BAGELS (B(e creative!) Algebraic Geometry Eating and Learning Seminar) meets on Monday, from 3:15 p.m. to 4:15 p.m., in 308 JWB. If you have any questions, please contact Allechar.
Abstract: The horizontal translation action of the real line on the complex upper half plane descends to an action of the circle group S^1 on the "unstable locus", or image of Im \tau > 1, in the complex modular curve. In this talk, we explain an analogous action of the open p-adic unit disk centered at 1 on the Katz moduli space, a p-adic analytic covering space of the unstable locus on the p-adic modular curve whose ring of functions can be constructed by p-adically interpolating the coefficients of classical modular forms. The analogy is richer than one might first expect, and leads to new perspectives on classical notions in the p-adic theory of modular curves and modular forms such as Dwork's equation \tau=\log q and Hida's space of ordinary p-adic modular forms, with implications for the p-adic representation theory of GL_2 and p-adic Galois groups.
Abstract: What do representations of subgroups of SL(2,C), chains of rational curves and quiver representations have in common? But a Dynkin diagram of course! (and, a triangulated category...) We'll talk about the classical McKay correspondence, and Bridgeland, King and Reid's modern take on it via Fourier-Mukai equivalences. This will be an excuse to present some interesting techniques in the study of derived categories. I'll follow parts of this survey by Craw.
Abstract: In this talk, I will introduce the finite monodromy problem for ODE and specialized to the integral Lame equation. Then I will state Dahmen's conjecture, which predicts the number of integral Lame equation with finite monodromy. Finally, I will explain how we reduce the problem to the zero counting of the SL(2,Z) modular form. The modular form computation will be the special case of Appendix A in here.
Abstract: Birational superrigidity was introduced as a measurement of nonrationality when people are studying rationality problems of Fano hypersurfaces in the projective spaces. On the other hand, the notion of K-stability came up in the study of existence of KE metrics on Fano manifolds, and in some recent studies, it also plays an important role in the study of Moduli problems of Fano varieties. The two notions seem to be very different according to their original definitions, however when we focus on Fano varieties of Picard number 1, they are both related to log maximal singularities where we examine the singularities of certain anti-canonical systems. In this talk, I will first define all these concepts, and then survey some results in the direction of finding the relation between birational superrigidity and K-stability.
Abstract: Lefschetz (1,1) Theorem is just the Hodge conjecture for Hodge class of degree 2. In this talk, I will introduce the technique of Lefschetz's orginal proof using Poincar\'e normal functions. By taking hyperplane sections, Lefschetz's original method would provide a sort of inductive proof of Hodge conjecture, but the Abel-Jacobi mapping is rarely surjective in higher dimensions which is the main technical obstruction to a complete generalization of Lefschetz's method. This induces the Griffiths' program approaching to the Hodge conjecture.
Abstract: In this talk I will discuss what it means for a set of varieties to be bounded. Roughly speaking, we want to give a proper algebraic notion of "being described by a finite set of parameters". I will present a few examples, together with some techniques to address boundedness questions.
Abstract: In this talk, I will first introduce some notations of stability conditions. Along the way I will define the Euler stability which was constructed by A.Bertram. We conjectured that for any class, the Euler stability will be eventually equal to the Gieseker stability. As an example, I will show the wall-crossings for the class of the structure sheaf of the twisted cubic curve, and the conjecture turns out to be true in this case.
Abstract: Multiplier ideals are ideal sheaves that are closely related to measuring singularities of divisors. In this talk, I will give a formal definition of multiplier ideals, along with some properties and applications.
Abstract: In this talk we will introduce the Grothendieck ring of varieties, prove some classical properties, and explore an alternative definition due to Bittner. As an application, we will sketch a proof related to a criterion of the rationality of curves and the rationality of the Weil Zeta function.
Abstract: Let X be a quasi-elliptic surfaces and A any ample line bundle on X. I will show that K_X+kA is base point free for k at least 3 and it is very ample for k at least 4.
Abstract: Generalized pair comes naturally from Kodaira canonical bundle formula. Suppose (X,B+M) is a generalized pair, if X itself has klt singularity, birkar proved the cone theorem for K_x+B+M. But if X is lc, the existence of contraction morphism is still not known. I proved under some good condition, the contraction morphism exists if X has lc singularity.
