# WIEN-BUDAPEST AG

## Fall 2021

The Algebraic Geometry and Differential Topology group of the Alfréd Rényi Institute of Mathematics and the Geometry and its Interfaces group of IST Austria organize a joint seminar, alternatingly in Budapest and Vienna. All are welcome to attend.

## Upcoming and past seminars

### Friday 15 October 2021, virtual seminar on Zoom

• 18:00--19:00 (CEST) Sergey Cherkis (The University of Arizona): The Geometry of Instantons on multi-Taub-NUT Space

This talk aims to give an overview of the bow construction of instantons on Asymptotically Locally Flat spaces. Most of these results are obtained in our joint work with Andres Larrain-Hubach and Mark Stern. We describe the emergence of a bow representation as a Dirac bundle associated with an instanton and how it comes equipped with a particular bow solution. This is what we call the Down transform. Then we describe the Up transform producing an instanton from any solution of the bow representation, and prove that it is inverse to the Down transform and that it is an isometry of the bow and instanton moduli spaces.

• 19:15--20:15 Richárd Rimányi (University of North Carolina at Chapel Hill): Enumerative geometry on Cherkis bow varieties

A singular subvariety in a smooth ambient space has a characteristic class in the cohomology of the ambient space. When the ambient space is the moduli space of interesting algebraic objects, then information on the characteristic classes can be turned to enumerative geometry results. This approach to enumeration is traditionally used when the ambient space is a Grassmannian or some related homogeneous space. In this talk we will introduce characteristic classes of singularities in more general ambient spaces: Cherkis bow varieties. In particular we will study the so-called elliptic stable envelope classes, and show their (conjectural) 3d mirror symmetry property.

### Friday 19 June 2020, virtual seminar on Zoom

• 2:00--4:30 Richard Wentworth (University of Maryland): WKB, Opers, and Limiting Configurations

The moduli space of SL(2) Higgs bundles on a closed Riemann surface is a hyperkaehler variety homeomorphic to the space of SL(2) flat connections. The end of the moduli space is parametrized by "limiting configurations" that are related to asymptotic spectral data for the Higgs bundle. Motivated by results in the exact WKB analysis of the Schrodinger equation, we will discuss the limiting behavior of complex projective structures (or "Opers"). A bridge between the various points of view is provided by Thurston's notion of a "pleated surface". This is joint work with Andreas Ott, Jan Swoboda, and Michael Wolf.

### Thursday 2 April 2020, virtual seminar on Zoom

• 2:00--4:30 Balázs Szendrői (University of Oxford): Hilbert schemes of points on singular surfaces: combinatorics, geometry, and representation theory

Given a smooth algebraic surface S over the complex numbers, the Hilbert scheme of points of S is the starting point for many investigations, leading in particular to generating functions with modular behaviour and Heisenberg algebra representations. I will explain aspects of a similar story for surfaces with rational double points, with links to algebraic combinatorics and the representation theory of affine Lie algebras. I will in particular explain our 2015 conjecture concerning the generating function of their Euler characteristics, and aspects of more recent work that lead to a very recent proof of the conjecture by Nakajima. Joint work with Gyenge and Némethi, respectively Craw, Gammelgaard and Gyenge.

### Thursday 31 October 2019, Vienna, Uni Wien (Oskar-Morgenstern-Platz 1, 12th floor, Sky Lounge)

• 2:00--4:30 Francois Loeser (Sorbonne Université): From the McKay correspondence to a motivic version of topological mirror symmetry

Groechenig, Wyss and Ziegler have recently proved a conjecture of Hausel and Thaddeus concerning an equality between stringy Hodge numbers of moduli spaces of Higgs bundles for $\mathrm{SL}_n$ and $\mathrm{PGL}_n$. A crucial ingredient in their approach is the use of $p$-adic integration in the fibres of the Hitchin fibration. I will start by presenting some old work with Jan Denef on the McKay correspondence (on resolutions of quotients by finite groups) that reappears in this context and then discuss a motivic version of the results of Groechenig, Wyss and Ziegler which is joint work with Dimitri Wyss.

