# Algebraic geometry and moduli seminar

## Spring Semester 2015

Note: The highlighted event marks the next occurring event and events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

Date / Time Speaker Title Location
* 18 February 2015
14:00-16:00
Prof. Dr. Paul Biran
ETH Zurich, Switzerland
The Lagrangian cubic equation HG G 19.2
* 20 February 2015
16:00-17:30
ETH Zurich
HG G 43
Abstract: Pixton recently has proposed a formula for the double ramification (DR) cycle, which describes the class of the locus of curves admitting a map to the projective line with specified ramification conditions along 0 and infinity. His formula takes the form of an inhomogenous class which is conjectured to vanish above some degree and whose top degree part coincides with the DR-cycle. We discuss a proof of the vanishing and its relation to other relations in the cohomology of the moduli space of curves.
25 February 2015
13:30-15:30
Ran Tessler
Hebrew University of Jerusalem
HG G 43
Abstract: We shall recall the definition of open descendent integrals in genus 0 (a joint work with Rahul Pandharipande and Jake Solomon). Then, after collecting some facts from related branches of geometry and topology, we will find an effective formula for the descendent integrals in terms of weighted sums over graphs. If time permits we shall briefly define the high genus descendents (joint with Solomon), show how to modify the combinatorial formula for these cases.
* 27 February 2015
16:00-17:15
Ran Tessler
Hebrew University of Jerusalem
HG G 43
Abstract: We shall recall the definition of open descendent integrals in genus 0 (a joint work with Rahul Pandharipande and Jake Solomon). Then, after collecting some facts from related branches of geometry and topology, we will find an effective formula for the descendent integrals in terms of weighted sums over graphs. If time permits we shall briefly define the high genus descendents (joint with Solomon), show how to modify the combinatorial formula for these cases.
4 March 2015
13:30-15:30
Johannes Schmitt
ETH Zürich
A compactification of the moduli space of self maps of CP1 HG G 43
18 March 2015
13:30-15:30
Jérémy Guéré
Humboldt Universität Berlin
HG G 43
Abstract: In this talk, I will first explain how to compute FJRW invariants in genus zero for a large class of polynomials which do not satisfy the concavity hypothesis. This should be seen as the counterpart of Gromov--Witten theory for hypersurfaces, in cases where convexity fails. A complete algebraic description of the FJRW virtual class has been recently provided by Polishchuk and Vaintrob and is based on matrix factorizations. I will sketch how to re-write this definition using a new characteristic class in K-theory, leading to the expression of the virtual class and to a mirror symmetry statement. At last, I will describe how the same ideas compute some FJRW invariants in arbitrary genus.
* 20 March 2015
16:00-17:15
Dr. Dan Petersen
University of Copenhagen
HG G 43
Abstract: The tautological rings of the spaces M_{g,n}^{rt} of n-pointed curves of genus g with rational tails are so far only understood for (g,n) small enough. I will report on ongoing joint work with Mehdi Tavakol, where we have obtained rather precise results when g=3 and n is arbitrary: we have a complete set of generators for the ideal of tautological relations, we can show that the tautological ring is Gorenstein, and we can identify the tautological ring with the part of the cohomology ring which "comes from" the trivial local system and the local system V_{111}.
25 March 2015
13:30-15:30
Prof. Dr. Nicolas Bergeron
Université Pierre et Marie Curie, Paris
HG G 43
Abstract: I will first give a quick introduction to arithmetic manifolds with emphasis on the one associated to orthogonal groups. I will then address the question of constructing classes in the cohomology of these manifolds. To do so we essentially have two basic methods at our disposal. The first, geometric, involves some special (totally geodesic) cycles, and the problem is to show that these contribute to the cohomology. The second -- via a formula of Matsushima that I will recall -- consists in studying if certain unitary representations, called "cohomological", do occur in the automorphic spectrum. I will describe these different methods and relate them.
* 27 March 2015
16:00-17:15
Nicolas Bergeron
Université Pierre et Marie Curie, Paris
HG G 19.2
Abstract: I will first briefly recall that the coarse moduli space of quasi-polarized K3 surfaces of genus g is isomorphic to some arithmetic manifold (orbifold in fact) and interpret the NL-cycles as special cycles. I will then sketch a proof that these special cycles generate the whole cohomology. The proof makes use of the recent stabilization of the twisted trace formula as well as related results of Arthur.
