Departement Mathematik

Algebraic geometry and moduli seminar

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Herbstsemester 2019

Datum / Zeit Referent:in Titel Ort
20. September 2019
16:00-17:15 		Johannes Schmitt
MPI Bonn
Details

Algebraic Geometry and Moduli Seminar

Titel Tautological rings of the moduli stacks of prestable curves
Referent:in, Affiliation Johannes Schmitt, MPI Bonn
Datum, Zeit 20. September 2019, 16:00-17:15
Ort HG G 43
Abstract The stacks of prestable curves parametrize algebraic curves C with at worst nodal singularities together with a finite set of marked smooth points of C. They contain, as open substacks, the classical moduli spaces of stable curves. In this talk, I will show how many results about the intersection theory of the moduli spaces of stable curves can be generalized to the larger stack of prestable curves. In particular, I define tautological classes and show how to intersect them and compute pullbacks and pushforward under natural morphisms. For the moduli stacks of genus zero curves, it turns out that all algebraic cycles are tautological and I discuss the linear relations between the tautological generators. I will point out the connection to previous work by Fulghesu and Oesinghaus. This is joint work with Younghan Bae.
Tautological rings of the moduli stacks of prestable curvesread_more
HG G 43
* 25. September 2019
16:00-17:15 		Prof. Dr. Tom Graber
California Institute of Technology
Details

Algebraic Geometry and Moduli Seminar

Titel Log stable maps and double ramification cycles
Referent:in, Affiliation Prof. Dr. Tom Graber, California Institute of Technology
Datum, Zeit 25. September 2019, 16:00-17:15
Ort ITS
Abstract I will describe how the modified product of double ramification cycles defined by Holmes, Pixton, and Schmitt naturally arises in localization calculations on the space of log stable maps. This suggests the possibility of using localization methods to find explicit tautological formulas for these classes which is work in progress with Ranganathan, Pandharipande, and Zvonkine.
Log stable maps and double ramification cyclesread_more
ITS
27. September 2019
16:00-17:15 		Dr. Dhruv Ranganathan
University of Cambridge
Details

Algebraic Geometry and Moduli Seminar

Titel Gromov-Witten theory and strict transforms
Referent:in, Affiliation Dr. Dhruv Ranganathan, University of Cambridge
Datum, Zeit 27. September 2019, 16:00-17:15
Ort HG G 43
Abstract Given a simple normal crossings pair (X,D), there are two ways of examining curves in X with prescribed tangency along D. The first is logarithmic Gromov-Witten theory, which is technically robust, has excellent properties, but has proved difficult to compute with. The second, more naively, considers the older relative stable maps theory of Jun Li’s for the various smooth pairs (X,E) where E is a component of D. In good cases, these can be packaged together to form their own virtual curve counting theory for (X,D). I will explain how these two are related, and the fundamental geometry, which concerns strict and total transforms of subvarieties of a toric variety under toric blowup. As an application, I will outline a proof of the local/logarithmic conjecture of van Garrel-Graber-Ruddat, subject to a positivity assumption. This talk is based on recent work with Nabijou, and ongoing speculations with Nabijou and Battistella.
Gromov-Witten theory and strict transformsread_more
HG G 43
9. Oktober 2019
13:30-14:45 		Dr. Maria Yakerson
Universität Osnabrück
Details

Algebraic Geometry and Moduli Seminar

Titel Infinite loop spaces in classical and motivic homotopy theory
Referent:in, Affiliation Dr. Maria Yakerson , Universität Osnabrück
Datum, Zeit 9. Oktober 2019, 13:30-14:45
Ort HG G 19.2
Abstract Classically in homotopy theory, infinite loop spaces are recognized as spaces with an additional structure: grouplike E_{infty}-spaces. The category of such spaces is equivalent to the category of connective spectra. Replacing topological spaces with smooth schemes, we end up in the realm of motivic homotopy theory, where an analogous statement was sought for since the theory has appeared. In this talk, we will recall the modern formulation of the classical recognition principle and discuss the motivic recognition principle. This is joint work with Elden Elmanto, Marc Hoyois, Adeel Khan and Vladimir Sosnilo.
Infinite loop spaces in classical and motivic homotopy theoryread_more
HG G 19.2
11. Oktober 2019
16:00-17:15 		Dr. Maria Yakerson
Universität Osnabrück
Details

