Forthcoming events in this series


Tue, 31 May 2022

15:30 - 16:30
L4

Hilbert scheme of points on manifolds and global singularity theory

Gergely Berczi
(Aarhus University)
Abstract

Global singularity theory is a classical subject which classifies singularities of maps between manifolds, and describes topological reasons for their appearance. I will start with explaining a central problem of the subject regarding multipoint and multisingularity loci, then give an introduction into some recent major developments by Kazarian, Rimanyi, Szenes and myself.

Tue, 24 May 2022

15:30 - 16:30
L3

Moment Polyptychs and the Equivariant Quantisation of Hypertoric Varieties

Ben Brown
(Edinburgh)
Abstract

We develop a method to investigate the geometric quantisation of a hypertoric variety from an equivariant viewpoint, in analogy with the equivariant Verlinde for Higgs bundles. We do this by first using the residual circle action on a hypertoric variety to construct its symplectic cut, resulting in a compact cut space which is needed for localisation. We introduce the notion of a moment polyptych associated to a hypertoric variety and prove that the necessary isotropy data can be read off from it. Finally, the equivariant Hirzebruch-Riemann-Roch formula is applied to the cut spaces and expresses the dimension of the equivariant quantisation space as a finite sum over the fixed-points. This is joint work with Johan Martens.

Tue, 10 May 2022

15:30 - 16:30
L4

Cohomological χ-independence for Higgs bundles and Gopakumar-Vafa invariants

Tasuki Kinjo
(University of Tokyo)
Abstract

In this talk, I will introduce the BPS cohomology of the moduli space of Higgs bundles on a smooth projective curve of rank r and degree d using cohomological Donaldson-Thomas theory. The BPS cohomology and the intersection cohomology coincide when r and d are coprime, but they are different in general. We will see that the BPS cohomology does not depend on d. This is a generalization of the Hausel-Thaddeus conjecture to non-coprime case. I will also explain that Toda's χ-independence conjecture (and hence the strong rationality conjecture) for local curves can be proved in the same manner. This talk is based on a joint work with Naoki Koseki and another joint work with Naruki Masuda.

Tue, 03 May 2022

15:30 - 16:30
L3

Quotients by Algebraic Foliations

Federico Bongiorno
(Imperial College London)
Abstract

Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on the non-empty open subset of algebraically smooth points, through which every invariant morphism factors uniquely. Some applications to quotients by connected groups will be discussed.
 

Fri, 19 Nov 2021

10:00 - 11:30
N3.12

Virtual classes via vanishing cycles

Tasuki Kinjo
(Kavli IPMU)
Abstract

[REMOTE TALK]

In this talk, we will propose a new construction of the virtual fundamental classes of quasi-smooth derived schemes using the vanishing cycle complexes. This is based on the dimensional reduction theorem of cohomological Donaldson—Thomas invariants which can be regarded as a variant of the Thom isomorphism. We will also discuss a conjectural approach to construct DT4 virtual classes using the vanishing cycle complexes.

Zoom link: https://us02web.zoom.us/j/86267335498?pwd=R2hrZ1N3VGJYbWdLd0htZzA4Mm5pd…

Tue, 02 Nov 2021

15:30 - 16:30
L4

Gromov-Witten invariants of blow-ups

Qaasim Shafi
(Imperial)
Abstract
Gromov-Witten invariants play an essential role in mirror symmetry and enumerative geometry. Despite this, there are few effective tools for computing Gromov-Witten invariants of blow-ups. Blow-ups of X can be rewritten as subvarieties of Grassmann bundles over X. In joint work with Tom Coates and Wendelin Lutz, we exploit this fact and extend the abelian/non-abelian correspondence, a modern tool in Gromov-Witten theory. Combining these two steps allows us to get at the genus 0 invariants of a large class of blow-ups.   
Tue, 12 Oct 2021

15:30 - 16:30
L5

The Mirror Clemens-Schmid Sequence

Alan Thompson
(Loughborough)
Abstract

I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed 
Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a "mirror P=W" conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting. This is joint work with Charles F. Doran.

