Bagels talks - Fall 2021
| Date | Speaker | Title |
| September 1st | No talk |
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| September 8th | You-Cheng | Intersection Theory
Abstract: Deformation to normal cone plays a prominent role in constructing intersection products. In this talk, I will introduce its construction and use it to define (refined) Gysin homomorphisms, which can be understood as the algebraic geometry version of cap product. As applications, I will talk about the ring structure on the Chow group of nonsingular variety and a naive idea for constructing virtual fundamental class of a given moduli space. |
| September 15th | Seungsu | Survey on Multiplier Ideals
Abstract: Multiplier ideals are the most fundamental object when it comes to measuring how severe a singularity is. In this talk, we will define the multiplier ideal of a divisor, see several properties of the ideal, and calculate it with various examples. Moreover, we will briefly talk about what happens when the varieties are over characteristic p field. |
| September 22nd | Peter | An Introduction to F-Singularities
Abstract: In this talk we’ll give a brief introduction to singularities defined via the Frobenius. In particular, we’ll discuss (globally) F-split and F-regular singularities, which are analogs of log terminal and log canonical singularities, respectively. |
| September 29th | Shih-Hsin | A Very Friendly Introduction to Jet Schemes
Abstract: For a long time after Nash innovated jet schemes and arc spaces, these objects have caught many researchers' interests and have been applied to handle some famous questions in birational geometry. In this talk, we'll give a brief introduction to both jet schemes and arc spaces, including some basic results, but will focus on jet schemes by computing some examples, the jet schemes of ADE singularities, to explain what motivates people to study this topic. If time is available, I would like to discuss my self-research on jet schemes. |
| October 6th | Jose | The volume function
Abstract: The volume function measures the asymptotic growth of sections of a big line bundle and plays a fundamental role in birational geometry. In this talk, I will present this function, show some properties about it, and introduce some questions related to it. |
| October 20th | Jack | From Representation theory to Algebraic geometry
Abstract: Geometric representation theory is a relatively modern approach to classifying representations of a reductive Lie group. The goal of such a field is to pass from the mainly analytic world of the classical theory to the geometric world by way of studying orbits of G on its Lie algebra and supports of modules arising from representations thereof. To make this precise we will cover the basics on the representation theory side and transform this to G-equivariant sheaves on the dual of the Lie algebra supported in the nilpotent cone. Time permitting we will discuss progress on a conjecture of Vogan which should lead to a geometric classification of representations of G. |
| October 27th | Marin | Constructing Moduli Spaces of Sheaves
Abstract: Classification of coherent sheaves on an algebraic variety is one of the central problems in algebraic geometry. This talk aims to explain the construction of the moduli spaces, following the work of Gieseker, Maruyama, and Simpson. We will also introduce the main tools used in the construction: Grothendieck's Quot scheme and Mumford's Geometric invariant theory. |
| November 3rd | No talk |
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| November 10th | Rahul | Eichler-Shimura Isomorphism and Deligne
Abstract: As is well-known, this correspondance between cusp forms and cohomology classes is the starting point of the derivation of the Ramanujan-Petersson conjecture from the Weil Conjectures. Deligne remarked in Bourbaki Seminar 1968/69 that the Shimura isomorphism could be written in the form of a Hodge decomposition: the cusp forms can be interpreted as the global sections of a sheaf $\omega^k\otimes\Omega^1$ on the compact Riemann Surface $\overline{S}=\Gamma\setminus \overline{\mathbb{H}}$. The aim of my talk is to first provide a sheaf theoretic interpretation of Automorphic forms and then outline a direct proof of the preceding isomorphism in the spirit of Hodge theory with degenerating coefficients. This was part of my Thesis at CMI made with extensive support from Infosys Excellence scholarship and KVPY-DST scholarship by the Department of Science and Technology, Govt. of India. |
| November 17th | No talk |
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| December 1st | Qingyuan | Elementary Examples in Birational Geometry
Abstract: In this talk, I will introduce some elementary (counter)examples in birational geometry, which illustrate: some peculiar behaviors of Mori cones, the sharpness of Matsusaka's big theorem, some peculiar behaviors of asymptotic linear systems and the volume function, and non-liftability of Calabi-Yau threefolds. |