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Autumn Semester 2020

Date / Time Speaker Title Location
19 August 2020
15:30-16:45 		Ajith Urundolil Kumaran
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title An operator formula for the Gromov-Witten theory of CP1 relative to 0 and infinity
Speaker, Affiliation Ajith Urundolil Kumaran, ETH Zürich
Date, Time 19 August 2020, 15:30-16:45
Location HG G 43
Abstract We generalize a theorem by Rahul Pandharipande and Andrei Okounkov regarding the Gromov-Witten theory of P^1 relative to 0 and \infty. This generalization extends their operator formula to the case where we allow negative contact orders. As an application of this formula we determine the big relative quantum ring of (P^1, 0 \cup \infty).
An operator formula for the Gromov-Witten theory of CP1 relative to 0 and infinityread_more
HG G 43
16 September 2020
15:00-16:15 		Dr. Johannes Schmitt
Universität Bonn
Details

Algebraic Geometry and Moduli Seminar

Title admcycles - a software for computations in the tautological ring of the moduli of curves
Speaker, Affiliation Dr. Johannes Schmitt, Universität Bonn
Date, Time 16 September 2020, 15:00-16:15
Location Zoom
Abstract Inside the cohomology ring of the moduli space of stable curves there is a subring, called the tautological ring, with generators described by decorated graphs. All standard operations (cup product, intersection numbers etc) have explicit, combinatorial descriptions in terms of these generators. In my talk, I discuss and demonstrate the software package admcycles (created jointly with J. van Zelm and V. Delecroix), which implements these operations, as well as many algorithms for computing interesting classes in the tautological ring. In particular, I will focus on the fundamental classes of strata of differentials and show some old and some new computer experiments resulting in (conjectural) formulas and relationships to other known classes. I'll finish by discussing some open questions and ongoing investigations in the area. The talk will be interactive: I'll share a link allowing you to perform and modify all the computations in your own web browser, and we'll have some live exercises for the audience.
admcycles - a software for computations in the tautological ring of the moduli of curvesread_more
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23 September 2020
15:00-16:15 		Younghan Bae
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title The Chow ring of the moduli space of genus zero prestable curves
Speaker, Affiliation Younghan Bae, ETH Zürich
Date, Time 23 September 2020, 15:00-16:15
Location Zoom
Abstract Let M_{g,n} be the moduli space of prestable, i.e. nodal, curves of genus g with n markings. M_{g,n} is a smooth algebraic stack locally of finite type over the base field. It has a well-defined Chow intersection theory by the work of A. Kresch. There are many results on the Chow group of various open subset of M_{0,n} by Keel, Lee-Pandharipande, Fulghesu, Oesinghaus etc. In this talk we completely determine the (rational) Chow group of M_{0,n} which generalizes almost all known results. First we prove that tautological classes span the Chow group of M_{0,n}. Second we prove that all tautological relations are additively generated by the WDVV relation and a degree one relation in M_{0,2}. If time permits, I will discuss some consequences on the Chow groups of arbitrary genus M_{g,n}. This is a joint work in progress with Johannes Schmitt.
The Chow ring of the moduli space of genus zero prestable curvesread_more
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25 September 2020
16:00-17:15 		Dr. Pierrick Bousseau
ETH-ITS
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Algebraic Geometry and Moduli Seminar

Title The skein algebra of the 4-punctured sphere from curve counting
Speaker, Affiliation Dr. Pierrick Bousseau, ETH-ITS
Date, Time 25 September 2020, 16:00-17:15
Location Zoom
Abstract The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the SL_2 character variety of a topological surface. I will explain how to realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov-Witten invariants of a log Calabi-Yau cubic surface. This leads to a proof of a previously conjectured positivity property of the bracelets bases of the skein algebras of the 4-punctured sphere and of the 1-punctured torus.
The skein algebra of the 4-punctured sphere from curve countingread_more
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30 September 2020
15:00-16:15 		Dr. Ulrike Riess
ETH-ITS
Details

