stanford algebraic geometry seminar 2019-20

Fridays 4-5 pm in 383-N (with exceptions)

Click on the title to see the abstract (if available). (For earlier talks in this seminar, click here. For related seminars, click here. For the department webpage for the algebraic geometry seminar, click here.) For more information, please contact Eric Larson, Ravi Vakil, or Isabel Vogt.

DateSpeakerTitle
September 27 12:30pm in 384-I Isabel Vogt (Stanford)Stability of normal bundles of space curves
October 4Jordan Ellenberg (Wisconsin)Heights of rational points on (some) stacks
October 11Stefan Schreieder (Munich)Stably irrational hypersurfaces of small slopes
October 18no seminar
October 25 2:30pm in 384-IJim Bryan (UBC)K3 surfaces with symplectic group actions, enumerative geometry, and modular forms
October 25 4pm in 383-NNathan Pflueger (Amherst)
November 1no seminar (WAGS at Utah)
November 8Rachel Pries (Colorado State)
November 15Eric Riedl (Notre Dame)
November 22Maddie Weinstein (Berkeley)
November 29no seminar (thanksgiving break)
December 6maybe no seminar
January 10
January 17no seminar (JMM)
January 24Jackson Morrow (Emory)
January 31Izzet Coskun (UIC)
February 7Jack Huizenga (Penn State)
February 14
February 21Daniel Litt (UGA)
February 28Anand Patel (Oklahoma State)
March 6
March 13
April 3Matt Baker (Georgia Tech)
April 10
April 17
April 24
May 1Rohini Ramadas (Brown) and Rob Silversmith (Northeastern)
May 8
May 15no seminar
May 22
May 29
June 5
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Posted in Uncategorized

abstracts for 2019-20 seminars

(The seminar webpage is here.)

Isabel Vogt (Stanford)

September 27, 2019

Stability of normal bundles of space curves

The normal bundle controls the deformation theory of a curve embedded in projective space. In this talk we study the stability of normal bundles of curves in P^3 using degeneration. This is joint work with Izzet Coskun and Eric Larson.


Jordan Ellenberg (Wisconsin)

October 4, 2019

Heights of rational points on (some) stacks

The study of rational points on algebraic stacks over global fields is in many respects very similar to the familiar world of Diophantine geometry of schemes. But one key element of that world is missing; a theory of heights. I will propose such a theory and explain how it recovers many already-in-use notions of complexity for points on stacks, while also generating new ones. I talked about this project in the Stanford number theory seminar in spring 2018; I will explain the basic ideas again and then talk about some examples we understand better now than we did then, such as points on the moduli stack of abelian varieties, and discuss some questions that remain open for future work.


Stefan Schreieder (Munich)

October 11, 2019

Stably irrational hypersurfaces of small slopes

We show that over any uncountable field of characteristic different from two, a very general hypersurface of dimension n>2 and degree at least log_2(n)+2 is not stably rational. This improves earlier results of Koll\’ar and Totaro, who proved the same result under a linear bound on the degree.


Jim Bryan (University of British Columbia)

October 25, 2019, 2:30 pm

K3 surfaces with symplectic group actions, enumerative geometry, and modular forms

The Hilbert scheme parameterizing n points on a K3 surface X is a holomorphic symplectic manifold with rich properties. In the 90s it was discovered that the generating function for the Euler characteristics of the Hilbert schemes is related to both modular forms and the enumerative geometry of rational curves on X. We show how this beautiful story generalizes to K3 surfaces with a symplectic action of a group G. Namely, the Euler characteristics of the “G-fixed Hilbert schemes” parametrizing G-invariant collections of points on X are related to modular forms of level |G| and the enumerative geometry of rational curves on the stack quotient [X/G] . These ideas lead to some beautiful new product formulas for theta functions associated to root lattices. The picture also generalizes to refinements of the Euler characteristic such as chi_y genus and elliptic genus leading to connections with Jacobi forms and Siegel modular forms.


Nathan Pflueger (Amherst)

October 25, 2019, 4 pm


Rachel Pries (Colorado State)

November 8, 2019


Eric Riedl (Notre Dame)

November 15, 2019


Maddie Weinstein (Berkeley)

November 22, 2019


Posted in seminars

abstracts for 2018-19 seminars

(The seminar webpage is here.)


June Huh (Institute for Advanced Study)

October 26, 2018

Distinguished Lecture 2 of 3: Kazhdan-Lusztig theory for matroids

There is a remarkable parallel between the theory of Coxeter groups (think of the symmetric group or the dihedral group) and matroids (think of your favorite graph or point configuration), based on their combinatorial cohomology theories. After giving an overview of the similarity, I will report on a cohomological approach to some conjectures in enumerative combinatorics. Joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang.


David Jensen (Kentucky)

November 2, 2018, 3:15-4:15 pm

Linear Systems on General Curves of Fixed Gonality

The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether theorem, which says that the space of linear systems of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods. This is joint work with Dhruv Ranganathan, based on prior work of Nathan Pflueger.


Sam Payne (UT Austin and MSRI)

November 2, 2018, 4:30-5:30 pm

Tropical methods for the Strong Maximal Rank Conjecture

I will present joint work with Dave Jensen using tropical methods on a chain of loops to prove new cases of the Strong Maximal Rank Conjecture of Aprodu and Farkas that are relevant to computing the Kodaira dimensions of the moduli spaces M_22 and M_23.  As time permits, I will also discuss relations to an analogous approach via limit linear series on chains of genus 1 curves, developed in the work of Liu, Osserman, Teixidor, and Zhang.


Margaret Bilu (Courant)

April 19, 2019, 2:30-3:30 pm

Motivic Euler products and motivic height zeta functions

The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties over k. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. I will focus on Manin’s conjecture and on its motivic analog: the latter predicts the behavior of moduli spaces of curves of large degree on some algebraic varieties. It may be formulated in terms of the generating series of the classes of these moduli spaces in the Grothendieck ring, called the motivic height zeta function. This will lead me to explain how some power series with coefficients in the Grothendieck ring can be endowed with an Euler product decomposition and how this can be used to give a proof of the motivic version of Manin’s conjecture for equivariant compactifications of vector groups.


Ronno Das (University of Chicago)

April 19, 2019, 4-5 pm

Points and lines on cubic surfaces

The Cayley-Salmon theorem states that every smooth cubic surface in CP^3 has exactly 27 lines. Their proof is that marking a line on each cubic surface produces a 27-sheeted cover of the moduli space M of smooth cubic surfaces. Similarly, marking a point produces a ‘universal family’ of cubic surfaces over M. One difficulty in understanding these spaces is that they are complements of incredibly singular hypersurfaces. In this talk I will explain how to compute the rational cohomology of these spaces. I’ll also explain how these purely topological theorems have (via the machinery of the Weil Conjectures) purely arithmetic consequences: the average smooth cubic surface over a finite field F_q contains 1 line and q^2 + q + 1 points.


Chenyang Xu

Thursday, May 23, 2019

Department Colloquium: Compact moduli spaces of varieties

Moduli space is ubiquitous in modern algebraic geometry. In this talk, I will discuss the recent progress on the construction of compact moduli spaces parametrising varieties whose Chern classes are positive. Ideas from differential geometry play an essential role.

