understanding of “computability”. In this talk, we present more details about our techniques to prove such anti-classification results.

In the search for classification or anti-classification results, we also place increasingly more stringent conditions on the family of transformations being considered for classification. Bernoulli shifts, which can be viewed as the most random type of transformations, have the strongest type of classification, a complete Borel invariant (namely measure-theoretic entropy). But what about classes of transformations that are more random than ergodic transformations, but less so than Bernoulli shifts, such as weakly mixing transformations, mixing transformations, and K-automorphisms?

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