Seminaire de la Tortue
Université de Genève, Section de Mathématiques
Villa
Battelle, 7 route de Drize, 1227 Carouge
29.03.2017 
from 14:00 in Villa
Battelle 

Wednesday 
Quantum groups and the cohomology of quiver varieties, III 



by Eric Vasserot (Université Pierre et Marie Curie / Université
Paris Diderot) 


We'll first give an
introduction to Nakajima's quiver varieties and their cohomology
and/or Ktheory.Next, when the quiver is of finite
type, we'll construct the action of yangians/quantum
affine algebras on these cohomology groups.
Finally, when the quiver is of Jordan type, we'll construct the action of the
elliptic Hall algebra. 


13.03.2017 
from 14:00 in Villa
Battelle 

Monday 
Quantum groups and the cohomology of quiver varieties, II 



by Eric Vasserot (Université Pierre et Marie Curie / Université
Paris Diderot) 


We'll first give an
introduction to Nakajima's quiver varieties and their cohomology
and/or Ktheory.Next, when the quiver is of finite
type, we'll construct the action of yangians/quantum
affine algebras on these cohomology groups.
Finally, when the quiver is of Jordan type, we'll construct the action of the
elliptic Hall algebra. 

10.03.2017 
from 14:00 in Villa
Battelle 
Friday 
Quantum groups and the cohomology of quiver varieties, I 
by Eric Vasserot (Université Pierre et Marie Curie / Université
Paris Diderot) 

We'll first give an
introduction to Nakajima's quiver varieties and their cohomology
and/or Ktheory.Next, when the quiver is of finite
type, we'll construct the action of yangians/quantum
affine algebras on these cohomology groups.
Finally, when the quiver is of Jordan type, we'll construct the action of the
elliptic Hall algebra. 
08.03.2017 
from 13:30 in Villa
Battelle 
Wednesday 
Introduction to the derived
category of sheaves. 
by Anton Fonarev (HSE
Moscow) 

23.02.2017 
from 14:30 in Villa
Battelle 
Wednesday 
The logarithmic deRham complex 
by Simone Chiarello (UniGe) 

I will prove that the deRham cohomology of a smooth
complex algebraic variety can be computed as the hypercohomology
of a suitable complex of sheaves, defined over a compactification
of the variety; this allows to put a mixed Hodge
structure on its cohomology. The exposition will be
enhanced by some downtoearth examples. 
08.03.2016 
from 10:30 in Villa
Battelle 
Tuesday 
Introduction to Higgs
Bundles 
by Simone Chiarello (UniGe) 

We will give an
introduction to the construction of the moduli spaces of Higgs bundles on a
Riemann surface based on the work of Hitchin.

03.03.2014 
from 10:30 in Villa
Battelle 
Monday 
Inversion of adjunction for
rational and Du Bois singularities 
by Sandor
Kovacs (University of Washington) 

We prove that Du Bois singularities
are invariant under small deformation and that the relationship of the
notions of rational and Du Bois singularities resembles that of canonical and
log canonical varieties. In particular, if a member of a family has Du Bois
singularities, then the total space of the family has rational singularities
near the given fiber. 
06.12.2013 
from 11:30 in Villa Battelle 
Friday 
Schur polynomials, tableau and the Littlewood–Richardson rule 2 
by Alexander Paunov (UniGe) 
05.12.2013 
from 16:15 in Villa
Battelle 

Thursday 
Holomorphic
maps between projective spaces are maximally singular 



by Laszlo Feher
(ELTE Budapest) 


We
conjecture that a nonlineal homorphic
map between projective spaces is maximally singular: Suppose that a contact singularity x can be represented
with polynomials of degree at most k. Suppose moreover that the expected
dimension of the locus of xtype singularities is nonnegative. Then for any holomorphic map of degree at
least k there is an xtype (or more complicated) singularity. 