15 April 2015
13:30-15:30
Dr. Alessandro Sisto
ETH Zürich
HG G 43
Abstract: The Mapping Class Group of a (real) surface is the group of its diffeomorphisms modulo the diffeomorphisms isotopic to the identity, and it is the "orbifold fundamental group" of its Moduli Space of hyperbolic metrics. The first part of the talk will mostly be dedicated to motivating the study of Mapping Class Groups. In the second part I will discuss pseudo-Anosov mapping classes, in various senses the most interesting elements of Mapping Class Groups, and then introduce the curve graph, a remarkably useful tool to study Mapping Class Groups. Pseudo-Anosovs are precisely the mapping classes acting on the curve graph with unbounded orbits.
* 17 April 2015
16:00-17:00
Dr. Martijn Kool
Utrecht
HG G 43
Abstract: Gromov-Witten invariants of an algebraic surface S with a smooth connected curve C in the canonical linear system are zero unless the curve class is a multiple d[C]. These invariants have been studied by Lee-Parker, Maulik-Pandharipande and others. They only depend on the the sign (-1)^\chi(\O_S) and the restriction map from cohomology of S to cohomology of C. For d=2, we give a product formula for higher descendent stable stable pair invariants of the 3-fold X=K_S. Using cosection localization, we find that only C^* fixed stable pairs which are thickened in the fibre direction (but not the surface direction) contribute. For arbitrary d, we calculate that part of the theory, which comes from thickenings in the fibre direction only. This is work in progress with R. P. Thomas.
* 17 April 2015
17:15-18:15
Dr. Thomas Hudson
POSTECH, Korea
HG G 43
Abstract: One of the first problems one has to address in Schubert calculus is how to express the fundamental classes of Schubert varieties. In the case of the Chow ring of the flag bundles associated to the classical groups, the Schubert classes were first computed by Fulton via a family of desingularizations known as Bott-Samelson resolutions. Starting from these same geometric inputs, the question can be restated and studied in the more general framework of the oriented cohomology theories of Levine and Morel. The aim of this talk is to present the geometry behind Fulton's original proof and to explain which difficulties one encounters in adapting them to algebraic cobordism, the universal oriented cohomology theory.
* 20 April 2015
15:30-16:30
Dr. Alessandro Sisto
ETH Zürich
HG F 33.1
Abstract: In this talk I will introduce the notion of Gromov-hyperbolicity of a metric space and prove that curve graphs are Gromov-hyperbolic. I will then discuss some aspects of the theory of group actions on Gromov-hyperbolic spaces, including the construction of elements with "interesting dynamics" (pseudo-Anosovs in the case of Mapping Class Groups), free subgroups and time permitting construction of quotients.
29 April 2015
13:30-15:30
Prof. Dr. Dragos Oprea
UC San Diego
HG G 43
Abstract: Generalized theta functions are sections of determinant line bundles over the moduli space of semistable bundles over a fixed smooth curve. The dimension of the space of generalized theta functions is given by the well-studied Verlinde numbers. As the curve varies in moduli, one obtains in this fashion the Verlinde vector bundles over the moduli space of smooth curves. The theory of conformal blocks can be used to extend the Verlinde bundles over the boundary of the moduli space. In these lectures, I will explain some of the ideas involved in the above constructions. In addition, I will explain a formula for the total Chern character of the Verlinde bundles in terms of tautological classes. This is based on joint work with Alina Marian, Rahul Pandharipande, Aaron Pixton and Dimitri Zvonkine.
6 May 2015
13:30-15:30
Prof. Dr. Dragos Oprea
UC San Diego
HG G 43
Abstract: Generalized theta functions are sections of determinant line bundles over the moduli space of semistable bundles over a fixed smooth curve. The dimension of the space of generalized theta functions is given by the well-studied Verlinde numbers. As the curve varies in moduli, one obtains in this fashion the Verlinde vector bundles over the moduli space of smooth curves. The theory of conformal blocks can be used to extend the Verlinde bundles over the boundary of the moduli space. In these lectures, I will explain some of the ideas involved in the above constructions. In addition, I will explain a formula for the total Chern character of the Verlinde bundles in terms of tautological classes. This is based on joint work with Alina Marian, Rahul Pandharipande, Aaron Pixton and Dimitri Zvonkine.