### Friday 27 September 2019, Rényi Institute, Budapest, Main Lecture Hall

• 2:00--3:00 Oscar Garcia-Prada (ICMAT Madrid): Higgs bundles and higher Teichmüller spaces

Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this talk, I will describe examples of such special components in moduli spaces of G-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into a semisimple Lie group G. These special components, which occur only for certain groups, generalize Teichmüller space (here G=PSL(2,R)), and are the main object of study of higher Teichmüller theory.

• 3:30--4:30 Róbert Szöke (ELTE Budapest): Some go and no-go theorem in geometric quantization

Geometric quantization intends to construct the quantum system (a Hilbert space and self-adjoint operators on it) out of a classical mechanical system (a symplectic manifold as the phase space and functions on it as physical observables). Although several well known no-go results show the limitations of this process, up till now this is the only method that (in lucky situations) does produce an actual Hilbert space and operators thereon. In the lecture we shall discuss what can be done when the phase space is the cotangent bundle of a compact Lie group or more generally of a compact Riemannian symmetric space.

### Thursday 4 July 2019, Rényi Institute, Budapest, Main Lecture Hall

• 2:00--3:00 George Lusztig (MIT): Total positivity in Lie groups

• 3:30--4:30 Mátyás Domokos (Rényi Institute): Degree bounds in noncommutative invariant theory

### Friday 22 March 2019, Vienna, Uni Wien (Oskar-Morgenstern-Platz 1, Room SR 16)

• 2:00--3:00 Olivier Schiffmann (Orsay): Moduli spaces of vector and Higgs bundles on curves and Lie theory

We will describe some Lie theoretic structures hidden in the enumerative geometry of moduli spaces of vector bundles or Higgs bundles on smooth projective curves. Time permitting, we will also introduce a family of Hecke operators acting on the cohomology of the moduli spaces (stacks) of vector bundles, and describe their relation with the cohomology rings.

• 3:30--4:30 Mikhail Shkolnikov (ISTA): Renormalization and functoriality of sandpiles

The abelian sandpile model was expected to be scale-invariant since its invention. Indeed, this property is the crucial part of the self-organised criticality paradigm whose illustration was the purpose to introduce the sandpile model. However, no precise renormalization procedure was known until very recently. In our joint work with Moritz Lang we suggest one: it goes through the extended sandpile group -- a tropical abelian variety defined over Z constructed from a discrete domain which is functorial under étale coverings. In this framework, the classical sandpile group is seen as a subgroup of integer points of the extended one and the renormalization is the truncation of the homomorphism associated to the domain inclusion.

### Friday 7 December 2018, Rényi Institute, Budapest, Main Lecture Hall

• 2:00--3:00 Olivier Wittenberg (CNRS, Orsay): On the integral Hodge conjecture for real varieties

We formulate an analogue of the integral Hodge conjecture for real algebraic varieties. As in the complex case, the real integral Hodge conjecture can fail, but is plausible for 1-cycles on varieties whose geometry is simple enough (rationally connected varieties, uniruled threefolds). Its study allows us to obtain new results about properties that are either classical (algebraicity of the homology of the real locus, approximation of loops by algebraic curves) or not so classical (existence of a real curve of even genus). This is joint work with Olivier Benoist.

• 3:30--4:30 András Stipsicz (Rényi Institute): Invariants of knots from knot Floer homology

We review the construction of various versions of knot Floer homologies, recall the main results concerning these invariants and show an extension of them using the double branched cover of the 3-sphere along the knot at hand.