Posted in seminars

stanford algebraic geometry seminar 2018-19

Fridays 4-5 pm in 383-N (with exceptions)

Click on the title to see the abstract (if available). (For earlier talks in this seminar, click here. For related seminars, click here. For the department webpage for the algebraic geometry seminar, click here.) For more information, please contact Michael Kemeny, Eric Larson, Pablo Solis, Ravi Vakil, or Isabel Vogt.

October 26 Distinguished Lecture 2 of 2June Huh (Institute for Advanced Study)Kazhdan-Lusztig theory for matroids
November 2, 3:15-4:15 pmDavid Jensen (Kentucky)Linear Systems on General Curves of Fixed Gonality
November 2, 4:30-5:30 pmSam Payne (UT Austin and MSRI)Tropical methods for the Strong Maximal Rank Conjecture
November 30Jack Hall (Arizona)GAGA theorems
January 25Arnav Tripathy (Harvard)A geometric model for complex analytic equivariant elliptic cohomology
February 1Eugene Gorsky (UC Davis)Soergel bimodules and Hilbert schemes
February 15Padmavathi Srinivasan (Georgia Tech)An arithmetic count of lines meeting four lines in P^3
Tuesday March 12 ColloquiumChristopher Hacon (Utah)On the classification of algebraic varieties
March 15Dan Bragg (UC Berkeley and MSRI)Supersingular twistor spaces
April 12, 2:30-3:30 pm, 384-I Akhil Mathew (Chicago and MSRI)A gentle approach to the de Rham-Witt complex
April 12, 4-5 pm, 383-NLaure Flapan (Northeastern and MSRI)Chow motives, L-functions, and powers of algebraic Hecke characters
April 13-14, BerkeleyWestern Algebraic Geometry Symposium (WAGS)
April 19, 2:30-3:30 pmMargaret Bilu (NYU)Motivic Euler products and motivic height zeta functions
April 19, 4-5 pmRonno Das (Chicago)Points and lines on cubic surfaces
Thursday April 25 ColloquiumXinwen Zhu (Caltech)
April 26Kristin DeVleming (UCSD and MSRI)Comparing compactifications of the moduli space of plane curves
May 3Juliette Bruce (UW Madison)Semi-Ample Asymptotic Syzygies
May 10Dan Erman (UW Madison)Limits of polynomials rings
Thursday May 23 ColloquiumChenyang Xu (MIT)Compact moduli spaces of varieties
May 24, 2:30-3:30 pmAnne-Sophie Kaloghiros (MSRI and Brunel University)Threefold Calabi-Yau pairs
May 24, 4:00-5:00 pmMilena Hering (MSRI and Edinburgh)Stability of Toric Tangent bundles
June 7Sam Payne (UT Austin)Top weight cohomology of M_{g,n}
Posted in Uncategorized

abstracts for 2017-18 seminars

(The seminar webpage is here.)


Sam Grushevsky (Stony Brook University)

September 29, 2017

Geometry of compactified moduli of cubic threefolds

Starting from considering the GIT compactification of the moduli of cubic threefolds, the “wonderful” compactification, which is smooth with normal crossing boundary, is constructed by an explicit sequence of blowups. We show that there exists a family of intermediate jacobians over the wonderful compactification. We compute the cohomology of the wonderful compactification by comparing it to the symplectic resolution. Based on joint works with Casalaina-Martin, Hulek, Laza


Felix Janda (University of Michigan)

October 6, 2017

Genus two curves on quintic threefolds

Virtual (Gromov-Witten) counts of maps from algebraic curves to quintic 3-folds in projective space have been of significant interest for mathematicians and physicists since the early 90s. While there are (very inefficient) algorithms for computing any specific Gromov-Witten invariant, explicit formulae are only known in genus zero and one. On the other hand, physicists have explicit conjectural formulas up to genus 51.

I will discuss a new approach to the Gromov-Witten theory of the quintic (using logarithmic geometry) which yields an explicit formula in genus two that agrees with the physicists’ conjecture.

This is based on joint works in progress with Q. Chen, S. Guo and Y. Ruan.


Remy van Dobben de Bruyn (Columbia)

October 20, 2017

Dominating varieties by liftable ones

Given a smooth projective variety over an algebraically closed field of positive characteristic, can we always dominate it by another smooth projective variety that lifts to characteristic 0? We give a negative answer to this question.


Jason Lo (Cal State Northridge)

October 27, 2017

The effect of Fourier-Mukai transforms on slope stability on elliptic fibrations

Slope stability is a type of stability for coherent sheaves on smooth projective varieties. On a variety where the derived category of coherent sheaves admits a non-trivial autoequivalence, it is natural to ask how slope stability `transforms’ to a different stability under the autoequivalence. This question also has implications for understanding the symmetries within various counting invariants. In this talk, we will give an answer to the above question for elliptic surfaces and threefolds under a Fourier-Mukai transform.


Daniel Litt (Columbia University)

December 1, 2017

Galois actions on fundamental groups

Let X be a variety over a field k, and let x be a k-rational point of X. Then the absolute Galois group of k acts on the etale fundamental group of X. If k is an arithmetically interesting field (i.e. a number field, a p-adic field, or a finite field), then this action reveals a great deal about the geometry of X; if X is a variety with an interesting fundamental group, this action reveals a great deal about the arithmetic of k.

This talk will discuss (1) joint work with Alexander Betts about the structure of Galois actions on fundamental groups, (2) how to describe invariants of these actions in terms of more geometric invariants of X, and (3) applications of this work to classical algebraic geometry, and, if time permits, arithmetic.


Pablo Solis (Stanford)

January 19, 2018

Hunting Vector Bundles on \mathbf{P}^1 \times \mathbf{P}^1

Motivated by Boij-Soderberg theory, Eisenbud and Schreyer conjectured there should be vector bundles on \mathbf{P}^1 \times \mathbf{P}^1 with natural cohomology and prescribed Euler characteristic. I’ll give some background on Boij-Soderberg theory, explain what natural cohomology means and prove the conjecture in “most” cases.


Izzet Coskun (University of Illinois at Chicago)

January 26, 2018

The geometry of moduli spaces of sheaves on surfaces

In this talk, I will discuss recent results concerning the Brill-Noether Theory of higher rank bundles on rational surfaces and stable cohomology of moduli spaces of sheaves. In joint work with Jack Huizenga, we characterize when the cohomology of a general stable sheaf on a Hirzebruch surface is determined by its Euler characteristic. We use these results to classify moduli spaces where the general bundle is globally generated. If time permits, I will discuss joint work with Matthew Woolf on the stable cohomology of moduli spaces on rational surfaces.


Katrina Honigs (Utah)

February 2, 2018

Fourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic

There are many results characterizing when derived categories of two complex surfaces are equivalent, including theorems of Bridgeland and Maciocia showing that derived equivalent Enriques or bielliptic surfaces must be isomorphic. The proofs of these theorems strongly use Torelli theorems and lattice-theoretic methods which are not available in positive characteristic. In this talk I will discuss how to prove these results over algebraically closed fields of positive characteristic (excluding some low characteristic cases). This work is joint with M. Lieblich and S. Tirabassi.


