# Algebraic geometry and moduli seminar

## Autumn 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
16 September 2015
13:30-15:00
Dr. Jérémy Guéré
Humboldt Universität, Berlin
HG G 43
Abstract: I will explain how to derive tautological relations in the moduli space of stable curves from a vanishing of the cohomology of some Koszul complex. The vanishing property comes from a result of Green on the study of base-point free linear systems and the Koszul complex comes from the algebraic definition of Witten r-spin class. Precisely, our tautological relations hold in the Chow ring of the moduli space of r-spin curves and do not rely on any semi-simplicity condition. I will also discuss the information we obtain on this Witten r-spin class, its generalization to some non-semi-simple cohomological field theories, and its application to the double ramification hierarchy recently introduced by Buryak.
18 September 2015
16:00-17:15
Dr. Margherita Lelli-Chiesa
SNS Pisa
HG G 43
Abstract: Severi varieties and Brill-Noether theory of curves on K3 surfaces is well understood. Quite little is known for curves lying on abelian surfaces. Given a general abelian surface S with polarization L of type (1,n), we will first prove non-emptiness and regularity of the Severi variety parametrizing d-nodal curves C in the linear system |L|. We will then study the gonality of the normalization of C: even in the smooth case, this is not constant when moving C in |L|. In the last part of the seminar we will focus on linear series of type g^r_d with r>=2. We will show that, as soon as the Brill-Noether number is negative and some other inequalities are satisfied, the locus |L|^r_d of smooth curves in |L| possessing a g^r_d is nonempty and has the expected dimension. As an application, we obtain the existence of a component of the Brill-Noether locus M^r_{g,d} having the expected codimension in the moduli space of curves M_g. This is a joint work with A. L. Knutsen and G. Mongardi.
23 September 2015
13:30-15:00
Prof. Dr. Rahul Pandharipande
ETHZ
Twisted canonical divisors HG G 43
30 September 2015
13:30-15:00
Dr. Javier Fresan
ETH Zürich
HG G 43
Abstract: From the viewpoint of an algebraic geometer, Feynman amplitudes are certain periods associated to graphs and the theory of motives provides new insights into the nature of these numbers. In the first talk I will introduce Feynman amplitudes, the motives associated to them and discuss the question whether they are mixed Tate or not. In the second talk I will report on a joint work with Omid Amini, Spencer Bloch and José Ignacio Burgos Gil in which we relate the quotient of Szymanzik polynomials appearing in Feynman amplitudes to the asymptotic behavior of the archimedean height in a family of stable rational curves. With (lots of) good will, this can be seen as a first step towards understanding QFT as a limit of string theory.
2 October 2015
16:00-17:15
Dr. Javier Fresan
ETH Zürich
HG G 43
Abstract: From the viewpoint of an algebraic geometer, Feynman amplitudes are certain periods associated to graphs and the theory of motives provides new insights into the nature of these numbers. In the first talk I will introduce Feynman amplitudes, the motives associated to them and discuss the question whether they are mixed Tate or not. In the second talk I will report on a joint work with Omid Amini, Spencer Bloch and José Ignacio Burgos Gil in which we relate the quotient of Szymanzik polynomials appearing in Feynman amplitudes to the asymptotic behavior of the archimedean height in a family of stable rational curves. With (lots of) good will, this can be seen as a first step towards understanding QFT as a limit of string theory.
14 October 2015
13:30-15:00
Dr. Paul Johnson
University of Sheffield
HG G 43
Abstract: The Hilbert scheme of n points on a complex surface is a smooth manifold of dimension 2n. Their topology has beautiful structure related to physics, representation theory, and combinatorics. For instance, Göttsche's formula gives a product formula for generating functions for their Betti numbers. Hilbert schemes of points on C^2/G, for G a finite group, are also smooth, and when G is abelian their topology is encoded in the combinatorics of partitions. When G is a subgroup of SL_2, the topology is well understood and in terms of cores and quotients of partitions. Following Gusein-Zade, Luengo and Melle-Hernández we study general abelian G, stating a conjectural product formula, and proving a homological stability result using a generalization of cores and quotients.
16 October 2015
16:00-17:15
Dr. Paul Johnson
University of Sheffield
HG G 43
Abstract: The Hilbert scheme of n points on a complex surface is a smooth manifold of dimension 2n. Their topology has beautiful structure related to physics, representation theory, and combinatorics. For instance, Göttsche's formula gives a product formula for generating functions for their Betti numbers. Hilbert schemes of points on C^2/G, for G a finite group, are also smooth, and when G is abelian their topology is encoded in the combinatorics of partitions. When G is a subgroup of SL_2, the topology is well understood and in terms of cores and quotients of partitions. Following Gusein-Zade, Luengo and Melle-Hernández we study general abelian G, stating a conjectural product formula, and proving a homological stability result using a generalization of cores and quotients.