Algebraic Geometry and Moduli Seminar

Titel Higher algebraic cobordism
Referent:in, Affiliation Dr. Maria Yakerson, Universität Osnabrück
Datum, Zeit 11. Oktober 2019, 16:00-17:15
Ort HG G 43
Abstract In this talk we discuss applications of the motivic recognition principle to the algebraic cobordism spectrum MGL, which is the motivic analogue of the complex cobordism spectrum MU. We will compute the infinite P^1-loop space of the spectrum MGL and obtain a geometric model for it. This result, together with the computation of suspensions of MGL, will allow us to get an explicit model for the category of MGL-modules. This is joint work with Elden Elmanto, Marc Hoyois, Adeel Khan and Vladimir Sosnilo.
Higher algebraic cobordismread_more
HG G 43
16. Oktober 2019
13:30-14:45 		Dr. Hyenho Lho
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Titel Tautological relations on the moduli space of stable maps
Referent:in, Affiliation Dr. Hyenho Lho, ETH Zürich
Datum, Zeit 16. Oktober 2019, 13:30-14:45
Ort HG G 19.2
Abstract Pixton conjectured a set of tautological relations on the moduli space of stable curves. We prove a set of relations on the moduli space of stable maps to X which naturally generalise the Pixton’s relations. While Pandharipande, Pixton and Zvonkine first proved the Pixton conjecture via the geometry of 3-spin curves, Janda later reproved the Pixton conjecture using equivariant Gromov-Witten theory of P1. We follow the Janda’s approach to study the relations on the moduli space of stable maps to X using equivariant Gromov-Witten theory of P1-bundle over X. This talk is based on the joint work in progress with Younghan Bae.
Tautological relations on the moduli space of stable mapsread_more
HG G 19.2
23. Oktober 2019
13:30-14:45 		Dr. Jérémy Guéré
Université de Grenoble
Details

Algebraic Geometry and Moduli Seminar

Titel Hodge-Gromov-Witten theory
Referent:in, Affiliation Dr. Jérémy Guéré, Université de Grenoble
Datum, Zeit 23. Oktober 2019, 13:30-14:45
Ort HG G 19.2
Abstract Hodge-Gromov-Witten theory of a smooth projective variety X deals with the cap product of the virtual fundamental cycle on the moduli space of stable maps to X with the Euler class of the Hodge vector bundle. I recently studied its deformation invariance to singular varieties and stated my results under the name `Regular Specialization Theorem'. In this first lecture, I will explain the origin of this theorem, its statement, and its proof. I will discuss as well an important application: a computation of genus-zero GW invariants for some hypersurfaces in weighted projective spaces which do not satisfy the so-called convexity property. It is a first step towards a mirror symmetry statement for these hypersurfaces.
Hodge-Gromov-Witten theoryread_more
HG G 19.2
25. Oktober 2019
16:00-17:15 		Dr. Jérémy Guéré
Université de Grenoble
Details

Algebraic Geometry and Moduli Seminar

Titel Costello's approach to Gromov-Witten theory
Referent:in, Affiliation Dr. Jérémy Guéré, Université de Grenoble
Datum, Zeit 25. Oktober 2019, 16:00-17:15
Ort HG G 43
Abstract In this second lecture, I will describe my proposal towards a general computation of Gromov-Witten invariants. Costello proved in 2003 how to express genus-g GW invariants of a projective variety X in terms of genus-0 GW invariants of the (g+1)-st symmetric power of X. In my recent preprint on Hodge-Gromov-Witten theory, I give a definition of genus-0 GW theory for some very singular DM stacks and a deformation invariance property. I will explain how to bound the two theorems above via Hironaka's resolution of singularities and how I intend to use them to compute GW invariants by virtual localization. As an illustration of this proposal, I will give the explicit example of genus-1 GW invariants of hypersurfaces. I will also discuss several research opportunities offered by this change in perspective.
Costello's approach to Gromov-Witten theoryread_more
HG G 43
30. Oktober 2019
13:30-14:45 		Dr. Pierrick Bousseau
ETHZ-ITS
Details