* 8 May 2015
15:45-16:45
Université Paris-Jussieu
Locus of abelian differentials with multiple zeros HG G 43
13 May 2015
13:30-15:30
Prof. Dr. Dragos Oprea
UC San Diego
HG G 43
Abstract: Generalized theta functions are sections of determinant line bundles over the moduli space of semistable bundles over a fixed smooth curve. The dimension of the space of generalized theta functions is given by the well-studied Verlinde numbers. As the curve varies in moduli, one obtains in this fashion the Verlinde vector bundles over the moduli space of smooth curves. The theory of conformal blocks can be used to extend the Verlinde bundles over the boundary of the moduli space. In these lectures, I will explain some of the ideas involved in the above constructions. In addition, I will explain a formula for the total Chern character of the Verlinde bundles in terms of tautological classes. This is based on joint work with Alina Marian, Rahul Pandharipande, Aaron Pixton and Dimitri Zvonkine.
15 May 2015
16:00-17:00
Rahul Pandharipande
ETH Zürich
HG G 43
Abstract: I will sketch a proposal for a compact moduli space of holomorphic/meromorphic differentials.
27 May 2015
13:30-14:45
Junliang Shen
ETH Zürich
HG G 43
Abstract: We describe the motivic theory of the generalized Kummer construction. Our method is to use a power structure over the Grothendieck ring of varieties relative to an abelian variety. As applications, we prove Goettsche's conjecture for geometrically ruled surfaces, and Gulbrandsen's conjecture for Euler characteristics. Moreover, we compute degree 0 motivic Donaldson-Thomas invariants for abelian geometries. This is a joint work with A. Morrison.
* 10 June 2015
13:30-14:30
Sven Prüfer
Augsberg
HG G 43
Abstract: In this talk I will describe how to extend some of Mirzakhani's ideas for calculating psi-class intersections via Weil--Petersson volumes of Deligne--Mumford spaces to Hurwitz numbers and ramification cycles. The main idea is to consider moduli spaces of coverings of bordered Riemann surfaces together with marked points on the boundary and then use symplectic reduction for suitable torus actions. This is still work in progress.
25 June 2015
13:30-14:45
Dr. Nick Sheridan
Princeton Univ.
HG G 43
Abstract: In 1991, string theorists Candelas, de la Ossa, Green and Parkes made a startling prediction for the number of curves in each degree on a generic quintic threefold, in terms of periods of a holomorphic volume form on a mirror manifold'. Givental and Lian, Liu and Yau gave a mathematical proof of this version of mirror symmetry for the quintic threefold (and many more examples) in 1996. In the meantime (1994), Kontsevich had introduced his homological mirror symmetry' conjecture and stated that it would unveil the mystery of mirror symmetry'. I will explain how to prove that the number of curves on the quintic threefold matches up with the periods of the mirror via homological mirror symmetry. I will also attempt to explain in what sense this is less mysterious' than the previous proof. The first talk will introduce the two versions of mirror symmetry: Hodge-theoretic (closed-string) mirror symmetry, and homological (open-string) mirror symmetry.The second talk will explain how open-string mirror symmetry implies closed-string mirror symmetry, via the open-closed string map. This is based on joint work with Sheel Ganatra and Tim Perutz.
2 July 2015
15:15-16:30
Dr. Nick Sheridan
Princeton Univ.
HG G 43
Abstract: In 1991, string theorists Candelas, de la Ossa, Green and Parkes made a startling prediction for the number of curves in each degree on a generic quintic threefold, in terms of periods of a holomorphic volume form on a mirror manifold'. Givental and Lian, Liu and Yau gave a mathematical proof of this version of mirror symmetry for the quintic threefold (and many more examples) in 1996. In the meantime (1994), Kontsevich had introduced his homological mirror symmetry' conjecture and stated that it would unveil the mystery of mirror symmetry'. I will explain how to prove that the number of curves on the quintic threefold matches up with the periods of the mirror via homological mirror symmetry. I will also attempt to explain in what sense this is less mysterious' than the previous proof. The first talk will introduce the two versions of mirror symmetry: Hodge-theoretic (closed-string) mirror symmetry, and homological (open-string) mirror symmetry.The second talk will explain how open-string mirror symmetry implies closed-string mirror symmetry, via the open-closed string map. This is based on joint work with Sheel Ganatra and Tim Perutz.

Organizers: Rahul Pandharipande