Algebraic geometry and moduli seminar

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Autumn Semester 2014

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
* 19 September 2014
16:15-17:00
Junliang Shen
ETH Zürich
Cobordism invariants of moduli space of stable pairs  HG G 19.1 
Abstract: Abstract: For a compact Calabi-Yau 3-fold, it is well known that the partition function of Pandharipande-Thomas invariants are rational and satisfies the q ~1/q symmetry. As a generalization, we consider the cobordism invariants obtained by pushing forward the virtual cobordism class of the moduli space to one point. The partition function of cobordism invariants are conjectured to be rational and satisfy a functional equation. I will explain the proof of the rationality for nonsingular projective toric 3-fold by descendent theory.
* 24 September 2014
14:00-15:00
Dr. Andrew Morrison
ETH Zürich
Lines on the Fermat quintic (motivically) HG G 19.2 
26 September 2014
16:00-17:00
Dr. Olivier Fabert
VU Amsterdam
Closed-string mirror symmetry for open manifolds  HG G 43 
Abstract: The classical mirror symmetry conjecture for closed Calabi-Yau manifolds X and Y relates the rational Gromov-Witten theory of X with the (extended) deformation theory of complex structures on Y (and vice versa). In my talk I will illustrate how this correspondence is supposed to generalize from closed to open manifolds. The main ingredient is a novel algebraic structure on the symplectic cohomology of an open symplectic manifold, defined by counting Floer curves with arbitrarily many cylindrical ends and varying conformal structure.
* 29 September 2014
16:00-17:00
Dr. Aaron Pixton
Harvard University
A conjectural formula for the double ramification cycle HG G 19.2 
* 30 September 2014
15:15-16:15
Rahul Pandharipande
ETH Zürich
Gromov-Witten theory, K3 surfaces, and Pixton's conjecture  HG F 26.1 
Abstract: I will discuss new calculations and conjectures made with G. Oberdieck related to the Gromov-Witten theory of K3 surfaces. A basic connection to Pixton's conjecture on DR cycles will be made and some related directions of inquiry will be pointed out.
* 1 October 2014
14:00-15:15
Dr. Dimitri Zvonkine
Jussieu
Givental's R-matrix action and Witten's r-spin class: explicit computations  HG G 19.2 
Abstract: The Givental-Teleman classification of semisimple cohomological field theories gives a recursive procedure that expresses in terms of tautological classes the Witten r-spin class shifted to any point of its Frobenius manifold outside the discriminant. The part of this expression that exceeds the degree of Witten's class gives a family of tautological relations on Mbar_{g,n}. For certain well-chosen points of the Frobenius manifold the recursive procedure can be solved explicitly. We will show the outcome of these computations and the tautological relations thus obtained.
* 1 October 2014
15:30-16:45
Felix Janda
ETH Zürich
r-spin relations on the moduli of curves (new connections)  HG G 19.2 
Abstract: By studying the behavior of the Givental-Teleman classification of semisimple cohomological field theories when approaching a non-semisimple point one can extract tautological relations of the moduli space of stable curves. I want to discuss the example of Witten's r-spin class for different r and show recent ideas on how to connect all of them to the 3-spin relations.
3 October 2014
16:00-17:00
Dr. Dimitri Zvonkine
Jussieu
Hurwitz numbers for real polynomials  HG G 43 
Abstract: There are n^{n-3} (properly normalized) complex degree n polynomials with n-1 fixed critical values. This can be found by establishing a one-to-one correspondence between these polynomials and marked trees, which are enumerated by the Cayley formula. The number of (properly normalized) real degree n polynomials with n-1 fixed real critical values is equal to the n-th Euler-Bernoulli number. This can be found by establishing a one-to-one correspondence between these polynomials and alternating permutations. The problem above can be generalized by allowing multiple critical values and fixing their ramification profiles. In the complex case this problem is solved; in the real case, however, the answer depends on the order of the critical values on the real line. Thus the question arises whether it is possible to attribute a sign to every real polynomial in such a way that the number of polynomials counted with signs is invariant under permutations of critical values. We construct a sign with this property and study the invariant thus obtained. This is joint work with I. Itenberg.