Algebraic Geometry and Moduli Seminar

Titel Refined curve counting on local CP2
Referent:in, Affiliation Dr. Pierrick Bousseau, ETHZ-ITS
Datum, Zeit 30. Oktober 2019, 13:30-14:45
Ort HG G 19.2
Abstract I will give an introduction to refined curve counting on Calabi-Yau 3-folds, focusing on the case of local CP2. This talk should be viewed as an advertisement for Longting’s talk on Friday.
Refined curve counting on local CP2read_more
HG G 19.2
1. November 2019
16:00-17:15 		Dr. Longting Wu
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Titel Holomorphic anomaly equation for (CP2, E), Part I
Referent:in, Affiliation Dr. Longting Wu, ETH Zürich
Datum, Zeit 1. November 2019, 16:00-17:15
Ort HG G 43
Abstract The holomorphic anomaly equation for local CP2 was established by Lho and Pandharipande. In this talk, we will show that a similar holomorphic anomaly equation also holds for Gromov-Witten theory of CP2 relative to a smooth cubic. This is based on joint work with Bousseau, Fan and Guo.
Holomorphic anomaly equation for (CP2, E), Part Iread_more
HG G 43
13. November 2019
13:30-14:45 		Dr. Longting Wu
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Titel Holomorphic anomaly equation for (CP2, E), Part II
Referent:in, Affiliation Dr. Longting Wu, ETH Zürich
Datum, Zeit 13. November 2019, 13:30-14:45
Ort HG G 19.2
Abstract The holomorphic anomaly equation for local CP2 was established by Lho and Pandharipande. In this talk, we will show that a similar holomorphic anomaly equation also holds for Gromov-Witten theory of CP2 relative to a smooth cubic. This is based on joint work with Bousseau, Fan and Guo.
Holomorphic anomaly equation for (CP2, E), Part IIread_more
HG G 19.2
15. November 2019
16:00-17:15 		Prof. Dr. Martin Ulrisch
Universität Frankfurt
Details

Algebraic Geometry and Moduli Seminar

Titel Tropical double ramification loci
Referent:in, Affiliation Prof. Dr. Martin Ulrisch, Universität Frankfurt
Datum, Zeit 15. November 2019, 16:00-17:15
Ort HG G 43
Abstract The double ramification locus in M_{g,n} can be defined in two different ways: as the locus of principal divisors with a given multiplicity profile or as the locus of curves admitting a map to the projective line of a given ramification profile over zero and infinity. When defining a tropical analogue of the double ramification locus, one observes that the two different ways of defining it algebraically lead to two different tropical analogues of the double ramification locus. This phenomenon is closely related to the realizability problem for tropical principal divisors and to the algebraic problem of compactifying the double ramification locus. In this talk I will give an introduction to this topic. I will explain both the structure of the two different tropical double ramification loci and the solution of the deformation-theoretic part of the realizability problem via admissible covers. Given time, I will also shed light on the analogous problem for strata of abelian differentials. This talk is based on joint work with D. Zakharov, as well as with M. Möller and A. Werner (in the case of abelian differentials).
Tropical double ramification lociread_more
HG G 43
20. November 2019
13:30-14:45 		Tim Buelles
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Titel Imprimitive Gromov-Witten theory of K3 surfaces and quasimodular forms
Referent:in, Affiliation Tim Buelles, ETH Zürich
Datum, Zeit 20. November 2019, 13:30-14:45
Ort HG G 19.2
Abstract Gromov-Witten theory of K3 surfaces in imprimitive curve classes comes with challenges that do not show up in the primitive theory. We describe some of these issues and present basic examples. The conjectural multiple cover behavior and the relation to quasimodular forms of higher level are discussed. This talk is based on joint work in progress with Younghan Bae.
Imprimitive Gromov-Witten theory of K3 surfaces and quasimodular formsread_more
HG G 19.2
27. November 2019
13:30-14:45 		Dr. Sam Molcho
Hebrew University
Details

