


Algebraic Geometry seminar
Wednesday, 15 March 2023, 11.30 am
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Venue: Ramanujan Hall
Host: Saurav Bhaumik
Speaker: Pinakinath Saha
Affilaition: IIT Bombay
Title: On (weak) Fano $G$BottSamelsonDemazureHansen varieties
Abstract: Let $G$ be a semisimple 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 BottSamelsonDemazureHansen variety (BSDHvariety for short) corresponding to $w.$ BottSamelsonDemazureHansenvarieties 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 $G$BottSamelsonDemazureHansen variety ($G$BSDHvariety 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}.
Geometry and Topology Seminar
Wednesday, 15 March 2023, 2.30 pm
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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.
Mathematics Colloquium
15 March 2023, 4 pm
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Venue: Ramanujan Hall
Host: Mayukh Mukherjee
Speaker: Arunima Ray
Affiliation: MPI, Bonn
Title: Embedding surfaces in 4manifolds
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 4dimensional manifolds has led to breakthroughs such as Freedman's proof of the 4dimensional Poincare conjecture. Important open questions on 4manifolds 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 4manifold 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.