* 8 October 2014
14:00-15:00
Prof. Dr. Boris Dubrovin
SISSA, Trieste
Quantum integrable systems, symplectic field theory, and Schur polynomials HG G 19.2 
10 October 2014
16:00-17:00
Dr. Paolo Rossi
Université de Bourgogne, France
Double ramification cycles and quantization of integrable hierarchies  HG G 43 
Abstract: Recently, inspired by symplectic field theory, Alexandr Buryak has defined a new construction of an integrable hierarchy of PDEs associated to a given cohomological field theory. Such construction involves the geometry of the double ramification cycle and, as opposed to SFT-type hierarchies, produces classical dispersive hamiltonian PDEs (among them KdV, ILW and extended Toda). In a joint work in progress we were then able to uncover some interesting properties of such "double ramification hierarchies", including effective recursions for the Hamiltonians and a simple formula for their quantization. In the talk I will present these results and some of their possible applications.
17 October 2014
16:00-17:00
Prof. Dr. Sergey Shadrin
University of Amsterdam
Gromov-Witten theory of the projective line  HG G 43 
Abstract: I'll explain how the Gromov-Witten theory of the projective line fits into the Chekhov-Eynard-Orantin (CEO) topological recursion scheme. It is a byproduct of a more general correspondence between the Givental theory and a local version of the CEO recursion. In the framework of CEO topological recursion physicists suggest a quantization procedure that allows to construct a differential operator called quantum spectral curve that vanishes a so-called wave function obtains as the principle specialization of the corresponding n-point function. In the case of the projective line this general principle appears to work perfectly, and I'll show some very nice combinatorics derived from the semi-infinite wedge formalism that stays behind it.
24 October 2014
16:00-17:00
Prof. Dr. Matthias Gaberdiel
ETH Zürich
Mathieu Moonshine HG G 43 
31 October 2014
16:00-17:00
Prof. Dr. Balazs Szendroi
Oxford University
Cohomological Donaldson-Thomas theory  HG G 43 
Abstract: I will explain the basics of the cohomological refinement of Donaldson-Thomas theory. I will give some sample local computations, and discuss some structures that only exist on cohomological DT. If time permits I will briefly mention an application in quantum cluster theory.
* 12 November 2014
15:15-16:15
Christoph Schiessl
ETH Zürich
Asymptotic behavior of Gromov-Witten invariants  HG G 19.2 
Abstract: I will discuss the asymptotic behavior of Gromov-Witten invariants in some concrete examples (P^2, P^3, P^1 x P^1, ...) using the explicit WDVV-relations. This is due to ideas of Itzykon/Francesco and Dubrovin.
14 November 2014
16:00-17:00
Dr. Benjamin Bakker
Humboldt Universität Berlin
Bounding torsion in geometric families of abelian varieties  HG G 43 
Abstract: A celebrated theorem of Mazur asserts that the order of the torsion part of the group of rational points of an elliptic curve over Q is absolutely bounded; it is conjectured that the same is true for abelian varieties over number fields K, though very little progress has been made in proving it. The natural geometric analog where K is replaced by the function field of a complex curve---known as the geometric torsion conjecture---is equivalent to the nonexistence of low genus curves in congruence towers of Siegel modular varieties. In joint work with J. Tsimerman, we prove the geometric torsion conjecture for abelian varieties with real multiplication. We will discuss a general method for ruling out low genus curves in locally symmetric varieties using hyperbolic geometry to bound Seshadri constants and also apply it to some related problems.
* 14 November 2014
17:00-18:00
Prof. Dr. Jim Bryan
UBC Vancouver
Donaldson-Thomas theory of local elliptic surfaces via the topological vertex  HG G 43 
Abstract: Donaldson-Thomas (DT) invariants of a Calabi-Yau threefold X are fundamental quantum invariants given by (weighted) Euler characteristics of the Hilbert schemes of X. We compute these invariants for the case where X is a so-called local elliptic surface --- it is the total space of the canonical line bundle over an elliptic surface. We find that the generating functions for the invariants admit a nice product structure. We introduce a new technique which allows us to use the topological vertex in this computation --- a tool which previously could only be used for toric threefolds. As a by product, we discover surprising new identities for the topological vertex. This is joint work with Martijn Kool, with an assist from Ben Young.