29.11.2013 
from 10:30 in Villa
Battelle 
Friday 
Localization
of Hirzebruch chi_ygenus 
by Andrzej
Weber (University of Warsaw) 

We study genera
of complex algebraic varieties. If a genus g satisfies g(E)=g(F)g(B)
for a fibration F>E>B with F being a
projective space then it is called rigid. It turns out that the universal
such genus is the Hirzebruch chi_ygenus.
The rigidity property has two important consequences: 2) the Hirzebruch genus can be
localized for varieties with torus action: g(X) is equal to the sum of local
contributions coming from the fixed points. The
situation when the action has discrete fixed points set is of particular
interest. We will give samples of computations and investigate positivity
property in particular cases. Also, we will discuss the underlying
construction on the level of Ktheory. 
15.11.2013 
from 10:30 in Villa
Battelle 
Friday 
Schur polynomials, tableau and the Littlewood–Richardson rule 1 
by Alexander Paunov (UniGe) 
11.11.2013 
from 10:30 in Villa
Battelle 
Monday 
Rimanyi's method of restriction
equations: computing Thom and "residual" polynomials for multisingularities 
by Natalia Kolokolnikova (UniGe) 



08.11.2013 
from 10:30 in Villa Battelle 
Friday 
Espace des arc des variétés sphériques et intégration motivique 
by Anne Moreau (Université de Poitiers) 

Résumé: Cet exposé porte sur des travaux (passés et en cours) en commun avec Victor Batyrev.
Nous nous intéressons à l'intégrale motivique sur l'espace des arcs d'une Gvariété sphérique QGorenstein X où G est un groupe
réductif connexe. Nous donnons une formule
pour la fonction de cordes
de X en terme de son éventail
colorié associé. Grâce à cette formule, nous établissons un nouveau critère
de lissité pour les variétés
horosphériques localement
factorielles. Nous conjecturons
que ce critère reste valable pour les variétés sphériques. 
27.09.2013 
from 10:30 in Villa
Battelle 
Friday 
On a problem
in real enumerative geometry 
by László
Fehér (ELTE, Budapest) 
12.06.2013 
from 14:30 in Villa
Battelle 
Wednesday 
Universal
polynomials for counting singular subvarieties 
by Yujong
Tzeng (Harvard) 

How many nodal
degree d plane curves are tangent to a given line? The celebrated CaporasoHarris recursion formula gives a complete answer
for any number of nodes, degrees, and all possible tangency conditions. In
this talk, I will report my recent work on the numbers of singular curves
with given tangency conditions on general surfaces, and singular subvarieties of higher dimensional subvarieties.
These numbers are given by universal polynomials, which generalizes Gottsche's conjecture. Unlike the nodal curve case, these
polynomials are not determined yet but it is possible to discuss their
asymptotic behavior and some properties of the generating function. 
15.03.2013 
from 14:30 in Villa
Battelle 
Friday 
Nekrasov's partition function and refined
DonaldsonThomas theory 
by Balázs
Szendrői (Oxford) 

01.03.2013 
from 11:15 in Villa
Battelle 
Friday 
Mock
theta functions and representations of affine Lie superalgebras 
by Victor Kac
(MIT) 

17.12.2012 
from 10:30 at Villa
Battelle 
Monday 
Damon's
theorem and Schur positivity for Thom polynomials
of contact singularities 
by Natalia Kolokolnikova (UniGe) 

Thom
polynomials of contact singularities have an important property: they can be expressed
in terms of relative Chern classes (this property
is known as Damon's theorem) and for this expression the Schur
positivity conjecture holds. Damon's theorem is a wellknown result, but the
references for the proof are hard to come by. I'll give a proof of this
theorem and a proof of the Schur positivity
conjecture in a way different from Pragacz's. 
10.12.2012 
from 10:30 at Villa
Battelle 
Monday 
Conormal bundles of Schubert varieties
and Yangian weight functions 
by Richárd
Rimányi (University of North Carolina; UniGe) 

There are
remarkable – more or less canonical – isomorphisms
between objects in geometry (e.g. equivariant cohomology of cotangent bundles of flag varieties) and
objects in quantum algebra. Using these bridges between geometry and algebra
we will present algebraic notions, such as conformal blocks, Rmatrices, Yangian weight functions, etc
in geometry. I will report on some joint works with Varchenko
and Tarasov, as well as results of Maulik and Okounkov. 
23.11.2011 
from 10:30 at Villa
Battelle 
Friday 
The
birthday problem, the chromatic polynomial, and Stanley conjecture 
by Alexander Paunov (UniGe) 

I will
present an interesting connection between the generalized birthday problem and
Stanley's epositivity conjecture. The talk will be focused on the properties
of clawfree graphs, colorings and extremal points
of chromatic functions. 
19.11.2012 
from 10:30 at Villa
Battelle 
Monday 
Moduli
space of Higgs bundles I. 
by Máté
Juhász (UniGe) 

In this talk
we will review the construction of the Quot scheme
and the moduli space of vector bundles, the first step towards the
construction of the moduli space of Higgs bundles. As time permits, a few
preliminary concepts concerning Higgs bundles shall be introduced as
well. The talk
presupposes a basic knowledge about GIT. 
09.11.2012 
from 10:00 at Villa
Battelle 
Friday 
The cohomology ring of Hilbert schemes for K3 surfaces II. 
by Zsolt
Szilágyi (UniGe) 