Abstract: This talk consists of two parts. First of all, we will discuss the behavior of multiplier ideals in char p reduction, which coincides with the test ideals for p>>0. Furthermore, one can recall that Kawamata-Viehweg vanishing theorem is not in general true for positive characteristic. In this talk, we will prove that we still have the vanishing theorem for globally F-regular varieties.
Abstract: Recently Birkar proved the famous BAB conjecture. A main tool in the proof is the theory of complements. In this talk, I will overview the notions involved in the formulation of the BAB conjecture and introduce what complements are. Then, I will focus on the case of surfaces, and highlight how complements are used in the proof of the conjecture.
Abstract: We will first introduce a problem known as strange duality for moduli of sheaves on surfaces, then bring in moduli of complexes and wall-crossing, and explain how they can help to understand strange dualityfor K3 surfaces.
Abstract: The purpose of this talk is to give a brief introduction to matrix factorizations. We show some examples and sketch the construction of the triangulated category of curved dg-sheaves. We also state Orlov's theorem on hypersurfaces and give examples.
Abstract: We will introduce the minimal log discrepancy, which is an invariant to measure the singularities of an algebraic variety. This object is a central invariant in the study of birational geometry. Then we will discuss some conjectures and known results.
Abstract: How much information can you determine about an algebraic variety from its cohomology? In certain cases, quite a bit! In this talk, we will prove the classical Torelli theorem which states that a complex algebraic curve is determined up to isomorphism by its second cohomology. A key player is the Jacobian, which I will introduce along the way. If time permits, we will discuss generalizations of the Torelli theorem to complex K3 surfaces, hyperkahler varieties and possibly generalizations to other number fields. I'll try to keep the prerequisites for this talk low, basic knowledge of complex algebraic curves should be plenty for the bulk of the talk.
Abstract: In "primitive cohomology and the tube mapping", Schnel constructs the generators of the rational first homology group of a closed Riemann surface using all invariant vanishing cycles under the monodromy action. We will construct the generators using the tube classes over only finitely many vanishing cycles. We will see that the generator we constructed is the same as the difference of two cones over the same vanishing cycles, that is to say, we glue two thimbles with the same boundary along the same boundary.
Abstract: In 1979, Goldfeld conjectured that the average rank of elliptic curves over the rational numbers is 1/2. Theoretical results towards boundedness of the average rank were proven, assuming the Generalized Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture. However, the data did not support these results. In a joint work with Arul Shankar, Manjul Bhargava provided an unconditional proof of boundedness by studying the n-Selmer group. In this talk, I will give an outline of their proof.
Abstract: Spectral sequences were developed in the 60s and since then they’ve become a central tool in algebra and geometry. We will review the definition and basic constructions, and the philosophy behind their application. I’ll then list a few examples and work out a very simple one in detail.
Abstract: In this talk, I will introduce a Frobenius variant of Seshadri constant defined by M. Mustata and K. Schwede, and then show that the lower bounds imply the global generation or very ampleness of the corresponding adjoint line bundle (in positive characteristic).
Abstract: We'll discuss how to reduce algebraic varieties in characteristic 0 modulo a prime p. Many problems are easier to solve in prime characteristic using the techniques of F-singularities, and solving a problem modulo all "sufficiently general" primes p often yields the solution in characteristic 0. We will discuss how this strategy can be used to show invariant rings of linearly reductive group actions are Cohen-Macaulay, as well as some containment results on symbolic powers.
Abstract: I will introduce what is Gromov-Witten invariants and sketch three approaches to study it. They are related to localization formula, integrable system, and Frobenius manifold separately. I will put more effort on the approach related to Frobenius manifold. If time permitted, I will explain how to write higher genus Gromov-Witten invariants in terms of invariants on Frobenius manifold.
Abstract: In this talk I will discuss Kawamata-Viehweg vanishing theorem, which gives in characteristic 0 a vanishing in cohomology involving the canonical divisor, and a nef and big divisor. I will also show a counter-example in positive characteristic, along with some partial result in the same context.
Abstract: Mirror symmetry predicts a relation between the stability manifold of some smooth variety (stack?) X and the moduli space of its mirror. We'll explore this principle in the case where X is a certain quotient of an elliptic curve. The mirror family is parameterized by the universal unfolding of a singularity, whose geometry is regulated by an extended affine root system.
Abstract: We will discuss the basic properties of ruled surfaces. Namely, we will see what is the invariant of the given ruled surfaces and how the divisors on the surface could be determined by the base curve and the fiber.