Tue, 03 Mar 2020

15:30 - 16:30
L4

Skein-triangulated representations of generalized braid categories

Timothy Logvinenko
(Cardiff University)
Abstract

The ordinary braid group ${\mathrm Br}_n$ is a well-known algebraic structure which encodes configurations of $n$ non-touching strands (“braids”) up to continious transformations (“isotopies”). A classical result of Khovanov and Thomas states that there is a natural categorical action of ${\mathrm Br}_n$ on the derived category of the cotangent bundle of the variety of complete flags in ${\mathbb C}^n$. 

In this talk, I will introduce a new structure: the category ${\mathrm GBr}_n$ of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. In the context of triangulated categories, it is natural to impose certain relations which result in the notion of a skein-triangulated representation of ${\mathrm GBr}_n$. A decade-old conjecture states that there is a skein-triangulated action of ${\mathrm GBr}_n$ on the cotangent bundles of the varieties of full and partial flags in ${\mathbb C}^n$. We prove this conjecture for $n = 3$. We also show that, in fact, any categorical action of ${\mathrm Br}_n$ can be lifted to a categorical action of ${\mathrm GBr}_n$, generalising a result of Ed Segal. This is a joint work with Rina Anno and Lorenzo De Biase.

Tue, 11 Feb 2020

15:30 - 16:30
L4

Ranks of cubic surfaces

Anna Seigal
(Oxford)
Abstract

There are various notions of rank, which measure the complexity of a tensor or polynomial. Cubic surfaces can be viewed as symmetric tensors.  We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven. We then explore the connection between the rank of a polynomial and the singularities of its vanishing locus, and we find the possible singular loci of a cubic surface of given rank. This talk is based on joint work with Eunice Sukarto.
 

Tue, 04 Feb 2020

15:30 - 16:30
L4

Genus one mirror symmetry

Dennis Eriksson
(Chalmers University)
Abstract

Mirror symmetry, in a crude formulation, is usually presented as a correspondence between curve counting on a Calabi-Yau variety X, and some invariants extracted from a mirror family of Calabi-Yau varieties. After the physicists Bershadsky-Cecotti-Ooguri-Vafa (henceforth BCOV), this is organised according to the genus of the curves in X we wish to enumerate, and gives rise to an infinite recurrence of differential equations. In this talk, I will give a general introduction to these problems, and present a rigorous mathematical formulation of the BCOV conjecture at genus one, in terms of a lifting of the Grothendieck-Riemann-Roch. I will explain the main ideas of the proof of the conjecture for Calabi-Yau hypersurfaces in projective space, based on the Riemann-Roch theorem in Arakelov geometry. Our results generalise from dimension 3 to arbitrary dimensions previous work of Fang-Lu-Yoshikawa.
 

This is joint work with G. Freixas and C. Mourougane.

Tue, 28 Jan 2020

15:30 - 16:30
L4

Donaldson-Thomas theory and tautological bundles on Hilbert schemes of points on surfaces

Noah Arbesfeld
(Imperial College)
Abstract

Tautological bundles on Hilbert schemes of points often enter into enumerative and physical computations. I will explain how to use the Donaldson-Thomas theory of toric threefolds to produce combinatorial identities that are expressed geometrically using tautological bundles on the Hilbert scheme of points on a surface. I'll also explain how these identities can be used to study Euler characteristics of tautological bundles over Hilbert schemes of points on general surfaces.

Wed, 15 Jan 2020

14:00 - 15:00
L3

Curve counting via stable objects in derived categories of Calabi-Yau 4-folds

Yalong Cao
(IPMU Tokyo)
Further Information

In a joint work with Davesh Maulik and Yukinobu Toda, we proposed a conjectural Gopakumar-Vafa type formula for the generating series of stable pair invariants on Calabi-Yau 4-folds. In this talk, I will present the recent joint work with Yukinobu Toda on how to give an interpretation of the above GV type formula in terms of wall-crossing phenomena in the derived category of coherent sheaves. 