Algebraic Geometry and Moduli Seminar

Title On the Kähler cone of irreducible symplectic orbifolds
Speaker, Affiliation Dr. Ulrike Riess, ETH-ITS
Date, Time 30 September 2020, 15:00-16:15
Location Zoom
Abstract In this talk I report on recent joint work with G. Menet: We generalize a series of classical results on irreducible symplectic manifolds to the orbifold setting. In particular we prove a characterization of the Kähler cone using wall divisors. This generalizes results of Mongardi for the smooth case. I will finish the talk by applying these results to study a concrete example.
On the Kähler cone of irreducible symplectic orbifoldsread_more
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2 October 2020
16:00-17:15 		Miguel Moreira
ETH Zürich
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Algebraic Geometry and Moduli Seminar

Title Orbifold Euler characteristics of the relative Hilbert scheme of points
Speaker, Affiliation Miguel Moreira, ETH Zürich
Date, Time 2 October 2020, 16:00-17:15
Location Zoom
Abstract Studying the moduli space of relative stable pairs on a product S x E of a surface S with a curve E leads naturally to the Hilbert scheme of points on a surface S relative to a smooth divisor D, introduced by Setayesh. Indeed, when E is an elliptic curve the (Satake) orbifold Euler characteristic of the relative Hilbert scheme is realized as a PT invariant. In this talk I’ll give a quick introduction to moduli spaces of relative sheaves and I’ll explain Setayesh’s computation of the Betti numbers of Hilb^n(S/D). The space Hilb^n(S/D) is an orbifold, and thus admits several different notions of Euler characteristics (Satake’s, topological, stringy, etc.). I’ll explain how to compute this hierarchy of Euler characteristics from Setayesh’s result. The answer turns out to have nice number theoretical properties.
Orbifold Euler characteristics of the relative Hilbert scheme of pointsread_more
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7 October 2020
15:00-16:15 		Tim Bülles
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Symmetries of stable pair invariants via derived equivalences - the STU model
Speaker, Affiliation Tim Bülles, ETH Zürich
Date, Time 7 October 2020, 15:00-16:15
Location Zoom
Abstract Curve counting on Calabi-Yau threefolds X via Pandharipande-Thomas stable pair invariants is much constrained by derived equivalences of X. These constraints are reflected by BPS counts in string theory compactified on X. An important example for physicists is the STU model, which admits remarkable BPS symmetries. I will discuss these symmetries via stable pairs and connect to derived equivalences given by Seidel-Thomas spherical twists.
Symmetries of stable pair invariants via derived equivalences - the STU modelread_more
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9 October 2020
16:00-17:15 		Dr. Sam Molcho
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title An overview of logarithmic stable maps
Speaker, Affiliation Dr. Sam Molcho, ETH Zürich
Date, Time 9 October 2020, 16:00-17:15
Location Zoom
Abstract In this expository talk I will discuss the relationship between the various compactifications of the moduli space of relative stable maps. I will explain how all existing approaches -- Jun Li's, B. Kim's, Abramovich-Chen-Gross-Siebert's and, time permitting, Ranganathan's -- can be phrased in the language of logarithmic geometry, and how using this language systematically allows one to compare the different spaces. In particular, I will describe how the logarithmic perspective gives access to the tropicalization of the spaces, which are certain combinatorial objects much simpler than the original spaces, but which already contain all information necessary for the comparison.
An overview of logarithmic stable mapsread_more
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14 October 2020
15:00-16:15 		Dr. Michel van Garrel
University of Warwick
Details

Algebraic Geometry and Moduli Seminar

Title Scattering diagram of (CP2,E), part I
Speaker, Affiliation Dr. Michel van Garrel, University of Warwick
Date, Time 14 October 2020, 15:00-16:15
Location Zoom
Abstract The pair (P^2,E), for E a smooth anticanonical divisor, is an example of a log K3 surface. Its enumerative geometry is a log analogue of the enumerative geometry of K3 surfaces and consists in counting curves with maximal tangency condition along E. This study is surprisingly intricate and was initiated by N. Takahashi 20 years ago followed by some progress by Gathmann. After staying dormant for many years, there recently has been significant progress by combinations of Bousseau, Choi, Fan, Gabele, Guo, Katz, N. Takahashi, Wu, You etc. At the heart of this lies the scattering diagram of (P^2,E), which was computed by Carl-Pumperla-Siebert. In many ways, it is a combinatorial condensate of (P^2,E). Its enumerative geometry is encoded by the wallcrossing functions and mirror symmetry for (P^2,E) by the algebra of theta functions. The purpose of these two talks is to introduce this scattering diagram and in particular explain its relationship to enumerative geometry as developed in the works of Bousseau and Gabele. Time permitting, I will talk about some mirror symmetry aspects of it as well.
Scattering diagram of (CP2,E), part Iread_more
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* 19 October 2020
16:00-17:15 		Dr. Michel van Garrel
University of Warwick
Details