Algebraic Geometry and Moduli Seminar

Titel Log geometry, tropicalization and compactifications of moduli spaces
Referent:in, Affiliation Dr. Sam Molcho, Hebrew University
Datum, Zeit 27. November 2019, 13:30-14:45
Ort HG G 19.2
Abstract This will be the first in a series of two talks on Jacobians of logarithmic curves. The Jacobian of the universal curve is an abelian variety over the moduli space of smooth curves, with deep connections to its geometry. However, the Jacobian does not extend to an abelian variety over the whole moduli space of stable curves. Nevertheless, in the setting of log geometry, an extension to a log smooth, proper group object -- the logarithmic Jacobian -- does exist. In this talk, after a brief introduction to the necessary notions from log geometry, I will discuss how log moduli problems have traditionally been used to construct compactifications of moduli spaces, such as the moduli space of curves. I will then sketch a procedure, called tropicalization, which allows one to study a log scheme or log moduli problem by assigning to it a certain combinatorial object, focusing on the case of curves. I will finish by discussing a specific example of such a combinatorial moduli problem, the tropical Jacobian of a tropical curve.
Log geometry, tropicalization and compactifications of moduli spacesread_more
HG G 19.2
29. November 2019
16:00-17:15 		Dr. Sam Molcho
Hebrew University
Details

Algebraic Geometry and Moduli Seminar

Titel The logarithmic Jacobian
Referent:in, Affiliation Dr. Sam Molcho, Hebrew University
Datum, Zeit 29. November 2019, 16:00-17:15
Ort HG G 43
Abstract In this talk, following ideas of Illusie and Kajiwara, Kato and Nakayama, I will introduce the log Jacobian of a family of nodal curves. Contrary to traditional compactifications obtained in log geometry, the logarithmic Jacobian is not an object that appears classically in algebraic geometry -- for instance, it is not representable by an algebraic stack. I will describe the underlying structure and main properties of the logarithmic Jacobian, such as properness, duality, and explicit presentations, and explain its connection with the tropical Jacobian, introduced in the first talk. Time permitting, I will discuss some recent applications to the theory of Neron models, and possible connections with double ramification cycles. The results of this talk are joint work with Jonathan Wise; results on duality of the log Jacobian were obtained jointly with M. Ulirsch, and the applications to Neron models are joint work with D. Holmes, G. Orrechia, and T. Poiret.
The logarithmic Jacobianread_more
HG G 43
* 11. Dezember 2019
13:30-14:45 		Prof. Dr. David Holmes
University of Leiden
Details

Algebraic Geometry and Moduli Seminar

Titel Tropical divisors and the double ramification cycle
Referent:in, Affiliation Prof. Dr. David Holmes, University of Leiden
Datum, Zeit 11. Dezember 2019, 13:30-14:45
Ort HG F 26.1
Abstract fter recalling some basics on logarithmic curves, we will explain in some detail a construction of Marcus and Wise of the stack of tropical divisors on a family of log curves. We will then explain how to see this as an incarnation of the double ramification cycle on the Picard stack over the stack of all prestable curves.
Tropical divisors and the double ramification cycleread_more
HG F 26.1
* 13. Dezember 2019
13:15-14:30 		Prof. Dr. David Holmes
Univ. of Leiden
Details

Algebraic Geometry and Moduli Seminar

Titel The universal double ramification cycle
Referent:in, Affiliation Prof. Dr. David Holmes, Univ. of Leiden
Datum, Zeit 13. Dezember 2019, 13:15-14:30
Ort HG G 19.2
Abstract We will present two equivalent constructions of a `universal’ double ramification cycle on the Picard stack over the stack of prestable curves. We will explain how all known instances of the DR cycle can be recovered from this universal one, and discuss some computations.
The universal double ramification cycleread_more
HG G 19.2
* 16. Dezember 2019
13:30-14:45 		Dr. Drew Johnson
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Titel How a "strange duality" influences universal series for Hilbert schemes of points
Referent:in, Affiliation Dr. Drew Johnson, ETH Zürich
Datum, Zeit 16. Dezember 2019, 13:30-14:45
Ort HG G 19.2
Abstract The Hilbert scheme of n points on a smooth, compact, complex surface parametrizes unordered n-tuples of points on the surface. Given a vector bundle on the surface, one can a obtain a sequence of "tautological" vector bundles on the Hilbert schemes, one for each value of n. The invariants of the tautological bundles can be assembled into a generating series. These generating series can be expressed in terms of certain "universal" series that don't depend on the surface or the vector bundle. Strange duality is a conjectural isomophism of vector spaces that suggests a surprising relationship between these series.
How a "strange duality" influences universal series for Hilbert schemes of pointsread_more
HG G 19.2
18. Dezember 2019
13:15-14:45 		Prof. Dr. Nicola Tarasca
Virgina Commonwealth University
Details

