• Past talks:

  • Saikat Majumdar (IIT B), Date : 7th April. Time 5--6PM.

Title: HIGHER ORDER YAMABE TYPE PROBLEMS

Abstract: Yamabe(1960) wanted to show that on a given compact Riemannian manifold of any dimension there always exists a (conformal) metric with constant scalar curvature. The solution of the Yamabe problem by Trudinger(1968), Aubin(1976) and Schoen(1984) highlighted the local and global nature of the problem and the unexpected role of the positive mass theorem of general relativity. I will survey the Yamabe problem. I will then consider the higher-order version of the Yamabe problem: “Given a compact Riemannian manifold, does there exists a conformal metric with constant Q-curvature”? The behaviour of Q-curvature under conformal changes of the metric is governed by certain conformally covariant powers of the Laplacian called the GJMS operator. I will present some of my results in this direction and mention some recent progress.

  • Vamsi Pingali (IISc Bangalore), Date: 17& 19th March. Time 5--6PM.

Title : An introduction to complex geometry and a conjecture of Calabi.

Abstract : I shall present a relatively gentle introduction to theory of complex manifolds, tell you why you ought to care, define Kahler metrics, and state a conjecture or two of Calabi. The pre-requisites needed are a working knowledge of smooth manifolds, linear algebra, and complex analysis.

  • Andrei Vesnin (Sobolev Institute of Mathematics, Novosibirsk), Date: 10th March, 5--6 PM.

Title: Hyperbolic polyhedra and hyperbolic knots: the right-angled case

Abstract: A polyhedron in a hyperbolic 3-space is said to be right-angled if all its dihedral angles are equal to pi/2. Three-dimensional hyperbolic manifolds constructed from right-angled polyhedra have many interesting properties [1]. Atkinson obtained low and upper bounds of volumes of right-angled polyhedral via vertex number [2]. We improve Atkinson’s upper bound in [3]. In [4] we describe an initial list of ideal (with all vertices at infinity) right-angled hyperbolic polyhedra. The obtained results imply that the right-angled knot conjecture from [5] holds for knots with small crossing number. Finally, we will discuss the relation of results from [4] with the maximum volume theorem from [6]. The talk is based on joint results with Andrey Egorov [3,4].

References.

[1] A. Vesnin, Right-angled polyhedra and hyperbolic 3-manifolds, Russian Mathematical Surveys 72 (2017), 335-374.

[2] C. Atkinson, Volume estimates for equiangular hyperbolic Coxeter polyhedra, Algebr. Geom. Topol. 9 (2009), 1225-1254.

[3] A. Egorov, A. Vesnin, Volume estimates for right-angled hyperbolic polyhedra, Rendiconti dell’Instituto di Matematica dell’Universita di Trieste 52 (2020), 565-576.

[4] A. Vesnin, A. Egorov, Ideal right-angled polyhedra in Lobachevsky space, Chebyshevskii Sbornik 21 (2020), 65-83.

[5] A. Champanerkar, I. Kofman, J. Purcell, Right-angled polyhedra and alternating links, arXiv:1910.13131.

[6] G. Belletti, The maximum volume of hyperbolic polyhedral, Trans. Amer. Math. Soc. 374 (2021), 1125-1153.

  • Kashyap Rajeevsarathy (IISER Bhopal) , Date: 3rd March. Time: 5--6 PM.

  • Title: Liftable mapping class groups of regular cyclic covers

  • Abstract: Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 1$. In this talk, we will consider the standard $k$-sheeted regular cover $p_k: S_{k(g-1)+1} \to S_g$ for $k \geq 2$, and analyze the subgroup $\mathrm{LMod}_{p_k}(S_g)$ of mapping classes that lift under the cover $p_k$. We will show that $\mathrm{LMod}_{p_k}(S_g)$ is the stabilizer subgroup of $\mathrm{Mod}(S_g)$ with respect to a collection of vectors in $H_1(S_g,\mathbb{Z}_k)$, and also derive a symplectic criterion for the liftability of a given mapping class under $p_k$. As an application of this criterion, we will obtain a normal series of $\mathrm{LMod}_{p_k}(S_g)$, which generalizes a well known normal series of the congruence subgroup $\Gamma_0(k)$ of $\mathrm{SL}(2,\mathbb{Z})$. Among other applications, we describe a procedure for obtaining a finite generating set for $\mathrm{LMod}_{p_k}(S_g)$ and examine the liftability of certain finite-order and pseudo-Anosov mapping classes.


  • Parameswaranan Sankaran (Chennai Mathematical Institute), Date: 24th February. Time: 5--6 PM.

Title : Geometric cycles in compact locally symmetric spaces.

Abstract : Suppose that G is a connected non compact linear semi simple Lie group and K a maximal compact subgroup of G. The space X:=G/K, which is diffeomorphic to Euclidean space, is a global symmetric space. If L is a torsion free uniform lattice in G, then the quotient of X by the action of L on the left is called a locally symmetric space. Topologically it is an Eilenberg-MacLane space K(L,1) and it is an important problem to understand its cohomology. This problem is related to understanding certain unitary representations which occur in the space of square-integrable functions on L\backslash G. We will explain the construction of geometric cycles due to Millson and Raghunathan which allows one to obtain non-trivial elements in the cohomology and discuss their implications to occurrence of certain unitary representations of G. If time permits, I will explain some recent developments.

  • William M. Goldman (University of Maryland), Date: 12th &17th February. Time: 5:30 -- 6:30 PM.

Title: The classification problem for geometric structures on manifolds

Abstract: In 1936 Charles Ehresmann proposed the question of which geometries in the sense of Klein exist locally on a fixed topology. For example, the nonexistnce of a metrically accurate world atlas suggests that the 2-sphere cannot support Euclidean geometry. The subject of such locally homogeneous geometric structures provided the context for Thurston's geometrization program for 3-manifolds (later proved by Perelman). In this talk I will present several successful classification problems.

In my second talk in the seminar (Wednesday 17 February) I will describe how this general class of classification problems leads to interesting dynamical systems.


  • Michael Kerber (TU Graz), Date: 10th February.

Title: The Persistent Homology Pipeline: Shapes, Computations, and Applications.

Abstract: The theory of persistent homology provides a multi-scale summary of homological features which is stable with respect to noise. These properties make homological algebra applicable to a growing range of application areas (in geometry and beyond) and give rise to the field of topological data analysis. This success of linking theory and applications has posed the challenge of computing persistence on large data sets. Typical questions in this context are: How can we efficiently build combinatorial cell complexes out of point cloud data? How can we compute the persistence summaries of very large cell complexes in a scalable way? Finally, how does the computed summary lead us to new insights into the considered application?

In my talk, I will introduce the field of topological data analysis in detail and discuss challenges, application areas and recent developments.


  • Ser Peow Tan (National University of Singapore), Date: 3rd February.

Title: Pseudomodular groups from hyperbolic jigsaws.


Abstract: we describe a hyperbolic jigsaw construction that produces non uniform lattices of PSL(2,R) and show how this can be used to construct infinitely many non-commensurable, pseudomodular groups ( non-arithmetic non uniform lattices with cusp set the rationals union infinity), thereby answering a question of Long and Reid.