Junliang Shen (ETH)

February 9, 2018

K3 categories, cubic 4-folds, and the Beauville-Voisin conjecture

We discuss recent progress on the connection between 0-cycles of holomorphic symplectic varieties and structures of K3 categories. We propose that there exists a sheaf/cycle correspondence for any K3 category, which controls the geometry of algebraically coisotropic subvarieties of certain holomorphic symplectic varieties. Two concrete cases will be illustrated in details:
(1) the derived category of a K3 surface (joint with Qizheng Yin and Xiaolei Zhao),
(2) Kuznetsov category of a cubic 4-fold (joint with Qizheng Yin).
If time permits, we will also discuss the connection to rational curves in cubic 4-folds.


Michael Viscardi (Berkeley)

February 16, 2018

Quantum cohomology and 3D mirror symmetry

Recent work on equivariant aspects of mirror symmetry has discovered relations between the equivariant quantum cohomology of symplectic resolutions and Casimir-type connections (among many other objects). We provide a new example of this theory in the setting of the affine Grassmannian, a fundamental space in the geometric Langlands program. More precisely, we identify the equivariant quantum connection of certain symplectic resolutions of slices in the affine Grassmannian of a semisimple group G with a trigonometric Knizhnik-Zamolodchikov (KZ)-type connection of the Langlands dual group of G. These symplectic resolutions are expected to be symplectic duals of Nakajima quiver varieties, so that our result is an analogue of (part of) the work of Maulik and Okounkov in the symplectic dual setting.


Roya Beheshti (Washington U. St. Louis)

February 23, 2018

Moduli spaces of rational curves on hypersurfaces

I will talk about the geometry of moduli spaces of rational curves on Fano hypersurfaces and discuss some results concerning their dimension and birational geometry.


Yongbin Ruan (Michigan)

March 9, 2018

The structure of higher genus Gromov-Witten invariants of quintic 3-fold

The computation of higher genus Gromov-Witten invariants of quintic 3–fold (or compact Calabi-Yau manifold in general) has been a focal point of research of geometric and physics for more than twenty years. A series of deep conjectures have been proposed via mirror symmetry for the specific solutions as well as structures of its generating functions. Building on our initial success for a proof of genus two conjecture formula of BCOV, we present a proof of two conjectures regarding the structure of the theory. The first one is Yamaguchi-Yau’s conjecture that its generating function is a polynomial of five generators and the other one is the famous holomorphic anomaly equation which governs the dependence on four out of five generators. This is a joint work with Shuai Guo and Felix Janda.


Andrei Calderaru (Wisconsin)

April 13, 2018

Computing a categorical Gromov-Witten invariant

In his 2005 paper “The Gromov-Witten potential associated to a TCFT” Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a cyclic A-inifinity algebra or category. Applying his construction to the derived category of sheaves of a complex projective variety provides a definition of higher genus B-model Gromov-Witten invariants, independent of the BCOV formalism. This has several advantages. Due to the categorical invariance of these invariants, categorical mirror symmetry automatically implies classical mirror symmetry to all genera. Also, the construction can be applied to other categories like categories of matrix factorization, giving a direct definition of FJRW invariants, for example.

In my talk I shall describe the details of the computation (joint with Junwu Tu) of the invariant, at g=1, n=1, for elliptic curves. The result agrees with the predictions of mirror symmetry, matching classical calculations of Dijkgraaf. It is the first non-trivial computation of a categorical Gromov-Witten invariant.


Daniel Halpern-Leistner (Cornell)

April 20, 2018

What is wall-crossing?

I will discuss recent progress in understanding geometric invariant theory from an “intrinsic” perspective. This leads to a conceptually clean meta-principle for how to study the birational geometry of moduli spaces as well as a universal wall-crossing formula for the integrals of tautological K-theory classes on these moduli spaces. I will apply this perspective to the example of Bridgeland semistable complexes on an algebraic surface. The result is a relatively straightforward construction of K-theoretic Donaldson invariants, along with wall-crossing formulas for these invariants which are new-ish (conjecturally equivalent to Mochizuki’s cohomological wall crossing formulas in the context where both are defined).


Sean Howe (Stanford)

April 27, 2018

Motivic random variables and random matrices

As first shown by Katz-Sarnak, the zero spacing of L-functions of smooth plane curves over finite fields approximate the infinite random matrix statistics observed experimentally for the zero spacing of the Riemann-Zeta function (arbitrarily well by first taking the size of the finite field to infinity, and then the degree of the curve to infinity). The key geometric inputs are a computation of the image of the monodromy representation and Deligne’s purity theorem, which ensures that only the zeroth cohomology group of irreducible local systems will contribute asymptotically to the statistics. In this talk, we explain how higher order terms (i.e. the lower weight part of cohomology) can be computed starting from a simple heuristic for the number of points on a random smooth plane curve.


Isabel Vogt (MIT)

May 4, 2018, 3-4 pm

Interpolation problems for curves in projective space

In this talk we will discuss the following question: When does there exist a curve of degree d and genus g passing through n general points in \mathbb{P}^r?


Eric Larson (MIT)

May 4, 2018, 4:30-5:30 pm

The Maximal Rank Conjecture

We find the Hilbert function of a general curve of genus g, embedded in \mathbb{P}^r via a general linear series of degree d. Note that Isabel Vogt’s talk earlier this afternoon is a pre-requisite for this talk.


Dori Bejleri (Brown)

May 18, 2018, 3-4 pm

Stable pair compactifications of the moduli space of degree one del Pezzo surfaces via elliptic fibrations

A degree one del Pezzo surface is the blowup of \mathbb{P}^2 at 8 general points. By the classical Cayley-Bacharach Theorem, there is a unique 9th point whose blowup produces a rational elliptic surface with a section. Via this relationship, we can construct a stable pair compactification of the moduli space of anti-canonically polarized degree one del Pezzo surfaces. The KSBA theory of stable pairs $(X,D)$ is the natural extension to dimension 2 of the Deligne-Mumford-Knudsen theory of stable curves. I will discuss the construction of the space of interest as a limit of a space of weighted stable elliptic surface pairs and explain how it relates to some previous compactifications of the space of degree one del Pezzo surfaces. This is joint work with Kenny Ascher.


Francois Greer (Stony Brook)

May 18, 2018, 4:30-5:30 pm

Elliptic fibrations in the presence of singularities

The Gromov-Witten generating series for an elliptic fibration is expected to have modular properties by mirror symmetry. When the homology class in the base is irreducible and the total space is smooth, we obtain a classical modular form for the full modular group. If the base class is reducible, we expect the series to be quasi-modular. If the fibration does not admit a section, then the modular form has higher level. Both of these relaxations are related to the presence of singularities in the geometry.


Soren Galatius (Stanford)

May 25, 2018, 3-4 pm, Room TBA

M_g, M_g^{trop}, GRT, and Kontsevich’s complex of graphs

I will report on recent joint work with Melody Chan and Sam Payne on the cohomology of M_g in degree 4g-6. It is known that the rational cohomology vanishes above this degree. We prove that the rational cohomology in this degree is non-trivial for all g \geq 7 and that its dimension grows faster than 1.324^g + constant, making it asymptotically larger than the entire tautological ring and disproving a recent conjecture of Church-Farb-Putman and an older conjecture of Kontsevich. Our proof relates the weight filtration on compactly supported cohomology of M_g with the moduli space of tropical curves and with the cohomology of Kontsevich’s graph complex. We then use a theorem of Willwacher to construct an injection of the Grothendieck-Teichmüller Lie algebra into H^{2g}_c(M_g).


