abstracts for 2014-15 seminars

(The seminar webpage is here.)

Jarod Alper (Australian National University)
September 26, 2014

Associated forms in classical invariant theory

There is an interesting map which assigns to a homogeneous form $f$ on $\mathbb{C}^n$ of degree $d$ with non-vanishing discriminant, a certain form on $\mathbb{C}^n$ of degree $n(d-2)$, which is the Macaulay inverse system of the Milnor algebra of $f$.  It was conjectured in a recent paper by M. Eastwood and A. Isaev that all absolute classical invariants of forms on $\mathbb{C}^n$ of degree $d$ can be extracted from those of forms of degree $n(d-2)$ via this map.  This surprising conjecture was motivated by the well-known Mather-Yau theorem for isolated hypersurface singularities.  I will report on joint work with A. Isaev which settles this conjecture in full generality and proves a stronger statement in the case of binary forms.

Jim Bryan (University of British Columbia)
October 24, 2014

Donaldson-Thomas theory of local elliptic surfaces via the topological vertex

Donaldson-Thomas (DT) invariants of a Calabi-Yau threefold X are fundamental quantum invariants given by (weighted) Euler characteristics of the Hilbert schemes of X. We compute these invariants for the case where X is a so-called local elliptic surface — it is the total space of the canonical line bundle over an elliptic surface. We find that the generating functions for the invariants admit a nice product structure. We introduce a new technique which allows us to use the topological vertex in this computation — a tool which previously could only be used for toric threefolds. As a by product, we discover surprising new identities for the topological vertex. This is joint work with Martijn Kool, with an assist from Ben Young.

October 31, 2014

Interpolation and vector bundles on curves

We aim to address the following: When is there a (smooth) curve of degree $d$ and genus $g$ passing through $n$ general points in $\mathbb{P}^r$. Generalizations ask for the dimension of such curves, or replace the point incidence conditions with higher dimensional linear spaces. We will start by relating these statements to a property of the normal bundle of curves in projective space. Next, we will show how to address these questions for $r = 3$ and $d \geq g + 3$. The demonstrated techniques generalize significantly and lead to an answer to our question for $d \geq g + r$. This is joint work with E. Larson and D. Yang.

Yefeng Shen (Stanford)
November 7, 2014

WDVV equations and Ramanujan identities

The occurrence of modular forms and quasi-modular forms in Gromov-Witten theory is an interesting phenomenon. I will present the following work joint with Jie Zhou. We show that the WDVV equations for elliptic orbifolds are equivalent to the Ramanujan identities for some modular groups. We then apply this to prove the genus zero Gromov-Witten correlation functions for all elliptic orbifolds are quasi-modular forms. Combining with the tautological relations on the moduli space of pointed curves, we also obtain the modularity for all genera. This generalizes an earlier result of Milanov–Ruan and solves a modularity conjecture for the Gromov-Witten theory of the elliptic orbifold curve with four $\mathbb{Z}_2$-orbifold points.

Yiwei Shi (University of Chicago)
November 14, 2014

The Shafarevich conjecture for K3 surfaces

Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves defined over K and with good reduction outside of S is finite. In 1983, Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfaces. Building on the work of Faltings and Andre, we prove the stronger unpolarized Shafarevich conjecture for K3 surfaces.

Sabin Cautis (University of British Columbia)
November 21, 2014

Categorical Heisenberg actions on Hilbert schemes of points

We define actions of certain Heisenberg algebras on the Hilbert schemes of points on ALE surfaces. This lifts constructions of Nakajima and Grojnowski from cohomology to K-theory and derived categories of coherent sheaves. This action can be used to define Lie algebra actions (using categorical vertex operators) and subsequently braid group actions and knot invariants.

François Charles (Paris Sud and MIT)
December 5, 2014, 1:15 pm (unusual time!)

Geometric boundedness results for K3 surfaces

Tate’s conjecture for divisors on algebraic varieties can be rephrased as a finiteness statement for certain families of polarized varieties with unbounded degrees. In the case of abelian varieties, the geometric part of these finiteness statements is contained in Zarhin’s trick. We will discuss such geometric boundedness statements for K3 surfaces over arbitrary fields and holomorphic symplectic varieties, with application to direct proofs of the Tate conjecture for K3 surfaces that do not involve the Kuga-Satake correspondence.

Dan Edidin (Missouri)
January 9, 2015

Strong regular embeddings and the geometry of hypertoric stacks

We explain how the notion of “strong regular embeddings” can be used to compare the geometry of a stack to that of a regularly embedded substack. This theory can be applied to understand the relationship between singular hypertoric varieties and singular Lawrence toric varieties. While this talk is about stacks, the motivating ideas come from simple observations about invariant rings for actions of finite groups.