Tue, 03 Dec 2019

15:45 - 16:45
L4

Combinatorial Lefschetz theorems beyond positivity

Karim Adiprasito
(Hebrew University)
Abstract

The hard Lefschetz theorem is a fundamental statement about the symmetry of the cohomology of algebraic varieties. In nearly all cases that we systematically understand it, it comes with a geometric meaning, often in form of Hodge structures and signature data for the Hodge-Riemann bilinear form.

Nevertheless, similar to the role the standard conjectures play in number theory, several intriguing combinatorial problems can be reduced to hard Lefschetz properties, though in extreme cases without much geometric meaning, lacking any existence of, for instance,  an ample cone to do Hodge theory with.

I will present a way to prove the hard Lefschetz theorem in such a situation, by introducing biased pairing and perturbation theory for intersection rings. The price we pay is that the underlying variety, in a precise sense, has itself to be sufficiently generic. For instance, we shall see that any quasismooth, but perhaps nonprojective toric variety can be "perturbed" to a toric variety with the same equivariant cohomology, and that has the hard Lefschetz property.

Finally, I will discuss how this applies to prove some interesting theorems in geometry, topology and combinatorics. In particular, we shall see a generalization of a classical result due to Descartes and Euler: We prove that if a simplicial complex embeds into euclidean 2d-space, the number of d-simplices in it can exceed the number of (d-1)-simplices by a factor of at most d+2.

Tue, 26 Nov 2019
15:30
L4

Degenerate Morse theory and quivers

Frances Kirwan
(Oxford)
Abstract


This talk is an update on joint work with Geoff Penington on extending Morse theory to smooth functions on compact manifolds with very mild nondegeneracy assumptions. The only requirement is that the critical locus should have just finitely many connected components. To such a function we associate a quiver with vertices labelled by the connected components of the critical locus. The analogue of the Morse–Witten complex in this situation is a spectral sequence of multicomplexes supported on this quiver which abuts to the homology of the manifold.

Tue, 19 Nov 2019

15:30 - 16:30
L4

3264 Conics in A Second

Bernd Sturmfels
(Berkeley and MPI Leipzig)
Abstract

Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given 
instance. This lecture illustrates how these two fields complement each other, especially in the light of emerging new applications. We start with a gem from
the 19th century, namely the 3264 conics that are tangent to five given conics in the plane. Thereafter we turn to current problems in statistics, with focus on 
maximum likelihood estimation for linear Gaussian covariance models.
 

Tue, 12 Nov 2019

15:30 - 16:30
L4

A motivic DT/PT correspondence via Quot schemes

Andrea T. Ricolfi
(SISSA)
Abstract

Donaldson-Thomas invariants of a Calabi-Yau 3-fold Y are related to Pandharipande-Thomas invariants via a wall-crossing formula known as the DT/PT correspondence, proved by Bridgeland and Toda. The same relation holds for the “local invariants”, those encoding the contribution of a fixed smooth curve in Y. We show how to lift the local DT/PT correspondence to the motivic level and provide an explicit formula for the local motivic invariants, exploiting the critical structure on certain Quot schemes acting as our local models. Our strategy is parallel to the one used by Behrend, Bryan and Szendroi in their definition and computation of degree zero motivic DT invariants. If time permits, we discuss a further (conjectural) cohomological upgrade of the local DT/PT correspondence.
Joint work with Ben Davison.
 

Tue, 05 Nov 2019

15:30 - 16:30
L4

Hilbert schemes of points of ADE surface singularities

Balazs Szendroi
(Oxford)
Abstract

I will discuss some recent results around Hilbert schemes of points on singular surfaces, obtained in joint work with Craw, Gammelgaard and Gyenge, and their connection to combinatorics (of coloured partitions) and representation theory (of affine Lie algebras and related algebras such as their W-algebra). 

