Date
| Speaker
| Title/Abstract
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| August 29th | Organizational Meeting |
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| September 5th | Labor Day (No talk) |
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| September 12th | Qingyuan Xue | Title: Moduli of Stable Surfaces in Arbitrary Characteristic.
Abstract: The moduli theory in positive and mixed characteristic is still
mysterious to people, especially after Kollar showed the failure of the
properness of the moduli of stable threefolds. However, for surfaces the
situation is much better. With the help of the recent development of MMP
in mixed characteristic ([BMP+20]), people now have a quite satisfactory
answer to the moduli of stable surfaces over Z. In this talk, I will
introduce the moduli problem and present known results, and focus on the
proof of properness if time permits.
|
| September 19th | Seungsu Lee | Title: On the behavior of F-signature on the Nef cone
Abstract: F-signature plays a crucial role when measuring singularities of varieties in positive characteristics. For example, if R is a local ring, s(R) = 1 implies R is regular, and 0 < s(R) < 1 implies R is strongly F-regular which is a char p analog of klt singularities. For a globally F-regular variety X, the F-signature of an ample invertible sheaf is defined as the F-signature of the section along the invertible sheaf over X. In this talk, we will discuss the F-signature is well-defined and is (locally Lipschitz) continuous on the ample cone, and how the signature extends to the boundary of the cone.
|
| September 26th | Jose Yanez | Title: Computing birational automorphism groups and movable cones on Calabi-Yau varieties
Abstract: A key tool to understanding the birational geometry of Calabi-Yau varieties is to have a description of their birational automorphism group and the movable cone. In this talk, I'll show how to compute these for a family of examples and suggest some further open questions.
|
| October 3rd | Shih-Hsin Wang | Title: Some Smoothness on Jet Schemes
Abstract: After Nash's observation in 1995, people started to investigate what information on the jet schemes and the arc space of an algebraic variety involve an understanding of its singularities.
In this talk, I would like to take an alternative point of view and focus
on the smoothness of jet schemes. Beginning with a basic result about the
jet scheme of an smooth variety, we will work through a description of the
sheaves of Kähler differentials of the jet schemes studied by Tommaso de
Fernex and Roi Docampo. Then we will apply this result to prove the
smoothness of fibers over any liftable jet.
|
| October 10th | Fall Break (No talk) | Title:
Abstract:
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| October 17th | Yi-Heng Tsai | Title: Kashiwara's equivalence
Abstract: The theory of D-modules gives an algebraic point of view of the theory of linear partial differential equations. Recently, the theory of D-modules has found applications in many branches of mathematics (for example, mixed Hodge modules). Kashiwara' equivalence is one of the important results in the theory of D-modules that establishes an equivalence of the triangulated categories $D^b_{qc} (D_X) and D^{b,X}_{qc} (D_Y)$ for any closed embedding from $X$ to $Y$. In this talk, I will briefly recall some basic definitions about D-modules, and then give a proof of Kashiwara's equivalence. As applications, I will talk about the base change theorem for D-modules.
|
| October 24th | Mahrud Sayrafi (Univ. of Minnesota) | Title: Computing in the Derived Category of a Toric Variety
Abstract: A classical question in algebraic geometry is the study of vector bundles on algebraic varieties. In 1956, Grothendieck proved that any vector bundle P^1 splits as a direct sum of line bundles. In contrast, for n>3 there are indecomposable vector bundles of rank n-1 on P^n. This is an introductory talk about the same question on a toric variety. I will talk about making the bounded derived category of coherent sheaves explicit using commutative algebra, and will explain how to use this machinery to check whether a vector bundle on certain toric varieties splits as a direct sum of line bundles.
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| October 31th | No Talk | Title:
Abstract:
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| November 7th | No Talk | Title:
Abstract:
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| November 14th | Rahul Ajit | Title: Hodge theory and Hard Lefschetz Theorem.
Abstract: I'll present a quick introduction Hodge decomposition for Kahler manifolds. Then, if time permits, I want to ramble about Symplectic Hodge theory and some applications. I wish to make this talk as elementary as possible!
|
| November 21st | Yu-Ting Huang | Title: An Introduction to Volume Functions of Line Bundles
Abstract: The volume function is an important invariant that characterizes the asymptotic behavior of H^0(X,mL) as m\to\infty. It has been used in lots of areas of pure math.
In the first part of this talk, I will define the volume function and go through its essential properties. For the second part, I will introduce Cutkosky’s construction to show that the volume can be irrational. Lastly, I will use Fujita's approximation theorem to prove that the limsup in the definition of volume is actually lim.
|
| November 28th |
Yotam Svoray
| Title: Invariants of non-isolated hypersurface singularities.
Abstract: A key tool in understanding (complex analytic) hypersurface singularities is to study what properties are preserved under special deformations. For example, the relationship between the Milnor number of an isolated singularity and the number of A_1 points. In this talk we will discuss the transversal discriminant of a singular hypersurfaces whose singular locus is a smooth curve, and how it can be applied in order to generalize a classical result by Siersma, Pellikaan, and de Jong which studies morsifications of such singularities. In addition, we will present some applications to the study of Yomdin-type isolated singularities.
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| December 5th |
Daniel Apsley | Title: Fourier-Mukai Transforms and Abelian Varieties.
Abstract: Fourier-Mukai transforms are useful tools for studying relationships between the derived categories of projective varieties. After giving examples and stating some useful theorems, we will apply these tools to abelian varieties. In this situation we will deduce a classical result of Mukai, stating that abelian varieties and their duals are in fact derived equivalent.
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