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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Date/Time : 17 Apr 2024 15:30
Venue : S17 #05-12 (SR4)
Speaker : Shilpak Banerjee, Indian Institute of Technology Tirupati

A fundamental theme in dynamics is the classification of systems up to appropriate equivalence relations. For instance, the equivalence relation of topological conjugacy preserves the qualitative behavior of topological dynamical systems. Smale’s celebrated program proposes to classify topological or smooth dynamical systems up to topological conjugacy. In Ergodic Theory the isomorphism problem dates back to von Neumann’s foundational paper and asks to classify measure-preserving transformations up to measure isomorphism.

These classification problems not only turn out to be hard but sometimes even to be impossible. In this talk, we give an overview of classification as well as anti-classification results and present some related projects.

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Date/Time : 08 Apr 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Chris Leininger, Rice University

A billiard trajectory in a polygon in the Euclidean or hyperbolic plane is the path of a particle in the polygon, following a geodesic line until it hits the side, reflects off, and continues. This trajectory gives rise to a symbolic coding, its “bounce sequence”, which is (generically) a biinfinite sequence recording the sides encountered (which we assume are labeled). In joint work with Erlandsson, Duchin, and Sadanand, we prove that generically the “bounce spectrum” (i.e. the set of all bounce sequences of biinfinite trajectories), determines the shape of the polygon. The “non-generic” case is also understood, and we can completely determine when two polygons have the same bounce spectrum. In the talk, I’ll explain this theorem and describe how it is related to non-positively curved cone metrics on surfaces.

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Date/Time : 03 Apr 2024 15:30
Venue : S17 #05-12 (SR4)
Speaker : DO Thai Duong, National University of Singapore

We will present some properties of the higher complex Sobolev spaces. These spaces share several similar properties with the complex Sobolev space; in particular, we can build their corresponding Vigny’s functional capacity and prove the Moser-Trudinger inequality for these spaces. We also discuss the relation of these spaces with the complex Monge-Ampere equation. This talk is based on my joint work with Nguyen Duc-Bao.

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Date/Time : 27 Mar 2024 14:30
Venue : S17 #05-12 (SR4)
Speaker : Wu Hao, National University of Singapore

Let X be a compact Riemann surface and L be a positive line bundle on it. We study the conditional zero expectation of all the holomorphic sections of L^n which do not vanish on D for some fixed open subset D of X. We prove that as n tends to infinity, the zeros of these sections are equidistributed outside D with respect to some probability measure. This gives rise to a surprising forbidden set.

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