Evan O’Dorney (Harvard)
January 16, 2015

Canonical rings of $\mathbf{Q}$-divisors on $\mathbf{P}^1$

The canonical ring $S_D = \bigoplus_{d\geq 0} H^0(X, \lfloor dD\rfloor)$ of a divisor $D$ on a projective curve $X$ is a natural object of study; when $D$ is a $\mathbf{Q}$-divisor, it has connections to projective embeddings of stacky curves and rings of modular forms. I will speak about my results from last summer’s Emory REU concerning the generators and relations of $S_D$ for the simplest curve $X = \mathbf{P}^1$. When $D$ contains at most two points, I give a complete description of $S_D$; for general $D$, I give bounds on the generators and relations. I have also proved that the generators (for at most five points) and a Groebner basis of relations between them (for at most four points) depend only on the coefficients in the divisor $D$, not its points or the characteristic of the ground field, and I conjecture that the minimal system of relations varies in a similar way.

Mark Shoemaker (Utah)
January 23, 2015

A proof of the LG/CY correspondence via the crepant resolution conjecture

Given a homogeneous degree five polynomial $W$ in the variables $X_1$, …, $X_5$, we may view $W$ as defining a quintic hypersurface in $\mathbf{P}^4$ or alternatively, as defining a singularity in $[ \mathbf{C}^5/\mathbf{Z}_5]$ where the group action is diagonal. In the first case, one may consider the Gromov-Witten invariants of $\{ W=0 \}$. In the second case, there is a way to construct analogous invariants, called FJRW invariants, of the singularity. The LG/CY correspondence states that these two sets of invariants are related. In this talk I will explain this correspondence and its relation to a much older conjecture, the crepant resolution conjecture (CRC). I will sketch a proof that the CRC is equivalent to the LG/CY correspondence in certain cases using a generalization of the “quantum Serre-duality” of Coates-Givental. This work is joint with Y.-P. Lee and Nathan Priddis.

January 30, 2015, 3:20-4:20

Moduli of degree 4 K3 surfaces revisited

For low degree K3 surfaces there are several way of constructing and compactifying the moduli space (via period maps, via GIT, or via KSBA). In the case of degree 2 K3 surface, the relationship between various compactifications is well understood by work of Shah, Looijenga, and others. I will report on work in progress with K. O’Grady which aims to give similar complete description for degree 4 K3s.

Jack Hall (ANU)
January 30, 2015, 3:30-4:30

A generalization of Luna’s etale slice theorem

Let $X$ be an affine $\mathbb{C}$-variety with an action of a reductive group $G$. A beautiful theorem of Luna is that if $x\in X(\mathbb{C})$ has closed $G$-orbit, then there is a $G_x$-invariant and affine subvariety $W \hookrightarrow X$ containing $x$, where $G_x$ is the stabilizer of $x$, such that the induced map $W\times^{G_x} G \to X$ is $G$-etale. This result is very useful in applications. Unfortunately, direct extensions of Luna’s result to non-affine, singular varieties are not possible. By slightly weakening the conclusion, however, I will describe a generalization of Luna’s etale slice theorem covering these situations, and many more. This is joint work with Alper (ANU) and Rydh (KTH).

Alessandro Chiodo (Jussieu)
February 13, 2015

Néron models of Picard groups by Picard groups

The Néron model provides a universal extension over a discrete valuation ring R of the degree-zero part Pic0CK of the Picard group of a smooth curve CK over K=Frac(R). It is natural to exploit the (relative) Picard functor Pic0CR of a regular (semi)stable reduction CR to describe its Néron model N(Pic0CK). The group Pic0CR is not separated in general, but the Néron model N(Pic0CK) equals Pic0CR modulo the closure of the zero section of Pic0CK (Raynaud, 1970). In some very special cases, we obtain N(Pic0CK) without passing through the quotient by simply singling out the identity component of Pic0CR, that is the group of line bundles of degree 0 on all irreducible components. In general, the quotient does not possess a similar modular interpretation but this talk shows that the Néron model does represent a separated Picard functor of degree-0 line bundles on all irreducible components as soon as we adopt a stack-theoretic stable model of CK.