05.11.2012 
from 10:30 at Villa
Battelle 
Monday 
The cohomology ring of Hilbert schemes for K3 surfaces I. 
by Zsolt
Szilágyi (UniGe) 

The talk is
based on the article "The cup product of Hilbert schemes for K3
surfaces" by M. Lehn and Ch. Sorger and its
aim is to present their construction of the cohomology
ring of Hilbert schemes. This construction motivated the one by Costello and Grojnowski presented in the previous talks. 
26.10.2012 
from 10:00 at Villa
Battelle 
Friday 
The cohomology of the Hilbert scheme of points via Cherednik algebras II 
by Emanuel Stoica (UniGe) 

In part I of
the talk, we gave a rather schematic outline of the clever construction of
the cohomology ring of Hilbert scheme of points on
a surface, based on Cherednik algebras, following
Costello and Grojnowski. In part II, we will try to
clarify the construction with more explanations and details. 
22.10.2012 
from 10:30 at Villa
Battelle 
Monday 
The cohomology of the Hilbert scheme of points via Cherednik algebras I. 
by Emanuel Stoica (UniGe) 

We will
outline the clever construction of the cohomology
ring of the Hilbert scheme of points on a smooth surface based on Cherednik algebras, following Costello and Grojnowski. 
18.05.2012 
from 10:30 at Villa
Battelle 
Friday 
Combinatorial
interpretation of ecoefficients of chromatic symmetric functions 
by Alexander Paunov (UniGe) 

04.05.2012 
from 10:30 at Villa
Battelle 
Friday 
Schurpositivity and 3+1 conjecture 
by Alexander Paunov (UniGe) 

30.04.2012 
from 11:00 at Villa
Battelle 
Monday 
Chromatic
functions and epositivity 
by Alexander Paunov (UniGe) 

13.12.2011 
from 13:30 at Villa
Battelle 
Tueday 
Topological
recursion relations in enumerative geometry 
Lecture
4: Some hints towards the proof 

by Bertrand Eynard (IPhT, CEA Saclay; UniGe) 

The topology
of moduli spaces of Riemann surfaces embedded into a "target
space", can be partially understood by computing intersection numbers of
various homology classes. Enumerative geometry aims at computing those
intersection numbers, or more precisely generating series for families of
intersection numbers. Recently it was realized that many enumerative geometry
problems can be solved by a universal topological
recursion (recursion on the Euler characteristics). This
minicourse is a basic introduction to those concepts. Outline: Lecture 1:
Introduction to enumerative geometry, GromovWitten
invariants, Intersection numbers, Kontsevich
integral 
06.12.2011 
from 13:30 at Villa
Battelle 
Tuesday 
Topological
recursion relations in enumerative geometry 
Lecture
3: The remodelling conjecture, introduction to
mirror symmetry, statement of the conjecture 

by Bertrand Eynard (IPhT, CEA Saclay; UniGe) 

The topology
of moduli spaces of Riemann surfaces embedded into a "target space",
can be partially understood by computing intersection numbers of various
homology classes. Enumerative geometry aims at computing those intersection
numbers, or more precisely generating series for families of intersection
numbers. Recently it was realized that many enumerative geometry problems can be solved by a universal topological recursion
(recursion on the Euler characteristics). This minicourse is a basic introduction to those concepts. Outline: Lecture 1:
Introduction to enumerative geometry, GromovWitten
invariants, Intersection numbers, Kontsevich
integral 
02.12.2011 
from 14:00 at Villa
Battelle 
Friday 
Minicourse
on JeffreyKirwan reduction theorem 
Lecture
8: JeffreyKirwan reduction theorem, circle case 

by Zsolt
Szilágyi (UniGe) 

JeffreyKirwan formula for symplectic
reduction via symplectic cut, circle case. 
29.11.2011 
from 13:30 at Villa
Battelle 
Tueday 
Topological
recursion relations in enumerative geometry 
Lecture 2:
Recursion relations for WeilPetersson volumes, for
Hurwitz numbers, for the GromovWitten theory of
C^3. 

by Bertrand Eynard (IPhT, CEA Saclay; UniGe) 