Algebraic Geometry and Moduli Seminar

Titel Vertex algebras of CohFT-type
Referent:in, Affiliation Prof. Dr. Nicola Tarasca, Virgina Commonwealth University
Datum, Zeit 18. Dezember 2019, 13:15-14:45
Ort HG G 43
Abstract This talk will focus on geometric realizations of non-commutative algebras. I will discuss how representations of conformal vertex algebras encode information about the geometry of algebraic curves. The starting point is the Virasoro uniformization, which provides an incarnation of the Virasoro algebra in the tangent space of a tautological line bundle on the moduli space of coordinatized curves. After briefly reviewing vertex algebras, I will discuss how their representations yield new vector bundles on moduli spaces of curves and new cohomological field theories. This is joint work with Chiara Damiolini and Angela Gibney.
Vertex algebras of CohFT-typeread_more
HG G 43
15. Januar 2020
13:30-14:45 		Prof. Dr. Dimitri Zvonkine
CNRS and Université Versailles
Details

Algebraic Geometry and Moduli Seminar

Titel The KP hierarchy via the Buryak-Rossi construction
Referent:in, Affiliation Prof. Dr. Dimitri Zvonkine, CNRS and Université Versailles
Datum, Zeit 15. Januar 2020, 13:30-14:45
Ort HG G 19.2
Abstract The Buryak-Rossi construction associates a hamiltonian integrable hierarchy to any cohomological field theory. In particular, the trivial CohFT yields the KdV hierarchy, while Witten's r-spin class conjecturally yields the r-KdV hierarchy up to a change of coordinates (a Miura transformation). The KP hierarchy does not exactly fit into this framework: it is not hamiltonian and the associated CohFT would have to have infinite rank. Nonetheless we suggest that the construction can be modified to include the KP hierarchy. Instead of a CohFT one takes the large r limits of Witten's r-spin class, which is somewhat reminiscent of the fact that r-KdV is a reduction of KP for any r. This is work in progress with Paolo Rossi.
The KP hierarchy via the Buryak-Rossi constructionread_more
HG G 19.2
17. Januar 2020
15:00-16:15 		Prof. Dr. Alexei Oblomkov
UMass Amherst
Details

Algebraic Geometry and Moduli Seminar

Titel Stationary Virasoro constraints for PT invariants with descendents
Referent:in, Affiliation Prof. Dr. Alexei Oblomkov, UMass Amherst
Datum, Zeit 17. Januar 2020, 15:00-16:15
Ort HG G 43
Abstract The talk is based on the joint paper with Miguel Moreira, Andrei Okounkov and Rahul Pandharipande. First I explain the formula for the correspondence between the stationary GW invariants and the stationary PT invariants with descendents. Next, I explain why this correspondence intertwines GW Virasoro constraints with PT Virasoro constraints. Since the stationary GW/PT correspondence with descendents is proven in the case of compact of toric varieties we obtain a proof of the PT Virasoro constraints in this case. In particular, we get new relations for the integrals of the tautological classes over the Hilbert scheme ponts on toric surfaces.The talk is a pre-talk for Miguel sequel talk.
Stationary Virasoro constraints for PT invariants with descendentsread_more
HG G 43
17. Januar 2020
16:30-17:45 		Miguel Moreira
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Titel Stable pairs in the line class of the cubic 3-fold
Referent:in, Affiliation Miguel Moreira, ETH Zürich
Datum, Zeit 17. Januar 2020, 16:30-17:45
Ort HG G 43
Abstract In this talk we will explain the complete computation of the theory of stable pairs with descendants in the line class of the cubic 3-fold. In this example the beautiful geometry of lines on cubic hypersurfaces plays an important role: the moduli spaces of stable pairs are smooth and can be described as the symmetric powers of the universal line. This is a first computation of stable pairs theory in the presence of odd cohomology. We verify that rationality and the functional equation hold, as well as the Virasoro constraints recently conjectured in joint work with A. Oblomkov, A. Okounkov and R. Pandharipande.
Stable pairs in the line class of the cubic 3-foldread_more
HG G 43

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