Date
| Speaker
| Title/Abstract
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| January 19th | Organizational Meeting |
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| January 26th | Lingyao Xie | Title: Minimal Model Program for Generalized Pairs
Abstract: Generalized pairs were introduced by Birkar and Zhang in [BZ16] in their study of effective Iitaka fibrations, and later became a central topic in modern day birational geometry. For exmaple, the theory of generalized pairs is used to prove the Borisov-Alexeev-Borisov conjecture [Bir19,Bir21]. In this talk I will explain why we care about generalized pairs by showing that they naturally appear in the classification of algebraic varieties. Then I will discuss several recent fundamental results about generalized pairs. e.g Cone theorem ([Hancon-Liu21]), Contraction theorem ([Xie22]), Existence of flips ([Liu-Xie22]). If time permits, I will give the sketch of the proofs, and say something about Koll\'ar's gluing theory, which is a key ingredient in our proofs. As an interesting corollary, we show that glc singularities are Du Bois.
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| February 2nd | Qingyuan Xue | Title: Generalized cone and rationality theorem
Abstract: In this talk, I will introduce the generalized cone theorem (by Choi and Gongyo) which describes the cone of curves that are movable in codimension l. As for its proof, I will focus on a generalized rationality theorem, which is the key step.
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| February 23rd | Seungsu Lee | Title: Global F-regular varieties are log Fano.
Abstract: In this talk, we will discuss globally F-regular/F-split pairs and will show that global F-regular varieties are log Fano, which is proved by Schwede and Smith (https://arxiv.org/pdf/0905.0404.pdf). Furthermore, we will discuss an open question in this direction.
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| April 13th | Yu-Ting Huang | Title: Schubert Calculus on Grassmann Bundle
Abstract: Schubert calculus is a technique developed in the nineteenth century that plays an important role in enumerative geometry and algebraic combinatorics. Hilbert's 15th problem put Schubert's calculus on a rigorous foundation. In this talk, I will introduce the construction of Schubert calculus on Grasmmanian and its generalization on Grassmann bundles. Then we will see their application in enumerative geometry.
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| April 20th | Yi-Heng Tsai | Title: Multiplier Ideals and V-filtration
Abstract: Given an effective divisor on a smooth complex algebraic variety X, we have a decreasing filtration on O_X associated to the multiplier ideals of cD, where c is a rational number. In this talk, following the approaches in "multiplier ideals, V-filtration and spectrum", written by N. Budur and M. Saito, I will establish the equivalence between the filtration given by the multiplier ideal and the V-filtration on X. One direct application is that the equivalence gives another proof of a theorem L. Ein, R Lazarsfeld, K.E. Smith and D. Varolin that the jumping numbers of multiplier ideals in (0,1] are roots of the Bernstein-Sato polynomial up to sign. To achieve the goal, I will briefly review the definitions of multiplier ideals and D-modules, and then compute the V-filtration corresponding to a normal crossing divisor.
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