Thu, 31 Oct 2019

16:30 - 17:30
L1

Complete Complexes and Spectral Sequences (COW Seminar)

Evangelos Routis
(Warwick)
Abstract

The space of complete collineations is an important and beautiful chapter of algebraic geometry, which has its origins in the classical works of Chasles, Schubert and many others, dating back to the 19th century. It provides a 'wonderful compactification' (i.e. smooth with normal crossings boundary) of the space of full-rank maps between two (fixed) vector spaces. More recently, the space of complete collineations has been studied intensively and has been used to derive groundbreaking results in diverse areas of mathematics. One such striking example is L. Lafforgue's compactification of the stack of Drinfeld's shtukas, which he subsequently used to prove the Langlands correspondence for the general linear group. 

In joint work with M. Kapranov, we look at these classical spaces from a modern perspective: a complete collineation is simply a spectral sequence of two-term complexes of vector spaces. We develop a theory involving more full-fledged (simply graded) spectral sequences with arbitrarily many terms. We prove that the set of such spectral sequences has the structure of a smooth projective variety, the 'variety of complete complexes', which provides a desingularization, with normal crossings boundary, of the 'Buchsbaum-Eisenbud variety of complexes', i.e. a 'wonderful compactification' of the union of its maximal strata.
 

Thu, 31 Oct 2019

14:45 - 15:45
L3

Classifying Fine Compactified Universal Jacobians (COW seminar)

Nicola Pagani
(Liverpool)
Abstract

A fine compactified Jacobian is a proper open substack of the moduli space of simple sheaves. We will see that fine compactified Jacobians correspond to a certain combinatorial datum, essentially obtained by taking multidegrees of all elements of the compactified Jacobian. This picture generalizes to flat families of curves. We will discuss a classification result in the case when the family is the universal family over the moduli space of curves. This is a joint work with Jesse Kass.

Thu, 31 Oct 2019

13:30 - 14:30
L3

Simplicity of Tannakian Categories (COW Seminar)

Martin Gallauer
(Oxford)
Abstract

Let A be a Tannakian category. Any exact tensor functor defined on A is either zero, or faithful. In this talk, I want to draw attention to a derived analogue of this statement (in characteristic zero) due to Jack Hall and David Rydh, and discuss some remarkable consequences for certain classification problems in algebraic geometry.

Tue, 29 Oct 2019

15:30 - 16:30
L4

Isotropic motives

Alexander Vishik
(Nottingham)
Abstract

The idea of isotropic localization is to substitute an algebro-geometric object (motive)
by its “local” versions, parametrized by finitely generated extensions of the ground field k. In the case of the so-called “flexible” ground field, the complexity of the respective “isotropic motivic categories” is similar to that of their topological counterpart. At the same time, new features appear: the isotropic motivic cohomology of a point encode Milnor’s cohomological operations, while isotropic Chow motives (hypothetically) coincide with Chow motives modulo numerical equivalence (with finite coefficients). Extended versions of the isotropic category permit to access numerical Chow motives with rational coefficients providing a new approach to the old questions related to them. The same localization can be applied to the stable homotopic category of Morel- Voevodsky producing “isotropic” versions of the topological world. The respective isotropic stable homotopy groups of spheres exhibit interesting features.

Tue, 22 Oct 2019

15:30 - 16:30
L4

Stability conditions and spectral networks

Fabian Haiden
(Oxford)
Abstract

Stability conditions on triangulated categories were introduced by Bridgeland, based on ideas from string theory. Conjecturally, they control existence of solutions to the deformed Hermitian Yang-Mills equation and the special Lagrangian equation (on the A-side and B-side of mirror symmetry, respectively). I will focus on the symplectic side and sketch a program which replaces special Lagrangians by "spectral networks", certain graphs enhanced with algebraic data. Based on joint work in progress with Katzarkov, Konstevich, Pandit, and Simpson.