21 October 2015
13:30-15:00
Dr. Penka Georgieva
Université Paris-Jussieu
HG G 43
Abstract: We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the quintic threefold. Our approach to the orientability problem is based entirely on the topology of real bundle pairs over symmetric surfaces. This allows us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces. In reasonably regular cases, these invariants can be used to obtain lower bounds for counts of real curves of arbitrary genus. Joint work with A. Zinger.
23 October 2015
16:00-17:15
Amitai Zernik
Hebrew University of Jerusalem
HG G 43
Abstract: The Atiyah-Bott localization formula has become a valuable tool for computation of symplectic invariants given in terms of integrals on the moduli spaces of holomorphic stable maps. In contrast, the open'' moduli spaces, of stable maps of marked Riemann surfaces with boundary, have boundaries, and these must be taken into account in order to apply fixed point localization. Homological perturbation for twisted $A_{\infty}$ algebras allows one to write down expressions which effectively eliminate the boundaries in genus zero, so one can define equivariant invariants and compute them using localization. These invariants specialize to the open Gromov-Witten invariants, and in particular produce new combinatorial expressions for Welschinger's signed counts of real rational plane curves in terms of summation over certain even-odd diagrams. Time permitting, we'll discuss the two-sided information flow with the intersection theory of Riemann surfaces with boundary (mapping to a point), which lends evidence to a conjectural generalization of the localization formula to higher genus. Joint work with Jake Solomon.
30 October 2015
16:00-17:15
Prof. Dr. Johannes Walcher
Universität Heidelberg
HG G 43
Abstract: I will report on work in progress with Richard Eager and Sam Selmani, in which we adapt the spectral network machinery of Gaiotto-Moore-Neitzke to give a B-model description of the (known) BPS spectrum of some simple Calabi-Yau three-folds.
4 November 2015
13:30-15:00
Prof. Dr. Ionut Ciocan-Fontanine
Uniersity of Minnesota
HG G 43
Abstract: 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, focusing primarily on the case of complete intersections in projective space. Some of its numerical consequences will be discussed as well. This is joint work with Bumsig Kim.
11 November 2015
13:30-15:00
Dr. Dustin Ross
University of Michigan
HG G 43
Abstract: In this three-part lecture series, I will describe Givental's axiomatic framework for studying GW theory, along with various problems to which the setup is particularly well-suited. The first two lectures will be devoted to a discussion of Givental's formalism, along with a survey of applications including mirror symmetry, the crepant transformation conjecture, and quasi-map wall-crossing. This survey spans twenty years of research in GW theory. In the third lecture, I will discuss recent work with Emily Clader in which we apply a number of the previously discussed tools to generalize the Landau-Ginzburg/Calabi-Yau correspondence beyond hypersurfaces.
13 November 2015
16:00-17:15
Prof. Dr. Kristian Ranestad
University of Oslo
HG G 43
Abstract: The discriminant of a P3 of rank 4 quadrics is a quartic surface with (in general ) 10 nodes. If the quadrics are real, how many nodes lie on the semidefinite locus? Examples and many pictures. (Report on work with Ottem, Sturmfels and Vincent)
18 November 2015
13:30-15:00
Dr. Dustin Ross
University of Michigan
Curve counting and crepant resolutions HG G 43
25 November 2015
13:30-15:00
Prof. Dr. Renzo Cavalieri
HG F 33.5
Abstract: This three-talk series will focus on an approach to the study of open GW invariants, which is available when the target space is equipped with a well behaved torus action. In this case, open invariants may be "defined via Atyiah-Bott localization". This approach was pioneered by Katz-Liu, then generalized by Brini-Cavalieri and Ross. Such families of invariants are then "tested" in various ways: one can compare them with physical predictions, and/or observe how naturally they interact with the mathematical structures and features of Gromov-Witten theory. In the first talk of this series we will become familiar with how to define (and compute) open invariants via localization. In the second and third talks we will then focus of how to encode open invariants in terms of Givental formalism. Such language allows us to formulate, and verify in several families of examples, an "open crepant transformation conjecture" which is a natural generalization of the CCIT/Ruan closed statement (and follows from it). Finally we would like to discuss the unexpected relationship of the disk tensor (a central object in our open theory), with Iritani's theory of integral structures in quantum cohomology. All work reported has been carried out in collaboration with subsets of {Andrea Brini, Dusty Ross}.
