## WIEN-BUDAPEST AG## Fall 2021 |
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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.

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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.
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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.
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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.
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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.
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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.
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*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.*

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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.
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*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.
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*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. *

*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.*

*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. *

*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.
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*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.
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*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.*

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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.
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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.
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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.
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*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. *

* 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.*

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

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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. *

* 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.
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*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. *