Tathagata Basak (Iowa State University)

May 25, 2018, 4:30-5:30 pm

A complex ball quotient and the monster

We shall talk about an arithmetic lattice M in PU(13,1) acting on the the unit ball B in thirteen dimensional complex vector space. Let X be the space obtained by removing the hypersurfaces in B that have nontrivial stabilizer in M and then quotienting the rest by M. The fundamental group G of the ball quotient X is a complex hyperbolic analog of the braid group. We shall state a conjecture that relates this fundamental group G and the monster simple group and describe our results (joint with D. Allcock) towards this conjecture.

The discrete group M is related to the Leech lattice and has generators and relations analogous to Weyl groups. Time permitting, we shall give a second example in PU(9,1) related to the Barnes-Wall lattice for which there is a similar story.

Posted in seminars

stanford algebraic geometry seminar 2017-18

Fridays 4-5 pm in 383-N (with exceptions)

Click on the title to see the abstract (if available). (For earlier talks in this seminar, click here. For related seminars, click here. For the department webpage for the algebraic geometry seminar, click here.) For more information, please contact Jun Li, Michael Kemeny, or Ravi Vakil.

September 29 Samuel Grushevsky (Stony Brook University) Geometry of compactified moduli of cubic threefolds
October 6 Felix Janda (University of Michigan) Genus two curves on quintic threefolds
October 13 no seminar (WAGS weekend)
Weekend of October 14-15 Western Algebraic Geometry Symposium, at UCLA
October 20 Remy van Dobben de Bruyn (Columbia University) Dominating varieties by liftable ones
October 27  Jason Lo (Cal State Northridge) The effect of Fourier-Mukai transforms on slope stability on elliptic fibrations
November 10
November 17 (probably no seminar, Ravi away)
November 24 no seminar (Thanksgiving break)
December 1 Daniel Litt (Columbia University) Galois actions on fundamental groups
January 19 Pablo Solis (Stanford) Hunting Vector Bundles on \mathbf{P}^1 \times \mathbf{P}^1
January 26 Izzet Coskun (UIC) The geometry of moduli spaces of sheaves on surfaces
February 2 Katrina Honigs (Utah) Fourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic
February 9 Junliang Shen (ETH) K3 categories, cubic 4-folds, and the Beauville-Voisin conjecture
February 16 Michael Viscardi (Berkeley) Quantum cohomology and 3D mirror symmetry
February 23 Roya Beheshti (Washington U. St. Louis) Moduli spaces of rational curves on hypersurfaces
March 9 Yongbin Ruan (Michigan) The structure of higher genus Gromov-Witten invariants of quintic 3-fold
April 13 Andrei Caldararu (Wisconsin) Computing a categorical Gromov-Witten invariant
April 20 Daniel Halpern-Leistner (Cornell) What is wall-crossing?
April 27 Sean Howe Motivic random variables and random matrices
May 4 (3-4 pm, in 380-W) Isabel Vogt (MIT) Interpolation problems for curves in projective space
May 4 (4:30-5:30 pm, in 383-N) Eric Larson (MIT) The Maximal Rank Conjecture
May 11 Sheldon Katz (UIUC) (joint with physics)
May 18 (3-4 pm) Dori Bejleri (Brown) Stable pair compactifications of the moduli space of degree one del Pezzo surfaces via elliptic fibrations
May 18 (4:30-5:30 pm) Francois Greer (Stony Brook) Elliptic fibrations in the presence of singularities
May 25 (3-4 pm, location TBA) Soren Galatius (Stanford) M_g, M_g^{trop}, the Grothendieck-Teichmuller group, and Kontsevich’s complex of graphs
May 25 (4:30-5:30 pm) Tathagata Basak (Iowa State) A complex ball quotient and the monster
Posted in seminars

abstracts for 2016-17 seminars

(The seminar webpage is here.)

Michael Kemeny (Stanford)
October 7, 2016

Syzygies, scrolls and Hurwitz spaces

A famous conjecture of Mark Green predicts a close relationship between the geometry of a curve and the algebraic properties of its coordinate ring. Namely, the Clifford index of the curve should equal the length of the linear part of the resolution of its coordinate ring under the canonical embedding. A conjecture of Schreyer goes beyond this by specifying that the last piece of the linear part should moreover tell you whether or not the curve has a unique pencil of minimal degree. We will discuss a proof of Schreyer’s conjecture for general curves of prescribed gonality, obtained jointly with Gavril Farkas. Two of the key actors in this story are the scroll associated to a pencil and the geometry of Hurwitz space.  (poster)

Christian Schnell (Stony Brook University)
October 21, 2016

Pushforwards of pluricanonical bundles and morphisms to complex abelian varieties

In the past few years, people working on the analytic side of algebraic geometry have obtained two important new results: a version of the Ohsawa-Takegoshi extension theorem with sharp estimates (Blocki, Guan-Zhou), and the existence of canonical singular hermitian metrics on pushforwards of relative pluricanonical bundles (Berndtsson, Paun, Takayama, and others). In this talk, I will explore some consequences of their work for the study of morphisms to complex abelian varieties, including the recent proof of Iitaka’s conjecture over abelian varieties (Cao-Paun).  (The talk will be understandable without any background in analysis.)

Wenhao Ou (UCLA)
October 28, 2016

Fano varieties where all pseudoeffective divisors are also numerically effective

We recall that a divisor in a smooth projective variety is said to be numerically effective (or nef) if it meets each curve with non negative intersection number, and is called pseudoeffective if it is the limit of effective Q-divisor classes. Both of these properties are ways in which a divisor can be in some sense “positive”. A nef divisor is always pseudoeffective, but the converse is not true in general. A Fano varity is a special variety whose anti-canonical divisor is ample. From the Cone Theorem, it turns out that the geometry of a Fano variety is closely related to its nef divisors. In this talk, we will consider Fano varieties such that all pseudoeffective divisors are nef. Wiśniewski shows that the Picard number of such a variety is at most equal to its dimension. Druel classifies these varieties when these two numbers are equal. We classify the case when the Picard number is equal to the dimension minus 1.

Gopal Prasad (University of Michigan)
November 4, 2016

Isospectrality of compact locally symmetric spaces and weak commensurability of arithmetic groups

Quotients of symmetric spaces of semi-simple Lie groups by torsion-free arithmetic subgroups are particularly interesting Riemannian manifolds which can be studied by using diverse techniques coming from the theories of Lie Groups, Lie Algebras, Algebraic Groups and Automorphic Forms. In my talk, I will discuss a well-known problem which was formulated by Mark Kac as “Can one hear the shape of a drum?”, and its solution, for arithmetic quotients of symmetric spaces, obtained in a joint paper (in Publ Math IHES, vol 109) with Andrei Rapinchuk. For its solution, we introduced a notion of “weak commensurability” of arithmetic, and more general Zariski-dense, subgroups and derive very strong consequences of weak commensurability.

Donghai Pan (Stanford)
November 11, 2016

Galois cyclic covers of the projective line and pencils of Fermat hypersurfaces

Classically, there are two objects that are particularly interesting to algebraic geometers: hyperelliptic curves and quadrics. The connection between these two seemingly unrelated objects was first revealed by M. Reid, which roughly says that there’s a correspondence between hyperelliptic curves and pencil of quadrics. I’ll give a brief review of Reid’s work and then describe a higher degree generalization of the correspondence.