### Friday 9 November 2018, Erwin Schrödinger-Institut, Wien, Boltzmanngasse 9a

• 2:00--3:00 Grigory Mikhalkin (Geneva): Maximally writhed real algebraic knots and links

The Alexander--Briggs tabulation of knots in R^3 (started almost a century ago, and considered as one of the most traditional ones in classical Knot Theory) is based on the minimal number of crossings for a knot diagram. From the point of view of Real Algebraic Geometry it is more natural to consider knots in RP^3 rather than R^3, and use a different number also serving as a measure of complexity of a knot: the minimal degree of a real algebraic curve representing this knot. As it was noticed by Oleg Viro about 20 years ago, the writhe of a knot diagram becomes an invariant of knots in the real algebraic set-up, and corresponds to a Vassiliev invariant of degree 1. In the talk we'll survey these notions, and consider the knots with the maximal possible writhe for its degree. Surprisingly, it turns out that there is a unique maximally writhed knot in RP^3 for every degree d. Furthermore, this real algebraic knot type has a number of other characteristic properties, from the minimal number of diagram crossing points (equal to d(d-3)/2) to the minimal number of transverse intersections with a plane (equal to d-2). Based on a series of joint works with Stepan Orevkov.

• 3:30--4:30 Tim Browning (IST Austria): Free rational curves and the circle method

We use a function field version of the Hardy–-Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre's reformulation of the Batyrev–-Manin conjecture in terms of slopes with respect to the tangent bundle. This is joint work with Will Sawin.

### Friday 25 May 2018, Erwin Schrödinger-Institut, Wien, Boltzmanngasse 9a

• 2:00--3:00 Lothar Göttsche (ICTP Trieste): Virtual topological invariants of moduli spaces of sheaves on surfaces

Using arguments from theoretical physics, Vafa and Witten gave a generating function for the Euler numbers of moduli spaces of rank 2 coherent sheaves on algebraic surfaces. These moduli spaces are in general very singular, but they carry a perfect obstruction theory (they are virtually smooth). This gives virtual versions of many invariants of smooth projective varieties. Such virtual invariants occur everywhere in modern enumerative geometry, like Gromov-Witten invariants and Donaldson Thomas invariants, when attempting to make sense of the predictions from physics. We conjecture that the Vafa-Witten formula is true for the virtual Euler numbers. We confirm this conjecture in many examples. Then we give refinements of the conjecture, first to the chi_y genus and then to the elliptic genus, and finally to the cobordism class. If there is time, we will also mention the generalization to moduli spaces of rank 3 sheaves, where the physics prediction appears to be incorrect. Our approach is based on Mochizuki's formula which reduces virtual intersection numbers on moduli spaces of sheaves to intersection numbers on Hilbert schemes of points. This is joint work with Martijn Kool.

• 3:30--4:30 Philippe Michel (EPFL Lausanne): Applied l-adic cohomology

In this lecture we describe various recent applications of l-adic cohomology techniques (around the works of Deligne, Laumon and Katz) to core problems in analytic number theory: these include the counting of prime numbers in arithmetic progressions and the study of moment of L-functions. We will in particular highlight how the original problem and the methods in use to approach it shape the geometric question.

### Friday 20 April 2018, Rényi Institute, Budapest, Main Lecture Hall

• 2:00--3:00 Adrian Langer (University of Warsaw): Rigid representations of projective fundamental groups

I will talk about some results related to Simpson's conjecture: a rigid irreducible representation of the fundamental group of a smooth complex variety comes from some family of smooth projective varieties. I will sketch the proof of this conjecture in the case of rank 3 representations. This is joint work with Carlos Simpson.

• 3:30--4:30 András Némethi (Rényi Institute): Pairs of invariants of surface singularities

We discuss several invariants of complex normal surface singularities with a special emphasis on the comparison of analytic--topological pairs of invariants. Additionally, we list several open problems related with them.

### Friday 24 November 2017, Rényi Institute, Budapest, Main Lecture Hall

• 2:00--3:00 Angelo Vistoli (SNS Pisa): Chow rings of some moduli spaces of smooth curves

There is by now an extensive theory of rational Chow rings of moduli spaces of smooth curves. The integral version of these Chow rings is not as well understood. I will survey what is known, including some recent developments.

