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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 13:3015:00 
Dr. Jérémy Guéré Humboldt Universität, Berlin 
From Koszul cohomology to tautological relations in the moduli space of curves  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 basepoint free linear systems and the Koszul complex comes from the algebraic definition of Witten rspin class. Precisely, our tautological relations hold in the Chow ring of the moduli space of rspin curves and do not rely on any semisimplicity condition. I will also discuss the information we obtain on this Witten rspin class, its generalization to some nonsemisimple cohomological field theories, and its application to the double ramification hierarchy recently introduced by Buryak.  
18 September 2015 16:0017:15 
Dr. Margherita LelliChiesa SNS Pisa 
Severi varieties and BrillNoether theory of curves on abelian surfaces  HG G 43  
Abstract: Severi varieties and BrillNoether 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 nonemptiness and regularity of the Severi variety parametrizing dnodal 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 BrillNoether 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 BrillNoether 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 13:3015:00 
Prof. Dr. Rahul Pandharipande ETHZ 
Twisted canonical divisors  HG G 43  
30 September 2015 13:3015:00 
Dr. Javier Fresan ETH Zürich 
Feynman amplitudes and limits of heights I  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 16:0017:15 
Dr. Javier Fresan ETH Zürich 
Feynman amplitudes and limits of heights II  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 13:3015:00 
Dr. Paul Johnson University of Sheffield 
Topology of Hilbert schemes and combinatorics of partitions I  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 GuseinZade, Luengo and MelleHernández 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 16:0017:15 
Dr. Paul Johnson University of Sheffield 
Topology of Hilbert schemes and combinatorics of partitions II  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 GuseinZade, Luengo and MelleHernández 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 13:3015:00 
Dr. Penka Georgieva Université ParisJussieu 
Real GromovWitten theory in all genera  HG G 43  
Abstract: We construct positivegenus analogues of Welschinger's invariants for many real symplectic manifolds, including the odddimensional 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 codimensionone 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 16:0017:15 
Amitai Zernik Hebrew University of Jerusalem 
Fixed point expressions and open GromovWitten theory  HG G 43  
Abstract: The AtiyahBott 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 GromovWitten invariants, and in particular produce new combinatorial expressions for Welschinger's signed counts of real rational plane curves in terms of summation over certain evenodd diagrams. Time permitting, we'll discuss the twosided 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 16:0017:15 
Prof. Dr. Johannes Walcher Universität Heidelberg 
Exponential networks and representations of quivers  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 GaiottoMooreNeitzke to give a Bmodel description of the (known) BPS spectrum of some simple CalabiYau threefolds.  
4 November 2015 13:3015:00 
Prof. Dr. Ionut CiocanFontanine Uniersity of Minnesota 
Wallcrossing in quasimap theory  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 GromovWitten included, depending on the linearization in the GIT problem. I will present a wallcrossing 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 13:3015:00 
Dr. Dustin Ross University of Michigan 
Applications of Axiomatic GromovWitten Theory I  HG G 43  
Abstract: In this threepart lecture series, I will describe Givental's axiomatic framework for studying GW theory, along with various problems to which the setup is particularly wellsuited. 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 quasimap wallcrossing. 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 LandauGinzburg/CalabiYau correspondence beyond hypersurfaces.  
13 November 2015 16:0017:15 
Prof. Dr. Kristian Ranestad University of Oslo 
Quartic spectrahedra  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 13:3015:00 
Dr. Dustin Ross University of Michigan 
Curve counting and crepant resolutions  HG G 43  
25 November 2015 13:3015:00 
Prof. Dr. Renzo Cavalieri Colorado State University 
Open GromovWitten invariants for toric orbifolds I  HG F 33.5  
Abstract: This threetalk 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 AtyiahBott localization". This approach was pioneered by KatzLiu, then generalized by BriniCavalieri 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 GromovWitten 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 16:0017:15 
Prof. Dr. Hannah Markwig Universität des Saarlandes 
Counting curves in surfaces  the tropical and the Fock space approach  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 CooperPandharipande, via floor diagrams (which can be viewed as the combinatorial essence of a tropical curve count) (following BlockGoettsche). Our own contribution relates the tropical and the Fock space approach for descendant GromovWitten invariants. Joint work with Renzo Cavalieri, Paul Johnson and Dhruv Ranganathan  
2 December 2015 13:3015:00 
Dr. Dustin Ross Uniersity of Michigan 
Applications of Axiomatic GromovWitten Theory II  HG G 43  
Abstract: In this threepart lecture series, I will describe Givental's axiomatic framework for studying GW theory, along with various problems to which the setup is particularly wellsuited. 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 quasimap wallcrossing. 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 LandauGinzburg/CalabiYau correspondence beyond hypersurfaces.  
4 December 2015 15:4517:00 
Dr. Qingtao Chen ETH Zürich 
New progress in knot theory inspired by large N duality  HG G 43  
Abstract: I will first briefly review the history of large N duality and LabastidaMarinoOoguriVafa Conjecture. Then I will explain in detail how we obtain these congruent skein relations for (reformulated) colored HOMFLYPT 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 13:3015:00 
Prof. Dr. Renzo Cavalieri Colorado State University 
Open GromovWitten invariants for toric orbifolds II  HG G 43  
Abstract: This threetalk 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 AtyiahBott localization". This approach was pioneered by KatzLiu, then generalized by BriniCavalieri 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 GromovWitten 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 16:0017:15 
Prof. Dr. Renzo Cavalieri Colorado State University 
Open GromovWitten invariants for toric orbifolds III  HG G 43  
Abstract: This threetalk 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 AtyiahBott localization". This approach was pioneered by KatzLiu, then generalized by BriniCavalieri 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 GromovWitten 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 13:3015:00 
Dr. Dustin Ross University of Michigan 
Applications of Axiomatic GromovWitten Theory III  HG G 43  
Abstract: In this threepart lecture series, I will describe Givental's axiomatic framework for studying GW theory, along with various problems to which the setup is particularly wellsuited. 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 quasimap wallcrossing. 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 LandauGinzburg/CalabiYau correspondence beyond hypersurfaces. 
Organizers: Rahul Pandharipande
Archive: SS 17 AS 16 SS 16 AS 15 SS 15 AS 14 SS 14 AS 13 SS 13 AS 12 SS 12 AS 11
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