Giulia Sacca (Stony Brook)
December 2, 2016

Intermediate Jacobians and hyperkahler manifolds

In recent years, there have been an increasing number of connections between cubic fourfolds and hyperkahler manifolds. The first instance of this was noticed by Beauville-Donagi, who showed that the Fano varieties of lines on a cubic fourfolds X is holomorphic symplectic. The aim of the talk is to describe another instance of this phenomenon, which is carried out in joint work with Laza and Voisin. The resulting hyperkahler manifold is fibered in intermediate Jacobians and is deformation equivalent to O’Grady’s ten-dimensional example. I will also present a more recent proof of this result, which is obtained in joint work with Laza, Kollár, and Voisin.

Hannah Larson (Harvard)
January 13, 2017, 4-4:45 pm

Lines on hypersurfaces with certain normal bundles

Let X be a smooth hypersurface. The Fano scheme of lines F(X) is the parameter space of all lines L \subset X. Given such a line, the normal bundle of L in X controls the deformation theory of L in X, and thus provides local information about F(X) near L. Being a vector bundle on {\mathbf{P}}^1, the normal bundle of L in X always splits as a direct sum of line bundles. In this talk, we consider natural subschemes of F(X) parameterizing lines L whose normal bundle in X has a certain splitting type.

Gavril Farkas (Humboldt University)
January 13, 2017, 5-6 pm

K3 surfaces of genus 14 via cubic fourfolds

In a celebrated series of papers, Mukai established structure theorems for polarized K3 surfaces of all genera g<21, with the exception of the case g=14. Using Hassett's identification between the moduli space of polarized K3 surfaces of genus 14 and the moduli space of special cubic fourfolds of discriminant 26, we establish the rationality of the universal K3 surface of genus 14. The proof relies on a degenerate version of Mukai's structure theorem for K3 surfaces of genus 8. This is joint work with Verra.

Ashvin Swaminathan (Harvard)
Thursday, January 19, 2017, 12:15-1 pm in 384-I

Inflection Points of Linear Systems on Families of Curves

It is a classic theorem in enumerative geometry that a general plane curve of degree d has exactly 3d(d-2) flex points (these are inflection points at which the tangent line has contact of order 3). Given this result, there are two natural generalizations to consider: (1) what can we say about inflection points of higher contact order, and (2) what happens when we look at such inflection points in families of curves acquiring a singularity? In this talk, I will discuss joint work with Anand Patel, in which we develop a method for answering these more general questions. Moreover, I will describe how to apply our method to tackle three interesting problems: (1) counting hyperflexes in a general pencil of plane curves, (2) describing the analytic-local behavior of the divisor of flexes in a family of plane curves acquiring a nodal singularity, and (3) computing the divisors of Weierstrass points of arbitrary order on the moduli space of curves.

Jake Levinson (Michigan)
January 20, 2017, 3:45-4:45 pm

Boij-Söderberg Theory for Grassmannians

Boij-Söderberg theory is a structure theory for syzygies of graded modules: a near-classification of the possible Betti tables of such modules (these tables record the degrees of generators in a minimal free resolution). One of the surprises of the theory was the discovery of a “dual” classification of sheaf cohomology tables on projective space.

I’ll tell part of this story, then describe some recent extensions of it to the setting of Grassmannians. Here, the algebraic side concerns modules over a polynomial ring in kn variables, thought of as the entries of a k \times n matrix. The goal is to classify “GL(k)-equivariant Betti tables”, recording the syzygies of equivariant modules, and to relate them to sheaf cohomology tables on the Grassmannian Gr(k,n). This work is joint with Nic Ford and Steven Sam.

Ben Bakker (Georgia)
January 20, 2017, 5-6 pm

A global Torelli theorem for singular symplectic varieties

Holomorphic symplectic manifolds are the higher-dimensional analogs of K3 surfaces and their local and global deformation theories enjoy many of the same nice properties.  By work of Namikawa, some aspects of the story generalize to singular symplectic varieties, but the lack of a well-defined period map means the moduli theory is ill-defined.  In joint work with C. Lehn, we consider locally trivial deformations—deformations along which the singularities don’t change — and show that in this context most of the results from the smooth case extend.  In particular, we prove a version of the global Torelli theorem and derive some applications to the geometry of birational contractions of moduli spaces of vector bundles on K3 surfaces.

Srikanth Iyengar (Utah)
January 27, 2017

A local Serre duality for modular representations of finite groups (and group schemes)

This talk will be about the representations of a finite group (or a finite group scheme) G defined over a field k of positive characteristic. In recent work, Dave Benson, Henning Krause, and Julia Pevtsova, and I discovered that stable module category of finite dimensional representations of G has local Serre duality. My plan is to explain what this means and also present some of the ideas, mostly from commutative algebra, that go into its proof.

Steven Sam (Wisconsin)
February 3, 2017

Secant varieties of Veronese embeddings

Given a projective variety X over a field of characteristic 0, and a positive integer r, we study the rth secant variety of Veronese re-embeddings of X. In particular, I’ll explain recent work which shows that the degrees of the minimal equations (and more generally, syzygies) defining these secant varieties can be bounded in terms of X and r independent of the Veronese embedding. This is based on http://arxiv.org/abs/1510.04904 and http://arxiv.org/abs/1608.01722.

Emily Clader (San Francisco State)
February 10, 2017

Double Ramification Cycles and Tautological Relations

Tautological relations are certain equations in the Chow ring of the moduli space of curves. I will discuss a family of such relations, first conjectured by A. Pixton, that arises by studying moduli spaces of ramified covers of the projective line. These relations can be used to recover a number of well-known facts about the moduli space of curves, as well as to generate very special equations known as topological recursion relations. This is joint work with various subsets of S. Grushevsky, F. Janda, X. Wang, and D. Zakharov.

Ionut Ciocan-Fontanine (Minnesota)
February 17, 2017

Wall-crossing in quasimap theory

Quasimap theory is concerned with curve counting on certain GIT quotients. In fact, one has a family of curve counting theories, with Gromov-Witten included, depending on the linearization in the GIT problem. I will present a wall-crossing formula, in all genera and at the level of virtual classes, as the size of the linearization changes. Some numerical consequences will be discussed as well. The talk will focus primarily on the recently established case of complete intersections in projective space (in this case, stable quasimaps coincide with stable quotients). This is joint work with Bumsig Kim.

Dusty Ross (SFSU)
February 24, 2017

Genus-One Landau-Ginzburg/Calabi-Yau Correspondence

First suggested by physicists in the late 1980’s, the Landau-Ginzburg/Calabi-Yau correspondence studies a relationship between spaces of maps from curves to the quintic 3-fold (the Calabi-Yau side) and spaces of curves with 5th roots of their canonical bundle (the Landau-Ginzburg side). The correspondence was put on a firm mathematical footing in 2008 when Chiodo and Ruan proved a precise statement for the case of genus-zero curves, along with an explicit conjecture for the higher-genus correspondence, which is determined from genus-zero data alone. In this talk, I will begin by describing the motivation and the mathematical formulation of the LG/CY correspondence, and I will report on recent work with Shuai Guo that verifies the higher-genus correspondence in the case of genus-one curves.