Daniel Murfet (USC)
February 20, 2015

Computing with hypersurfaces

Associated to an isolated hypersurface singularity is a triangulated category of matrix factorisations. A decade ago Khovanov and Rozansky figured out how to model knots in a bicategory built out of these triangulated categories. Inspired by their construction I will describe how matrix factorisations can be used to construct a model of the polymorphic lambda calculus of Girard, an abstract functional programming language which underlies, for example, Haskell. Beyond the sheer amusement of representing programs by objects of triangulated categories, there are motivations from within logic and computer science that I will also discuss.
Of course I will explain all the logic/comp-sci terms!

Srinivas (Tata Institute for Fundamental Research)
February 27, 2015, 3:20-4:20

Etale motivic cohomology and algebraic cycles

This talk will report on joint work with A. Rosenschon. There are examples (which I’ll briefly discuss) showing that the torsion and co-torsion of Chow groups are complicated, in general, except in the “classical” cases (divisors and 0-cycles, and torsion in codimension 2); also the integral Hodge conjecture is known to fail. Instead, we may (following Lichtenbaum)
consider the etale Chow groups, which coincide with the usual ones if we use rational coefficients; we show that they have better “integral” properties if we work over the complex numbers. In contrast, they can have infinite torsion in some arithmetic situations (the usual Chow groups are conjectured to be finitely generated).

Sándor Kovács (Washington)
February 27, 2015, 4:30-5:30

Projectivity of the moduli space of stable log-varieties

Stable log-varieties are the higher dimensional analogues of stable pointed curves and their moduli spaces are generalizations of $\bar M_{g,n}$. Just as in the curve case, extending the moduli problem to include stable objects leads to a compactification of the moduli space of the smooth ones. (These are known facts due to work of Koll​ár, Viehweg, Koll​ár-Shepherd-Barron, Alexeev, Keel-Mori, Hacon-McKernan-Xu and others). In the higher dimensional case there are many issues that come up that do not appear in dimension one. Furthermore, the log case is far from a straightforward generalization of the absolute case. In particular, Koll​ár’s powerful Ampleness Lemma does not apply in the log case. This talk is an overview report on recent and ongoing work, joint with Zsolt Patakfalvi, in which we prove a number of results: a generalization of Koll​ár’s Ampleness Lemma that works in the log case, various positivity results on pushforwards of relative pluricanonical sheaves and their determinants and as applications of these we prove that any complete moduli space of stable log-varieties is projective and we establish log-subadditvity of Kodaira dimension of log canonical fiber spaces.

Jie Zhou (Perimeter Institute)
April 3, 2015

Counting higher genus curves in local $\mathbf{P}^2$ via mirror symmetry

I will talk about the modularity of the generating series of Gromov-Witten invariants of local $\mathbf{P}^2$ by using mirror symmetry.

The moduli space involved in the mirror side of local $\mathbf{P}^2$ is naturally identified with a modular curve of genus zero. This implies that the mirror of the generating series, as sections of line bundles over the modular curve, are modular forms. The Fricke involution acting on the modular curve exchanges the asymptotic behaviour at the cusps of the modular forms. It induces a duality of the underling physics theories, in a way similar to the electro-magnetic duality in Seiberg-Witten theory.

The talk is based on a joint work with M. Alim, E. Scheidegger and S.-T. Yau.

Y.P. Lee (Utah)
April 10, 2015

A + B model in conifold transitions for Calabi–Yau threefolds

Witten named two topological field theories for Calabi–Yau threefolds A and B models respectively in his explanation of mirror symmetry. In this talk, A model is the Gromov–Witten theory and B model the variation of Hodge structure. All known examples of (simply-connected) Calabi–Yau threefolds are connected by a special kind of surgery, called (extremal) transition, of which the basic case is the conifold transition.

In this talk, I will explain a phenomenon of partial exchange of A and B models when the Calabi–Yau threefold undergoes a conifold transition. This suggests the possibility of an A+B theory which is invariant under transitions and is therefore equivalent for all (simply-connected) Calabi–Yau threefolds.

This talk is based on joint work with H.-W. Lin and C.-L. Wang.

Anand Patel (Boston College)
April 17, 2015

Algebraic cohomology of Hurwitz spaces

I will introduce the general problem of determining Chow rings of the Hurwitz spaces parametrizing branched covers of the projective line. I will present some conjectures as well as recent progress in the degree 3 case (joint with R. Vakil). This talk may be of interest to topologists as well – I will discuss ongoing work on the stabilization of the algebraic cohomology of Hurwitz spaces as the degree and genus parameters tend to infinity.