Algebraic Geometry and Moduli Seminar

Title Scattering diagram of (CP2,E), part II
Speaker, Affiliation Dr. Michel van Garrel, University of Warwick
Date, Time 19 October 2020, 16:00-17:15
Location Zoom
Abstract The pair (P^2,E), for E a smooth anticanonical divisor, is an example of a log K3 surface. Its enumerative geometry is a log analogue of the enumerative geometry of K3 surfaces and consists in counting curves with maximal tangency condition along E. This study is surprisingly intricate and was initiated by N. Takahashi 20 years ago followed by some progress by Gathmann. After staying dormant for many years, there recently has been significant progress by combinations of Bousseau, Choi, Fan, Gabele, Guo, Katz, N. Takahashi, Wu, You etc.
At the heart of this lies the scattering diagram of (P^2,E), which was computed by Carl-Pumperla-Siebert. In many ways, it is a combinatorial condensate of (P^2,E). Its enumerative geometry is encoded by the wallcrossing functions and mirror symmetry for (P^2,E) by the algebra of theta functions. The purpose of these two talks is to introduce this scattering diagram and in particular explain its relationship to enumerative geometry as developed in the works of Bousseau and Gabele. Time permitting, I will talk about some mirror symmetry aspects of it as well.
Scattering diagram of (CP2,E), part IIread_more
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21 October 2020
15:00-16:15 		Dr. Longting Wu
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title A recursive formula for Gromov-Witten invariants of local Fano threefolds
Speaker, Affiliation Dr. Longting Wu, ETH Zürich
Date, Time 21 October 2020, 15:00-16:15
Location Zoom
Abstract In this talk, we will consider the Gromov-Witten invariants of three Fano threefolds O_{P^2}(-1), O_{P^2}(-2) and O_{P^1\times P^1}(-1,-1). We will present a simple uniform recursive formula which completely determine their (primary) Gromov-Witten invariants for all genera. This is a work in progress with Pierrick Bousseau.
A recursive formula for Gromov-Witten invariants of local Fano threefoldsread_more
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23 October 2020
16:00-17:15 		Prof. Dr. Nicola Tarasca
Virgina Commonwealth University
Details

Algebraic Geometry and Moduli Seminar

Title Incidence varieties in the projectivized k-th Hodge bundle
Speaker, Affiliation Prof. Dr. Nicola Tarasca, Virgina Commonwealth University
Date, Time 23 October 2020, 16:00-17:15
Location Zoom
Abstract The k-th Hodge bundle parametrizes stable curves together with sections of the k-th power of their dualizing sheaf. By imposing conditions on orders of zeros and poles of k-differentials, one obtains a natural stratification of the projectivized k-th Hodge bundle. The image of such strata in moduli spaces of stable pointed curves has recently been the focus of many studies. I will show how working on the projectivized k-th Hodge bundle offers some advantages. In the holomorphic case, the study of the strata can be reduced to the study of some incident varieties inside the projectivized k-th Hodge bundle over the moduli space of stable pointed curves. Specifically, the incidence variety Z_n is defined as the closure of the locus of smooth curves of genus g with n marked points together with the class of a k-differential vanishing at all the n marked points. The boundary of these varieties can be studied via the space of twisted k-differentials. I will present a graph formula for the restriction of the incident varieties Z_n over the locus of pointed curves with rational tails. Also, in the range n <= k, the same formula holds over stable pointed curves. I will conclude with some applications. Joint work with Iulia Gheorghita.
Incidence varieties in the projectivized k-th Hodge bundleread_more
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28 October 2020
15:00-16:15 		Dr. Maria Yakerson
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Algebraic K-theory I
Speaker, Affiliation Dr. Maria Yakerson, ETH Zürich
Date, Time 28 October 2020, 15:00-16:15
Location Zoom
Algebraic K-theory I
Zoom
* 2 November 2020
14:30-15:45 		Dr. Jérémy Guéré
Université de Grenoble
Details