Dhruv Ranganathan (MIT)
April 14, 2017

A Brill-Noether theorem for curves of a fixed gonality

The Brill-Noether varieties of a curve C parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension. When C is general in moduli, these varieties are well understood: they are smooth, irreducible, and have the “expected” dimension. As one ventures deeper into the moduli space of curves, past the locus of general curves, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill–Noether theorem, which determines the dimensions of the Brill–Noether varieties on a general curve of fixed gonality, i.e. “general inside a chosen special locus”. The proof blends recent advances in tropical linear series theory, Berkovich geometry, and ideas from logarithmic Gromov-Witten theory. This is joint work with Dave Jensen.

Erik Carlsson (UC Davis)
April 21, 2017

Geometry behind the shuffle conjecture

The original “shuffle conjecture” of Haglund, Haiman, Loehr, Ulyanov, and Remmel predicted a striking combinatorial formula for the bigraded character of the diagonal coinvariant algebra in type A, in terms of some fascinating parking functions statistics. I will start by explaining this formula, as well as the ideas that went into my recent proof of this conjecture with Anton Mellit, namely the construction of a new algebra which has many elements in common with DAHA’s, and which has been expected to have a geometric construction. I will then describe a current project with Eugene Gorsky and Mellit, in which we have discovered the desired action of this algebra on the torus-equivariant K-theory of a certain smooth subscheme of the flag Hilbert scheme, which parametrizes flags of ideals of finite codimension in C[x,y].

Arnav Tripathy (Harvard)
May 12, 2017

Motivic Donaldson-Thomas invariants of K3 times an elliptic curve

I’ll describe a new chapter in the enumerative geometry of the K3 surface and its product with an elliptic curve in a long line of extensions starting from the classic Yau-Zaslow formula for counts of rational nodal curves. In particular, I’ll describe a string-theoretic prediction for the threefold’s motivic Donaldson-Thomas invariants given the Hodge-elliptic genus of the K3, a new quantity interpolating between the Hodge polynomial and the elliptic genus.

John Lesieutre (UIC)
May 19, 2017

A projective variety with discrete, non-finitely generated automorphism group

I will outline the construction of a projective variety over \mathbf{C} for which the group of automorphisms is discrete, but not finitely generated. I’ll also discuss a related example of a variety over \mathbf{R} with infinitely many non-isomorphic \mathbf{C}/\mathbf{R}-forms.

Yuchen Liu (Princeton)
May 26, 2017

Construction of hyperbolic cyclic covers

A complex algebraic variety is Brody hyperbolic if there are no non-constant holomorphic maps from the complex plane to the variety. As shown by Duval, Shiffman, Zaidenberg and many other people, degeneration methods are very useful in constructing Brody hyperbolic varieties. Using degeneration to the normal cone, we construct lots of examples of Brody hyperbolic cyclic covers when their branch loci have sufficiently large degree. If time permits, I will also discuss a different approach to construct hyperbolic double covers of \mathbf{P}^2 and Hirzebruch surfaces with the smallest possible degree of branch loci.

Posted in seminars

stanford algebraic geometry seminar 2016-17

 

Fridays 4-5 pm in 383-N (with exceptions)

Click on the title to see the abstract (if available). (For earlier talks in this seminar, click here. For related seminars, click here. For the department webpage for the algebraic geometry seminar, click here.) For more information, please contact Jun Li, Michael Kemeny, or Ravi Vakil.

October 7 Michael Kemeny Syzygies, scrolls and Hurwitz spaces
October 14 no seminar (WAGS weekend)
October 15-16 weekend Enrico Arbarello, Sapienza Universita di Roma/Stony Brook, Emily Clader, San Francisco State University, Luis Garcia, University of Toronto, Diane Maclagan
University of Warwick, Sandra Di Rocco, KTH, and Brooke Ullery, University of Utah
Western Algebraic Geometry Symposium (at the Colorado State University)
October 21 Christian Schnell (Stony Brook University) Pushforwards of pluricanonical bundles and morphisms to complex abelian varieties
October 28 Wenhao Ou (UCLA) Fano varieties where all pseudoeffective divisors are also numerically effective
November 4 Gopal Prasad (University of Michigan) Isospectrality of compact locally symmetric spaces and weak commensurability of arithmetic groups
November 11 Donghai Pan (Stanford) Galois cyclic covers of the projective line and pencils of Fermat hypersurfaces
November 25 no seminar (Thanksgiving break)
December 2 Giulia Sacca (Stonybrook University) Intermediate Jacobians and hyperKahler manifolds
January 13, 4-4:45 pm Hannah Larson (Harvard) Lines on hypersurfaces with certain normal bundles
January 13, 5-6 pm Gavril Farkas (Humboldt University) K3 surfaces of genus 14 via cubic fourfolds
Thursday, January 19, 12:15-1 pm in 384-I
Ashvin Swaminathan (Harvard) Inflection Points of Linear Systems on Families of Curves
January 20, 3:45-4:45 pm Jake Levinson (Michigan) Boij-Soderberg Theory for Grassmannians
January 20, 5-6 pm Ben Bakker (Georgia) A global Torelli theorem for singular symplectic varieties
January 27 Srikanth Iyengar (Utah) A local Serre duality for modular representations of finite groups (and group schemes)
February 3 Steven Sam (Wisconsin) Secant varieties of Veronese embeddings
February 10 Emily Clader (SFSU) Double ramification cycles and tautological relations
February 17 Ionut Ciocan-Fontanine (Minnesota) Wall-crossing in quasimap theory
February 24 Dusty Ross (SFSU) Genus-One Landau-Ginzburg/Calabi-Yau Correspondence
April 7 no seminar
April 8-9 Western Algebraic Geometry Symposium (at UBC)
April 14 Dhruv Ranganathan (MIT) A Brill-Noether theorem for curves of a fixed gonality
April 21 Erik Carlsson (Davis) Geometry behind the shuffle conjecture
May 12 Arnav Tripathy (Harvard) Motivic Donaldson-Thomas invariants of K3 times an elliptic curve
May 19 John Lesieutre (UIC) A projective variety with discrete, non-finitely generated automorphism group
May 26 Yuchen Liu (Princeton) Construction of hyperbolic cyclic covers
Posted in seminars

abstracts for 2015-16 seminars

(The seminar webpage is here.)

Mihnea Popa (Northwestern University)
September 25, 2015

Positivity for Hodge modules and geometric applications

M. Saito’s theory of Hodge modules provides a powerful generalization of classical Hodge theory that is beginning to find basic applications to birational geometry. One of the main reasons for this is that generalizations of Hodge bundles and de Rham complexes arising in this context satisfy analogues of well-known vanishing and positivity theorems. I will review a number of such results that have been obtained recently, and how they can be applied to deduce new statements regarding holomorphic one-forms and families of varieties of general type. Much of this work is joint with C. Schnell. This talk is intended for a wide algebro-geometric audience, and will contain a review of necessary background from Hodge theory.