* 19 November 2014
14:00-15:00
Junliang Shen
ETH Zürich
Virtual cobordism classes and Chern numbers  HG G 19.2 
Abstract: Given a projective scheme with a perfect obstruction theory, we get virtual Chern numbers by integrating Chern classes of the virtual tangent bundle against the (Chow) virtual fundamental class. A result by Ciocan-Fontanine and Kapranov says that these numbers actually are the Chern numbers of some integer-coefficient cobordism class. We show that this class is just the virtual cobordism class which can be constructed in the corresponding algebraic cobordism group similarly as Li-Tian and Behrend-Fantechi. Finally, as an application, we discuss the relation between virtual Chern numbers and descendent invariants in Pandharipande-Thomas theory. This is related to my talk in September.
21 November 2014
16:00-17:00
PD Dr. Emanuel Scheidegger
Université de Fribourg
Topological string, Hodge theory and a new Lie algebra  HG G 43 
Abstract: After a review of the B-model in topological string theory, we will give a Hodge-theoretic reformulation of the holomorphic anomaly equation and explain a natural Lie algebra structure.
* 26 November 2014
15:30-16:30
Ran Tessler
Hebrew Univ.
Computation of all open genus 0 gravitational descendents  HG G 19.2 
Abstract: We recall the definition of open gravitational descendents in g=0. We then prove a topological recursion relation, which allows us to calculate all open g=0 descendents. Based on a joint work with R. Pandharipande and J. Solomon.
28 November 2014
16:00-17:00
Dr. Alexander Alexandrov
Université de Fribourg
Open intersection numbers, matrix models and integrability  HG G 43 
Abstract: In my talk I will discuss a family of matrix models, which describes the generating functions of intersection numbers on moduli spaces both for open and closed Riemann surfaces. Linear (Virasoro\W-constraints) and bilinear (KP\MKP integrable hierarchies) equations follow from the matrix model representation in a standard way.
* 3 December 2014
14:00-16:00
Prof. Dr. Lothar Göttsche
ICTP, Trieste
Hilbert schemes of points on surfaces  HG G 19.2 
Abstract: In the lecture we review some well-known facts about Hilbert schemes of points on surfaces: We introduce the Hilbert scheme of points Hilb^n(S) on projective a algebraic surface. These parametrize finite subschemes of length n on S, i.e. generically sets of n points on S. We compute their Betti numbers and introduce the action of the the Heisenberg algebra on the direct sum of the cohomologies of all Hilb^n(S).
5 December 2014
16:00-17:00
Prof. Dr. Lothar Göttsche
ICTP, Trieste
Refined curve counting: Hilbert schemes, tropical geometry and Fock space  HG G 43 
Abstract: The Severi degrees count the number of plane curves of degree d with delta nodes though d(d+3)/2-delta general points. One can study this problem more generally for a linear system on a general algebraic surface. We introduce refined curve counting invariants in two ways: via Hilbert schemes of points and via tropical geometry. For toric surfaces they interpolate between Severi degrees and Welschinger invariants in real algebraic geometry. We use tropical geometry to compute them as vacuum expectation values for some operators on a Fock space.
12 December 2014
16:00-17:00
Prof. Dr. Marcos Marino
Université de Genève
Spectral theory and mirror symmetry  HG G 43 
Abstract: Recent work in string theory has revealed a surprising connection between the spectral theory of functional difference operators and local mirror symmetry. In this talk, I will give an overview of these developments by focusing on its mathematical consequences. I will associate a functional difference operator to any toric Calabi--Yau manifold by quantizing its mirror curve, and will state and discuss a precise conjecture expressing the Fredholm determinant of this operator in terms of the enumerative properties of the underlying Calabi--Yau manifold.

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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