• 3:30--4:30 Szilárd Szabó (Technical University, Budapest): On the irregular Hitchin map

After an introduction to moduli spaces of irregular parabolic Higgs bundles on curves, we turn to a description of the irregular Hitchin fibration in the case of semi-simple polar parts. The description is given in terms of a Hilbert scheme of curves on a certain ruled surface and its relative Picard scheme. We then describe a concrete example, obtained in recent joint work with P. Ivanics and A. Stipsicz, of a moduli space of dimension two, with particular emphasis on the dependence of the classes of the singular fibers on the parabolic weights. We also discuss failure of properness of the Hitchin map in the case of a non-closed orbit and a partial compactification obtained by the corresponding orbit closure.

### Friday 13 October 2017, Uni Wien (Oskar-Morgenstern-Platz 1, Room HS10)

• 2:00--3:00 Rahul Pandharipande (ETH Zürich): Classes on the moduli of K3 surfaces

I will discuss recent progress in the study of tautological classes on the moduli space of K3 surfaces --- questions related to Picard groups, Noether-Lefshetz loci, and a possible numerical theory. The subject has parallels to the study of the moduli of curves.

• 3:30--4:30 Tamás Hausel (IST Austria): Mirror symmetry with branes by equivariant Verlinde formulae

I will discuss an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL(2) Higgs moduli space on a Riemann surface. One side we have the components of the Lagrangian brane of U(1,1) Higgs bundles whose mirror was proposed by Nigel Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL(2) Higgs moduli space. The agreement arises from a mysterious Hard Lefschetz type functional equation. This gives strong computational evidence for Hitchin’s proposal. This is joint work with Anton Mellit and Du Pei.

### Friday 21 April 2017, Erwin Schrödinger-Institut, Boltzmanngasse 9a, Boltzmann Lecture Hall Map

• 2:00--3:00 Yuri Tschinkel (Courant Institute, New York): Rational points and rational varieties

I wil discuss connections between geometry and arithmetic, with implications for both sides.

• 3:30--4:30 Anton Mellit (IST Austria): Cohomology of character varieties

Character varieties are spaces that parametrize representations of fundamental groups of Riemann surfaces with punctures, with prescribed local monodromies around the punctures. Via Simpson's correspondence, they are diffeomorphic to moduli spaces of semistable Higgs bundles. A conjecture of Hausel, Letellier and Rodriguez-Villegas gives an explicit formula for the Betti numbers of these spaces, which hints on a connection with Hilbert schemes. In another development Gorsky, Oblomkov, Rasmussen and Shende conjecture a connection between Hilbert schemes and homological invariants of torus knots and links. The purpose of this talk is to show that certain cell decompositions of character varieties produce an explicit connection between the two conjectures, allows us to calculate the cohomologies in some examples, and implies the so-called curious hard Lefschetz property.

### Friday 10 March 2017, Rényi Institute, Main Lecture Hall

• 2:00--3:00 Gábor Farkas (Humboldt-Universität Berlin): Compact moduli of holomorphic differentials

The moduli space of holomorphic differentials (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. I will discuss a compactification of these strata in the moduli space of Deligne-Mumford stable pointed curves, which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the strata of holomorphic differentials and as a consequence, one can determine the cohomology classes of the strata. This is joint work with Rahul Pandharipande.

• 3:30--4:30 Tamás Szamuely (Rényi Institute): Homotopy groups of algebraic homogeneous spaces

According to a classical theorem of Elie Cartan, compact Lie groups have trivial second homotopy. I shall explain the analogue of this result in algebraic geometry (in all characteristics). I shall also show how to compute low-degree homotopy groups of homogeneous spaces of algebraic groups in terms of invariants of the group and the stabilizer. This is joint work with Cyril Demarche.