Tony Pantev (University of Pennsylvania)
October 2, 2015

Symplectic geometry, foliations, and potentials

I will explain how Lagrangian foliations in derived algebraic symplectic geometry give rise to global potentials. I will also give natural constructions of isotropic foliations on moduli
spaces and will discuss the associated potentials. I will give applications to the moduli of representations of fundamental groups and to non-abelian Hodge theory. This is based on joint works with Calaque, Katzarkov, Toen, Vaquie, and Vezzosi.  This talk is intended for a general algebro-geometric audience, and no advanced background will be assumed.

Alex Perry (Harvard)
October 30, 2015

Categorical joins

Homological projective duality is a powerful theory developed by Kuznetsov for studying the derived categories of varieties. It can be thought of as a categorification of
classical projective duality. I will describe a categorical version of the classical join of two projective varieties, and its relation to homological projective duality. I will discuss some applications to the structure of the derived categories of Fano varieties and to derived equivalences of Calabi-Yau varieties. This is work in progress with Alexander Kuznetsov.

Zijun Zhou (Columbia)
November 13, 2015

Relative orbifold Donaldson-Thomas theory and local gerby curves

In this talk I will introduce the generalization of relative Donaldson-Thomas theory to 3-dimensional smooth Deligne-Mumford stacks. We adopt Jun Li’s construction of expanded pairs and degenerations and prove an orbifold DT degeneration formula. I’ll also talk about the application in the case of local gerby curves, and its relationship to the work of Okounkov-Pandharipande and Maulik-Oblomkov.

Vikraman Balaji (Chennai Mathematical Institute)
November 20, 2015, 2:30-3:30

Degeneration of moduli of Higgs bundles on curves

In this this talk I will discuss the construction of a degeneration of the moduli space of Higgs bundles on smooth curves, as the smooth curve X degenerates to a nodal curve with a single node. As an application we get new compactifications of the Picard variety for smooth curves degenerating to irreducible nodal curves (with multiple nodes) which have analytic normal crossing singularities.

Jose Rodriguez (University of Chicago)
November 20, 2015

Numerically computing Galois groups with Bertini.m2

Galois groups are an important part of number theory and algebraic geometry. To a parameterized system of polynomial equations one can associate a Galois group whenever the system has k (finitely many) nonsingular solutions generically. This Galois group is a subgroup of the symmetric group on k symbols. Using random monodromy loops it has already been shown how to compute Galois groups that are the full symmetric group. In this talk, we show how to compute Galois groups that are proper subgroups of the full symmetric group. We conclude with an implementation using Bertini.m2, an interface to the numerical algebraic geometry software Bertini through Macaulay2. This is joint work with Jonathan Hauenstein and Frank Sottile.

Yukinobu Toda (Institute for the Physics and the Mathematics of the Universe, University of Tokyo)
December 4, 2015, 2:30-3:30 (because of the department party)

Non-commutative deformations and Donaldson-Thomas invariants

In this talk, I will show the existence of certain global non-commutative structures on the moduli spaces of stable sheaves on algebraic varieties, whose formal completion at a closed point gives the pro-representable hull of the non-commutative deformation functor of the sheaf developed by Laudal, Eriksen, Segal and Efimov-Lunts-Orlov. I will then introduce the generating series through integrations over Hilbert schemes of points on these NC structures. When the underlying variety is a Calabi-Yau 3-fold, and the moduli space of stable sheaves satisfy some assumptions, this generating series admits a product expansion described by generalized DT invariants. This formula explains the dimension formula of Donovan-Wemyss’s contractions algebras for floppable curves on 3-folds in terms of genus zero Gopakumar-Vafa invariants.

Aaron Landesman (Harvard)
Tuesday, January 5, 2016 in 384-H (Student Algebraic Geometry Seminar)

Interpolation of Projective Varieties

In this talk, we discuss interpolation of projective varieties through points. It is well known that one can find a rational normal curve in \mathbb P^n through n+3 general points. More recently, it was shown that one can always find nonspecial curves through the expected number of general points. We consider the generalization of this question to varieties of all dimensions and explain why rational normal scrolls satisfy interpolation. Time permitting, we’ll also discuss joint work with Anand Patel on interpolation for del Pezzo surfaces and present several interesting open interpolation problems. We’ll place particular emphasis on explaining the standard techniques used to solve interpolation problems: deformation theory, specialization, degeneration, and the gale transform.

Tonghai Yang (Wisconsin)
January 15, 2016, 2:30-3:30 pm (joint with Number Theory)

Generating series of arithmetic divisors on a Shimura variety of unitary type (n-1, 1)

In this talk we will define a generating series of arithmetic divisors on a Shimura variety of unitary type (n-1, 1), and give a rough idea how to prove that it is modular. This can be viewed as generalization of modular form of weight 2 of Heeger divisors used in Gross-Zagier formula and its generalization Yuan-Zhang-Zhang formula. If time permits, I will mention its application to a special case of Colmez conjecture and `integral’ generalization of the Gross-Zagier and Yuan-Zhang-Zhang formula. This is a joint work with Bruinier, Howard, Kudla, and Rapoport.

Sheldon Katz (UIUC)
January 15, 2016

Stable pair invariants of elliptically fibered Calabi-Yau threefolds and Jacobi forms

I explain the conjectural description of the generating function of stable pair invariants of elliptically fibered Calabi-Yau threefolds with fixed base class and variable fiber class in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base class.  This talk is based on joint work with Albrecht Klemm and Minxin Huang.

Jacob Tsimerman (Toronto)
January 22, 2016

Bounding 2-torsion in class groups

(joint with Bhargava, Shankar, Taniguchi, Thorne, and Zhao) Zhang’s conjecture asserts that for fixed positive integers m, n, the size of the m-torsion in the class group of a degree n number field is smaller than any power of the discriminant. In all but a handful of cases, the best known result towards this conjecture is the ”convex” bound given by the Brauer-Siegel Theorem. We make progress on this conjecture by giving a `”subconvex” bound on the size of the 2-torsion of the class group of a number field in terms of its discriminant, for any value of n. The proof is surprisingly elementary, and we give several applications of this result stemming from the case of cubic fields, including improved bounds on the number of A_4 fields, and on the number of integer points an elliptic curve can have. Along the way, we prove a surprising result on the shape of the lattice of the ring of integers of a number field. Namely, we show that such a lattice is very limited in how ‘skew’ it can be.

Vivek Shende (Berkeley)
January 29, 2016

Cluster varieties from Legendrian knots

We give a uniform geometric explanation for the existence of cluster structures on the positroid strata, character varieties, wild character varieties, etc.

The point is that such spaces can be identified as the moduli spaces of sheaves with microsupport in an appropriate Legendrian knot. In general, an exact Lagrangian with asymptotics in the knot gives rise to an algebraic torus chart on the moduli space. We construct such fillings for alternating knots, so get charts in correspondence with alternating representatives of the knot in question. Such can be enumerated by bicolored graphs, and we recover the various aspects of the bicolored graph theory — the boundary measurement map of Postnikov; the cluster structure on the positroid varieties, the analogous construction on the character varieties due to Fock and Goncharov — by computing the Floer homology between our fillings.

This talk presents joint work with David Treumann, Harold Williams, and Eric Zaslow.