27 November 2015
16:00-17:15
Prof. Dr. Hannah Markwig
Universität des Saarlandes
HG G 43
Abstract: Tropical geometry can be viewed as an efficient tool to organize degenerations. We review how curves in surfaces can be counted using tropical geometry. These techniques are related to the Fock space approach to count curves on surfaces initiated by Cooper-Pandharipande, via floor diagrams (which can be viewed as the combinatorial essence of a tropical curve count) (following Block-Goettsche). Our own contribution relates the tropical and the Fock space approach for descendant Gromov-Witten invariants. Joint work with Renzo Cavalieri, Paul Johnson and Dhruv Ranganathan
2 December 2015
13:30-15:00
Dr. Dustin Ross
Uniersity of Michigan
HG G 43
Abstract: In this three-part lecture series, I will describe Givental's axiomatic framework for studying GW theory, along with various problems to which the setup is particularly well-suited. The first two lectures will be devoted to a discussion of Givental's formalism, along with a survey of applications including mirror symmetry, the crepant transformation conjecture, and quasi-map wall-crossing. This survey spans twenty years of research in GW theory. In the third lecture, I will discuss recent work with Emily Clader in which we apply a number of the previously discussed tools to generalize the Landau-Ginzburg/Calabi-Yau correspondence beyond hypersurfaces.
4 December 2015
15:45-17:00
Dr. Qingtao Chen
ETH Zürich
HG G 43
Abstract: I will first briefly review the history of large N duality and Labastida-Marino-Ooguri-Vafa Conjecture. Then I will explain in detail how we obtain these congruent skein relations for (reformulated) colored HOMFLY-PT invariants from the idea from framed LMOV conjecture. I will also discuss the congruent skein relations of colored Jones polynomials and SU(n) invariants. Finally I will report a recent discovery of Volume Conjectures and cyclotomic expansion for SU(n) invariants. This is a joint work with K. Liu, P. Peng and S. Zhu.
9 December 2015
13:30-15:00
Prof. Dr. Renzo Cavalieri
HG G 43
Abstract: This three-talk series will focus on an approach to the study of open GW invariants, which is available when the target space is equipped with a well behaved torus action. In this case, open invariants may be "defined via Atyiah-Bott localization". This approach was pioneered by Katz-Liu, then generalized by Brini-Cavalieri and Ross. Such families of invariants are then "tested" in various ways: one can compare them with physical predictions, and/or observe how naturally they interact with the mathematical structures and features of Gromov-Witten theory. In the first talk of this series we will become familiar with how to define (and compute) open invariants via localization. In the second and third talks we will then focus of how to encode open invariants in terms of Givental formalism. Such language allows us to formulate, and verify in several families of examples, an "open crepant transformation conjecture" which is a natural generalization of the CCIT/Ruan closed statement (and follows from it). Finally we would like to discuss the unexpected relationship of the disk tensor (a central object in our open theory), with Iritani's theory of integral structures in quantum cohomology. All work reported has been carried out in collaboration with subsets of {Andrea Brini, Dusty Ross}.
11 December 2015
16:00-17:15
Prof. Dr. Renzo Cavalieri
HG G 43
Abstract: This three-talk series will focus on an approach to the study of open GW invariants, which is available when the target space is equipped with a well behaved torus action. In this case, open invariants may be "defined via Atyiah-Bott localization". This approach was pioneered by Katz-Liu, then generalized by Brini-Cavalieri and Ross. Such families of invariants are then "tested" in various ways: one can compare them with physical predictions, and/or observe how naturally they interact with the mathematical structures and features of Gromov-Witten theory. In the first talk of this series we will become familiar with how to define (and compute) open invariants via localization. In the second and third talks we will then focus of how to encode open invariants in terms of Givental formalism. Such language allows us to formulate, and verify in several families of examples, an "open crepant transformation conjecture" which is a natural generalization of the CCIT/Ruan closed statement (and follows from it). Finally we would like to discuss the unexpected relationship of the disk tensor (a central object in our open theory), with Iritani's theory of integral structures in quantum cohomology. All work reported has been carried out in collaboration with subsets of {Andrea Brini, Dusty Ross}.
16 December 2015
13:30-15:00
Dr. Dustin Ross
University of Michigan
HG G 43
Abstract: In this three-part lecture series, I will describe Givental's axiomatic framework for studying GW theory, along with various problems to which the setup is particularly well-suited. The first two lectures will be devoted to a discussion of Givental's formalism, along with a survey of applications including mirror symmetry, the crepant transformation conjecture, and quasi-map wall-crossing. This survey spans twenty years of research in GW theory. In the third lecture, I will discuss recent work with Emily Clader in which we apply a number of the previously discussed tools to generalize the Landau-Ginzburg/Calabi-Yau correspondence beyond hypersurfaces.

Organizers: Rahul Pandharipande

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