Research talks
- Gluing of Fourier-Mukai partners using tensor triangulated geometry, poster presentation at Higher Dimensional Algebraic Geometry: James 60, UCSD, January 2024
Abstract
We initiate studies of the relationship between birational geometry of smooth varieties and tensor structures of their derived categories. A crucial insight from tensor triangulated geometry is that we can view a variety as a specific tensor structure on its derived category and an upshot is that closer observations on such tensor structures allow us to glue its Fourier-Mukai partners to construct a smooth scheme locally of finite type, which we call the Fourier-Mukai locus. The geometry of the locus reflects geometric properties of the Fourier-Mukai partners in various contexts, such as abelian, toric, and Fano varieties, as well as birational operations among them such as flops. We further investigate relations with the DK hypothesis and methods to purely categorically characterize the locus.
- Symmetries on derived categories and reconstruction of elliptic curves, Eisenbud's Seminar on Commutative Algebra and Algebraic Geometry, UC Berkeley, Octorber 2023
Abstract
We know the structure of the derived category of coherent sheaves on an elliptic curve very well and moreover it has been known that an elliptic curve can be reconstructed from categorical structures of its derived category. In this talk, we will discuss a version of reconstruction of elliptic curves that was announced this year, using symmetry of derived categories coming from their autoequivalences. Moreover, if time permits, we will see possible generalization and obstructions to higher dimensional cases, which were discussed in my recent paper in relation with monoidal structures on derived categories.
Seminar talks
- The Hochschild-Kostant-Rosenberg isomorphism and cyclic action on the derived loop space, Derived Algebraic Geometry Student Seminar , UC Berkeley, November 2023
- Structures of the derived category on an elliptic curve, Eisenbud's Seminar on Commutative Algebra and Algebraic Geometry , UC Berkeley, Octorber 2023
Abstract
Although the definition of derived categories (of coherent sheaves on a variety) requires some understanding of homological algebra, derived categories themselves can be treated geometrically in many sense. In this talk, I will first talk about how and why people use derived categories in algebraic geometry and then I will briefly explain how we can indeed define derived categories. Then, as an example, we will explore the derived category on an elliptic curve. If time permits, I will also compare it with the derived category on a projective line.
- Geometry in derived categories, Stundent Seminar on Complex Geometry, UC Berkeley, April 2023
Abstracts
Today I will introduce the derived category of coherent sheaves on a variety as an algebraic invariant. We’ll see why it is a reasonable invariant and how we can make it work by using the language of triangulated categories. Then I will explain what kind of tools we have to compute or study derived categories, some of which such as stability conditions are related to analytic geometry, and as time permits, I will perform some explicit computations.
- Motivation, definition, and techniques for affine Grassmannians, Nadler's Geometric Representation Theory Seminar, UC Berkeley, January-February 2023
Abstracts
- First I will briefly talk about motivation of the affine Grassmannian and in particular we will focus on its relation with vector bundles over an algebraic curve. Then I will introduce three definitions of the affine Grassmannian. For each definition, the affine Grassmannian will be defined as a certain functor and we will see how to view it as a geometric object following ideas of a functor of points and ind-schemes.
- Last time I explained a motivation of the affine Grassmanian in the context of a classification of vector bundles over a curve and we defined the affine Grassmannian as (the fppf sheafification of) a functor from sending \(\mathbb C\)-algebras \(R\) to \(\operatorname{GL}_n(R((t)))/\operatorname{GL}_n(R[[t]])\). Today we’ll see how we can view this functor as a geometric object following ideas of a functor of points and ind-schemes. To see that the affine Grassmannian is indeed an ind-scheme, we’ll introduce the second definition using the lattice descriptions.
- Today, I will start with reviewing what we did last time, namely, showing the \(\mathbb C\)-points of the affine Grassmannian can be written as a union of an increasing sequence of projective schemes, and then wrap up my introductory talks with the third definition by parametrizing (fppf) vector bundles and their trivializations.
Undergraduate talks