Izzet Coskun (University of Illinois at Chicago)
April 15, 2016

Birational geometry of moduli spaces of sheaves on surfaces

In this talk, I will discuss recent developments in the birational geometry of moduli spaces of sheaves on surfaces motivated by Bridgeland stability conditions. After reviewing joint work with Arcara, Bertram, Huizenga and Woolf in the case of the projective plane, I will describe how to use Bridgeland stability conditions to construct nef divisors on moduli spaces of sheaves on surfaces. I will illustrate the theory with examples on the projective plane, the quadric surface and some general type surfaces. This is joint work with Jack Huizenga.

Zhiwei Yun (Stanford)
April 29, 2016

Intersection numbers and higher derivatives of L-functions for function fields

In joint work with Wei Zhang, we prove a higher derivative analogue of the Waldspurger formula and the Gross-Zagier formula in the function field setting under some unramifiedness assumptions. Our formula relates the self-intersection number of certain cycles on the moduli of Drinfeld Shtukas for GL(2) to higher derivatives of automorphic L-functions for GL(2).

Shouwu Zhang (Princeton)
Thursday, May 12, 2016, 4:30 pm, 380-W

Lecture 1 of Distinguished Lecture Series: Rational points: ABC conjecture and Szpiro’s conjecture

I will talk about ABC conjecture and its relation to Szpiro’s conjecture, Milnor’s proof of Szpiro’s inequality (before it was known as Szpiro’s conjecture) and its relation to hyperbolic geometry, and relation between Szpiro’s conjecture and Landau—Siegel conjecture on zeros of L-functions.

Shouwu Zhang (Princeton)
Friday, May 13, 2016, 2:30 pm, 383-N

Lecture 2 of Distinguished Lecture Series: Torsion points and preperiodic points: Manin—Mumford’s conjecture and its dynamical analogue

I will talk about techniques used in different proofs of Manin-Mumford’s conjecture and its analogue in dynamical system: p-adic rigid geometry (Raynaud), o-minimality geometry (Pila—Zannier), Arakelov geometry (Ullmo—Zhang), and perfectoid geometry (Xie).

Shouwu Zhang (Princeton)
Monday, May 16, 2016, 4 pm, 384-H

Lecture 1 of Distinguished Lecture Series: CM points and derivatives of L-dunctions: Andre—Oort’s conjecture and Colmez’ conjecture

I will talk about recent work of Tsimerman about reducing Andre—Oort’s conjecture to an averaged version of Colmez’ conjecture, and some related work on derivatives of L-functions by Zhiwei Yun and Wei Zhang using Drinfeld’s moduli of Shtukas, and by Xinyi Yuan using Shimura curves.

Ronen Mukamel (Rice)
May 20, 2016 (joint with Informal Geometry and Topology Seminar)

Equations for real multiplication in genus two and applications to Teichmuller curves

I will describe methods for computing with genus two curves whose Jacobians admit real multiplication and discuss applications of these methods to Teichmuller curves. This is joint work with Abhinav Kumar.

Colleen Robles (Duke)
May 27, 2016

Degenerations of Hodge structures

Two motivating questions from algebraic geometry are: how can a smooth projective variety degenerate? and what are the “relations” between two such degenerations? One way to gain insight into these questions is to ask the analogous questions of invariants associated with the smooth projective varieties. In this case, the invariant that we have in mind is a polarized Hodge structure. (And indeed detailed analysis of degenerations of polarized Hodge structures can be used to better understand degeneration of smooth projective varieties, and moduli spaces and their compactifications.)

I will explain how the work of Cattani, Kaplan and Schmid allows us to view a polarized limiting mixed Hodge structure (PLMHS) as a degeneration of a polarized Hodge structure. There is a notion of “polarized relation” between PLMHS that encodes information on how varieties may degenerate within a family. I will give a classification of PLMHS and their polarized relations in terms of Hodge diamonds (discrete data associated with a PLMHS), effectively answering the Hodge-theoretic analogs of the two motivating questions above. This is joint work with Matt Kerr.

Posted in Uncategorized

stanford algebraic geometry seminar 2015-16

 

Fridays 3:45-4:45 in 383-N (with exceptions)

Click on the title to see the abstract (if available).  (For earlier talks in this seminar, click here.  For related seminars, click here.  For the department webpage for the algebraic geometry seminar, click here.) For more information, please contact Jun Li or Ravi Vakil.

September 25 at 3 pm Mihnea Popa (Northwestern) Positivity for Hodge modules and geometric applications
October 2 Tony Pantev (U Penn) Symplectic geometry, foliations, and potentials
October 17-18 weekend Western Algebraic Geometry Symposium (at the University of Washington) Aravind Asok (USC), Yiwei She (Columbia), Rekha Thomas (UW), Nikolaos Tziolas (University of Cyprus), Alena Pirutka (Courant), Brian Osserman (UC Davis), Valery Alexeev (U Georgia)
October 23 no seminar
October 30 Alex Perry (Harvard) Categorical joins
November 13 Zijun Zhou (Columbia) Relative orbifold Donaldson-Thomas theory and local gerby curves
November 20 (2:30-3:30) Vikraman Balaji (Chennai Mathematical Institute) Degeneration of moduli of Higgs bundles on curves
November 20 (3:45-4:45) Jose Rodriguez (Chicago) Numerically computing Galois groups with Bertini.m2
November 27 no seminar (Thanksgiving break)
December 4 (2:30-3:30, because of the department party) Yukinobu Toda (Institute for the Physics and Mathematics of the Universe) Non-commutative deformations and Donaldson-Thomas invariants
January 5 (Student Algebraic Geometry Seminar outside speaker), in 383-H Aaron Landesman (Harvard) Interpolation of Projective Varieties
January 15 (joint Algebraic Geometry – Number theory seminar),  2:30-3:30 Tonghai Yang (Wisconsin) Generating series of arithmetic divisors on a Shimura variety of unitary type (n-1, 1)
January 15 Sheldon Katz (UIUC) Stable pair invariants of elliptically fibered Calabi-Yau threefolds and Jacobi forms
January 22 (joint Algebraic Geometry – Number theory seminar) Jacob Tsimerman (Toronto) Bounding 2-torsion in class groups
January 29 Vivek Shende (Berkeley) Cluster varieties from Legendrian knots
April 15 Izzet Coskun (UIC) Birational geometry of moduli spaces of sheaves on surfaces
April 29 Zhiwei Yun (Stanford) Intersection numbers and higher derivatives of L-functions for function fields
Thursday May 12, 4:30 pm in 380-W Shouwu Zhang (Princeton) Rational points: ABC conjecture and Szpiro’s conjecture (first in a Distinguished Lecture Series)
May 13, 2:30 pm in 383-N Shouwu Zhang (Princeton) Torsion points and preperiodic points: Manin—Mumford’s conjecture and its dynamical analogue (second in a Distinguished Lecture Series)
Monday May 16, 4 pm in 384-H Shouwu Zhang (Princeton) CM points and derivatives of L-dunctions: Andre—Oort’s conjecture and Colmez’ conjecture (third in a Distinguished Lecture Series)
May 20 (joint with the Informal Geometry and Topology Seminar) Ronen Mukamel (Rice) Equations for real multiplication in genus two and applications to Teichmuller curves
May 27 Colleen Robles (Duke) Degenerations of Hodge structures

Fall 2016 speakers:  Christian Schnell (Stony Brook), Giulia Sacca (Stony Brook), and (although he doesn’t know it yet) Michael Kemeny (Stanford)

Posted in seminars | Tagged