webcal

Wed, March 15, 2023
Public Access

Category: All

8:00am
9:00am
10:00am
11:00am	[11:30am] Pinakinath Saha, IIT Bombay Description: Algebraic Geometry seminar Wednesday, 15 March 2023, 11.30 am ============================= Venue: Ramanujan Hall Host: Saurav Bhaumik Speaker: Pinakinath Saha Affilaition: IIT Bombay Title: On (weak) Fano $G$-Bott-Samelson-Demazure-Hansen varieties Abstract: Let $G$ be a semi-simple simply connected algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $W$ be the Weyl group of $G$ with respect to $T$. For an arbitrary sequence $w$ of simple reflections in $W,$ let $Z_{w}$ be the Bott-Samelson-Demazure-Hansen variety (BSDH-variety for short) corresponding to $w.$ Bott-Samelson-Demazure-Hansen-varieties are an important tool in geometric representation theory. They were originally defined as desingularizations of Schubert varieties and were used to describe the geometry of Schubert varieties. There is a natural action of $B$ on $Z_{w}$ given by the left multiplication. Let $\widetilde{Z_{w}}:=G\times^{B}Z_{w}$ be the fibre bundle associated to the principal $B$-bundle $G\to G/B.$ We call it $G$-Bott-Samelson-Demazure-Hansen variety ($G$-BSDH-variety for short). In the first part of the talk, we will describe a basis of the Picard group of $G$-BSDH variety, which we will refer as the $\mathcal{O}(1)$-basis. Then we will characterize the nef, globally generated, ample and very ample line bundles on $G$-BSDH variety in terms of the $\mathcal{O}(1)$-basis. Finally, we will provide a characterization of (weak) Fano $G$-BSDH varieties. We introduce a few more notations for the second part of my talk. Let $G=SO(8n,\mathbb{C})\big/SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the end simple root of its Dynkin diagram. In the second part of the talk, we will discuss the projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle. The first part of the talk will be based on joint work with Saurav Bhaumik. Here is the link of the preprint: \url{https://arxiv.org/abs/2212.10366}. The second part of the talk will be based on joint work with Arpita Nayek. Here are the links of the preprints: \url{https://arxiv.org/abs/2207.01477} and \url{https://arxiv.org/abs/2302.00555}.
12:00pm
1:00pm
2:00pm	[2:30pm] Arghya Mondal, CMI, Chennai Description: Geometry and Topology Seminar Wednesday, 15 March 2023, 2.30 pm ============================= Venue: Ramanujan Hall Host: Sandip Singh Speaker: Arghya Mondal Affiliation: CMI, Chennai Title: A higher dimensional generalization of Margulis' construction of expander graphs Abstract: Expanders are a family of finite graphs whose vertex sizes go to infinity but edge sizes grow at most linearly in vertex sizes, while still remaining highly connected. The first explicit construction of such graphs was by Margulis using discrete groups having Property (T), a rigidity property defined in terms of unitary representations. In recent years various higher dimensional generalizations of expanders, replacing graphs by simplicial complexes of a fixed dimension, have been considered. We will discuss a group theoretic construction of one such generalization, which is an extension of Margulis' construction to higher dimensions.
3:00pm
4:00pm	[4:00pm] Arunima Ray, MPI, Bonn Description: Mathematics Colloquium 15 March 2023, 4 pm ==================== Venue: Ramanujan Hall Host: Mayukh Mukherjee Speaker: Arunima Ray Affiliation: MPI, Bonn Title: Embedding surfaces in 4-manifolds Abstract: Manifolds are fundamental objects in topology since they locally model Euclidean space. Within a given ambient manifold, we are often interested in finding embedded submanifolds, which would then enable cutting and pasting operations, such as surgery. The study of surfaces in 4-dimensional manifolds has led to breakthroughs such as Freedman's proof of the 4-dimensional Poincare conjecture. Important open questions on 4-manifolds can also be reduced to the question of finding certain embedded surfaces. I will consider the following question: When is a given map of a surface to a 4-manifold homotopic to an embedding? I will give a survey of related results, including the celebrated work of Freedman and Quinn, and culminating in a general surface embedding theorem, arising in joint work with Daniel Kasprowski, Mark Powell, and Peter Teichner.
5:00pm
6:00pm

Time:

11:30am

Location:

Ramanujan Hall, Department of Mathematics

Description:

Algebraic Geometry seminar

Wednesday, 15 March 2023, 11.30 am

=============================

Venue: Ramanujan Hall

Host: Saurav Bhaumik

Speaker: Pinakinath Saha

Affilaition: IIT Bombay

Title: On (weak) Fano $G$-Bott-Samelson-Demazure-Hansen varieties

Abstract: Let $G$ be a semi-simple simply connected algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $W$ be the Weyl group of $G$ with respect to $T$. For an arbitrary sequence $w$ of simple reflections in $W,$ let $Z_{w}$ be the Bott-Samelson-Demazure-Hansen variety (BSDH-variety for short) corresponding to $w.$ Bott-Samelson-Demazure-Hansen-varieties are an important tool in geometric representation theory. They were originally defined as desingularizations of Schubert varieties and were used to describe the geometry of Schubert varieties. There is a natural action of $B$ on $Z_{w}$ given by the left multiplication. Let $\widetilde{Z_{w}}:=G\times^{B}Z_{w}$ be the fibre bundle associated to the principal $B$-bundle $G\to G/B.$ We call it $G$-Bott-Samelson-Demazure-Hansen variety ($G$-BSDH-variety for short). In the first part of the talk, we will describe a basis of the Picard group of $G$-BSDH variety, which we will refer as the $\mathcal{O}(1)$-basis. Then we will characterize the nef, globally generated, ample and very ample line bundles on $G$-BSDH variety in terms of the $\mathcal{O}(1)$-basis. Finally, we will provide a characterization of (weak) Fano $G$-BSDH varieties. We introduce a few more notations for the second part of my talk. Let $G=SO(8n,\mathbb{C})\big/SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the end simple root of its Dynkin diagram. In the second part of the talk, we will discuss the projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle. The first part of the talk will be based on joint work with Saurav Bhaumik. Here is the link of the preprint: \url{https://arxiv.org/abs/2212.10366}. The second part of the talk will be based on joint work with Arpita Nayek. Here are the links of the preprints: \url{https://arxiv.org/abs/2207.01477} and \url{https://arxiv.org/abs/2302.00555}.