Algebraic Geometry and Moduli Seminar

Title Congruences on K-theoretic Gromov-Witten invariants
Speaker, Affiliation Dr. Jérémy Guéré, Université de Grenoble
Date, Time 2 November 2020, 14:30-15:45
Location Zoom
Abstract K-theoretic Gromov-Witten invariants of smooth projective varieties have been introduced by YP Lee, using the Euler characteristic of a virtual structure sheaf. In particular, they are integers. In this talk, I look at these invariants for the quintic threefold and I will explain how to compute them modulo 41, using the virtual localization formula under a finite group action, up to genus 19 and degree 40.
Congruences on K-theoretic Gromov-Witten invariantsread_more
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4 November 2020
15:00-16:15 		Dr. Maria Yakerson
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Algebraic K-theory II
Speaker, Affiliation Dr. Maria Yakerson, ETH Zürich
Date, Time 4 November 2020, 15:00-16:15
Location Zoom
Algebraic K-theory II
Zoom
6 November 2020
16:00-17:15 		Dr. Junliang Shen
MIT
Details

Algebraic Geometry and Moduli Seminar

Title Cohomology of the moduli of Higgs bundles, the Hausel-Thaddeus conjecture, and vanishing cycles
Speaker, Affiliation Dr. Junliang Shen, MIT
Date, Time 6 November 2020, 16:00-17:15
Location Zoom
Abstract We describe the cohomological structure of the moduli space of stable SL_n Higgs bundles on a curve following the topological mirror symmetry conjecture of Hausel-Thaddeus. A key point of the approach is to establish a connection between: (a) the moduli space of twisted Higgs bundles by an effective divisor of degree greater than 2g-2, and (b) the moduli space of K_C-Higgs bundles, using vanishing cycle functors. This is inspired by considerations in Donaldson-Thomas theory, which allows us to apply Ngô's support theorem, which has a simpler form in the case (a) (by Ngô, Chaudouard-Laumon, de Cataldo), to the case (b) which concerns hyper-Kähler geometries. In particular, this gives a new proof of the Hausel-Thaddeus conjecture proven previously by Gröchenig-Wyss-Ziegler via p-adic integration. We will also discuss connections to the P=W conjecture if time permits. Based on joint work with Davesh Maulik.
Cohomology of the moduli of Higgs bundles, the Hausel-Thaddeus conjecture, and vanishing cyclesread_more
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11 November 2020
15:00-16:15 		Dr. Maria Yakerson
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Algebraic K-theory III
Speaker, Affiliation Dr. Maria Yakerson, ETH Zürich
Date, Time 11 November 2020, 15:00-16:15
Location Zoom
Algebraic K-theory III
Zoom
13 November 2020
16:00-17:00 		Dr. Helen Jenne
Université de Tours
Details

Algebraic Geometry and Moduli Seminar

Title Double-dimer condensation and the dP3 Quiver
Speaker, Affiliation Dr. Helen Jenne, Université de Tours
Date, Time 13 November 2020, 16:00-17:00
Location Zoom
Abstract In the first half of this talk we will discuss a new result about the double-dimer model: under certain conditions, the partition function for double-dimer configurations of a planar bipartite graph satisfies a recurrence related to Dodgson condensation (also called the Desnanot-Jacobi identity). A similar identity for the number of dimer configurations of a planar bipartite graph was established nearly 20 years ago by Kuo. In the second half of the talk, we will describe an application of this condensation result to a problem in cluster algebras, which is ongoing joint work with Tri Lai and Gregg Musiker. In 2017, Lai and Musiker gave combinatorial interpretations for many toric cluster variables in the cluster algebra associated to the cone over the del Pezzo surface dP3. Specifically, they used Kuo condensation to show that most toric cluster variables have Laurent expansions agreeing with partition functions for dimer configurations. However, in some cases, the dimer model was not sufficient. We show that in these cases, the Laurent expansions agree with partition functions for double-dimer configurations.
Double-dimer condensation and the dP3 Quiverread_more
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13 November 2020
17:15-18:15 		Prof. Dr. Ben Young
University of Oregon
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Algebraic Geometry and Moduli Seminar