Simion Filip (Chicago)
May 1, 2015

Hodge theory and arithmetic in Teichmuller dynamics

The dynamics of a billiard ball in a polygon is a classical dynamical system for which many questions remain open. These questions are closely related to a natural action of the group SL(2,R) on the tangent bundle to the moduli space of Riemann surfaces. It can be viewed as a “complexified” geodesic flow. By recent results of Eskin and Mirzakhani, this action of SL(2,R) enjoys rigidity properties akin to Ratner’s theorems – in particular, orbit closures are submanifolds. In this talk, I will explain why these orbit closures are in fact algebraic varieties with interesting arithmetic properties. For instance, they parametrize algebraic curves with real multiplication and torsion conditions on (factors of) their Jacobian. These results depend on extending results about variations of Hodge structures to this special setting (in particular, giving a different approach to some results of Schmid). No background in any of the above topics will be assumed and I will provide the necessary introduction.

Monday May 4, 2015

Specialization of Quintic Threefolds to the Secant Variety

We consider the degeneration of quintic threefolds obtained by running semistable reduction on a general pencil of quintics specializing to the secant variety of a normal elliptic curve in projective 4-space. The question we would like to address is: what are the flat limits of the rational curves in this degeneration? In this talk, I will provide a good “first approximation” to the answer by describing the space of genus zero stable morphisms to the central fiber (as defined by J. Li).

Joe Rabinoff (Georgia Tech)
May 8, 2015

Integration on wide opens and uniform Manin-Mumford

Coleman’s effective version of Chabauty’s method of attacking the Mordell conjecture involves counting zeros of certain p-adic integrals on p-adic open discs, using a Newton polygon argument. Recently, Stoll extended the Chabauty–Coleman method using integration on open discs as well as open annuli, to prove a uniform Mordell conjecture for hyperelliptic curves of fixed genus and small Mordell–Weil rank. We use integration on so-called basic wide open subdomains, the p-adic analogue of a “pair of pants” in a Riemann surface, in order to prove a uniform Manin–Mumford result for curves of a fixed genus with sufficiently degenerate reduction type. This variant of the Chabauty–Coleman method is unique as it can be used to bound *geometric* points on a curve, not just rational points. Other ingredients include potential theory on Berkovich curves, the theory of linear systems on metric graphs, and a comparison of different p-adic integration theories. A special case of this method extends Stoll’s results to all curves of low Mordell–Weil rank.

This work is joint with Eric Katz and David Zureick-Brown. It complements David’s talk on April 13, but it is self-contained and assumes no prior knowledge of Berkovich spaces or the Chabauty–Coleman method.

Andrei Caldararu (Wisconsin)
May 15, 2015

Algebraic proofs of degenerations of Hodge-de Rham complexes

In the first half of the talk I shall present a new algebraic proof of a result of Deligne-Illusie about the degeneration of the Hodge-de Rham spectral sequence. The idea is to reduce the main technical point of their proof to a question about the formality of a derived intersection in an Azumaya space.

In the second half of the talk I shall discuss the main technical difficulty that arises when trying to extend these techniques to obtain a proof of a famous claim of Barannikov-Kontsevich. This claim, which was first proved by analytic methods by Sabbah, expresses the hypercohomology of the twisted de Rham complex in terms of computations with coherent sheaves. It is conceptually the analogue of the Hodge-de Rham degeneration statement for dg categories of matrix factorizations.

This is joint work with Dima Arinkin and Marton Hablicsek.

Alex Perry (Harvard)
May 29, 2015

Derived categories of Gushel–Mukai varieties

A Fano Gushel–Mukai (GM) variety is a Fano variety of Picard number 1, degree 10, and coindex 3. For such a variety $X$, there is a special subcategory $A_X$ of its derived category, which appears to be closely related to its birational geometry. I will discuss a duality operation that exchanges GM varieties of different dimensions and identifies their $A_X$ categories. As a special case, this gives a family of rational GM fourfolds whose category $A_X$ is equivalent to the derived category of a K3 surface. I will also discuss a relation between the derived categories of GM and cubic fourfolds. This is joint work with Alexander Kuznetsov.

Tom Coates (Imperial College London)
June 1, 2015, 4 pm (special algebraic and symplectic geometry seminar)

Mirror Symmetry and Fano Manifolds

Fano manifolds are basic building blocks in algebraic geometry, and the classification of Fano manifolds is a long-standing and important open problem. We explain a surprising connection between Mirror Symmetry and Fano classification. This is joint work with Akhtar, Corti, Galkin, Golyshev, Heuberger, Kasprzyk, Oneto, Petracci, Prince, and Tveiten.

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