Title The combinatorial PT-DT correspondence
Speaker, Affiliation Prof. Dr. Ben Young, University of Oregon
Date, Time 13 November 2020, 17:15-18:15
Location Zoom
Abstract I'm going to explain how the box-counting formula for Pandharipande-Thomas' invariants can be rephrased in terms of the double-dimer model with nodes. This allows us, using Jenne's work on the double-dimer model, to characterized the PT invariants' generating function as the solution to a certain recurrence. We can then show that the reduced DT invariants' generating function also satisfies this recurrence. This result, given the geometric PT-DT correspondence, shows that the box-counting formula in PT theory is actually computing PT invariants (without really doing any new geometry).
The combinatorial PT-DT correspondenceread_more
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18 November 2020
15:00-16:15 		Dr. Maria Yakerson
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Algebraic K-theory IV
Speaker, Affiliation Dr. Maria Yakerson, ETH Zürich
Date, Time 18 November 2020, 15:00-16:15
Location Zoom
Algebraic K-theory IV
Zoom
20 November 2020
16:00-17:15 		Dr. Miroslav Rapčák
UC Berkeley
Details

Algebraic Geometry and Moduli Seminar

Title Affine Yangian and Pandharipande-Thomas box counting
Speaker, Affiliation Dr. Miroslav Rapčák, UC Berkeley
Date, Time 20 November 2020, 16:00-17:15
Location Zoom
Abstract I will discuss new observations in representation theory of affine Yangians A (providing an universal description of a class of Coulomb-branch algebras due to Kodera and Nakajima) and Y (providing an universal description of a class of vertex operator algebras from my previous work). The crucial observation is that the vector space with a basis labelled by Pandharipande-Thomas box configurations admits a natural action of the algebra A defining a family of A-modules labeled by 3 partitions. This observation leads to a natural proposal for Y-modules labelled by 6 partitions solving an outstanding problem of finding such a generalization of the simpler 4-partition case. The talk is based on the work in progress with Davide Gaiotto.
Affine Yangian and Pandharipande-Thomas box countingread_more
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25 November 2020
15:00-16:15 		Dr. Sam Molcho
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Log geometry I
Speaker, Affiliation Dr. Sam Molcho, ETH Zürich
Date, Time 25 November 2020, 15:00-16:15
Location Zoom
Log geometry I
Zoom
27 November 2020
16:00-17:15 		Prof. Dr. Richard Thomas
Imperial College
Details

Algebraic Geometry and Moduli Seminar

Title Square root Euler classes and counting sheaves on Calabi-Yau 4-folds
Speaker, Affiliation Prof. Dr. Richard Thomas, Imperial College
Date, Time 27 November 2020, 16:00-17:15
Location Zoom
Abstract I will explain a nice characteristic class of SO(2n,C) bundles in both Chow cohomology and K-theory, and how to localise it to the zeros of an isotropic section. This builds on work of Edidin-Graham, Polishchuk-Vaintrob, Anderson and many others. This can be used to construct an algebraic virtual cycle (and virtual structure sheaf) on moduli spaces of stable sheaves on Calabi-Yau 4-folds. It recovers the real derived differential geometry virtual cycle of Borisov-Joyce but has nicer properties, like a torus localisation formula. Joint work with Jeongseok Oh (KIAS).
Square root Euler classes and counting sheaves on Calabi-Yau 4-foldsread_more
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4 December 2020
16:00-17:15 		Prof. Dr. Hsian-Hua Tseng
Ohio State University
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Algebraic Geometry and Moduli Seminar

Title Relative Gromov-Witten theory without log geometry
Speaker, Affiliation Prof. Dr. Hsian-Hua Tseng, Ohio State University
Date, Time 4 December 2020, 16:00-17:15
Location Zoom
Abstract We describe a new Gromov-Witten theory of a space relative to a simple normal-crossing divisor constructed using multi-root stacks.
Relative Gromov-Witten theory without log geometryread_more
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9 December 2020
15:00-16:15 		Dr. Sam Molcho
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Log geometry II
Speaker, Affiliation Dr. Sam Molcho, ETH Zürich
Date, Time 9 December 2020, 15:00-16:15
Location Zoom
Log geometry II
Zoom
11 December 2020
16:00-17:15 		Dr. Sam Molcho
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Log geometry III
Speaker, Affiliation Dr. Sam Molcho, ETH Zürich
Date, Time 11 December 2020, 16:00-17:15
Location Zoom
Log geometry III
Zoom
16 December 2020
15:00-16:15 		Dr. Sam Molcho
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Log geometry IV
Speaker, Affiliation Dr. Sam Molcho, ETH Zürich
Date, Time 16 December 2020, 15:00-16:15
Location Zoom
Log geometry IV
Zoom

Notes: events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

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