The topology
of moduli spaces of Riemann surfaces embedded into a "target space",
can be partially understood by computing intersection numbers of various
homology classes. Enumerative geometry aims at computing those intersection
numbers, or more precisely generating series for families of intersection
numbers. Recently it was realized that many enumerative geometry problems can be solved by a universal topological recursion
(recursion on the Euler characteristics). This minicourse is a basic introduction to those concepts. Outline: Lecture 1:
Introduction to enumerative geometry, GromovWitten
invariants, Intersection numbers, Kontsevich
integral 
25.11.2011 
from 14:00 at Villa
Battelle 
Friday 
Moduli
space of Higgs bundles 
by András
Szenes (UniGe) 

22.11.2011 
from 13:30 at Villa
Battelle 
Tuesday 
No
seminar today. 
18.11.2011 
from 14:00 at Villa
Battelle 
Friday 
Minicourse
on JeffreyKirwan reduction theorem 
Lecture
7: Symplectic reduction 

by Zsolt
Szilágyi (Unige) 

Finishing
the proof ABBV theorem. Symplectic reduction and symplectic cut. 
15.11.2011 
from 13:30 at Villa
Battelle 
Tuesday 
Topological
recursion relations in enumerative geometry 
Lecture
1: Introduction to enumerative geometry, GromovWitten
invariants, Intersection numbers, Kontsevich
integral 

by Bertrand Eynard (IPhT, CEA Saclay; UniGe) 

The topology
of moduli spaces of Riemann surfaces embedded into a "target
space", can be partially understood by computing intersection numbers of
various homology classes. Enumerative geometry aims at computing those
intersection numbers, or more precisely generating series for families of
intersection numbers. Recently it was realized that many enumerative geometry
problems can be solved by a universal topological
recursion (recursion on the Euler characteristics). This minicourse is a basic introduction to those concepts. Outline: Lecture 1:
Introduction to enumerative geometry, GromovWitten
invariants, Intersection numbers, Kontsevich
integral 
11.11.2011 
from 14:00 at Villa
Battelle 
Fridaz 
Minicourse
on JeffreyKirwan reduction theorem 
Lecture
6: AtiyahBottBerlineVergne localization theorem 

by Zsolt
Szilágyi (UniGe) 

Equivariant integration and ABBV
localization theorem 
08.11.2011 
"Lie group
and moduli space" seminar from 13h30. 
Tuesday 
For more
information visit: 
04.11.2011 
No seminar
today. 
Friday 

01.11.2011 
from 15:00 at Villa
Battelle 
Tuesday 
Minicourse
on Thom polynomials 
Lecture
5: Groebner basis 

by András
Szenes (UniGe) 

Groebner basis 
28.10.2011 
from 14:00 at Villa Battelle 
Friday 
Minicourse
on JeffreyKirwan reduction theorem 
Lecture
5: (Equivariant) characteristic classes, part 2. 

by Zsolt
Szilágyi (UniGe) 

Continuation
from last time: characteristic classes via ChernWeil
theory, equivariant characteristic classes 
25.10.2011 
No seminar
today. 
Tuesday 

21.10.2011 
from 14:00 at Villa
Battelle 
Friday 
Minicourse
on JeffreyKirwan reduction theorem 
Lecture
4: (Equivariant) characteristic classes 

by Zsolt
Szilágyi (UniGe) 

Cartan map (continued from last
time), characteristic classes of vector bundles, equivariant
characteristic classes 
18.10.2011 
from 13:30 at Villa
Battelle 
Tuesday 
Minicourse
on Thom polynomials 
Lecture 4:
Thom polynomials and commutative algebra 

by András
Szenes (UniGe) 

Resolutions,
Groebner bases and Thom polynomials 
14.10.2011 
from 14:00 at Villa
Battelle 
Friday 
Minicourse
on JeffreyKirwan reduction theorem 
Lecture
3: Cartan model 

by Zsolt
Szilágyi (UniGe) 

From Borel model to Cartan model
using Kalkman's trick, characteristic map, ChernWeil transgression, Cartan
isomorphism. 
11.10.2011 
from 13:30 at Villa
Battelle 
Tuesday 
Minicourse
on Thom polynomials 
Lecture
3: The Thom polynomials and syzygies 

by András
Szenes (UniGe) 

Hilbert
syzygy theorem, Hilbert functions and Hilbert polynomials, first construction
of Thom polynomials. 
07.10.2011 
from 14:00 at Villa Battelle 
Friday 
JeffreyKirwan reduction formula II. 
by Zsolt
Szilágyi (UniGe) 

Second talk
on equivariant cohomology. This talk is
part of a series of talks, aiming to explain the JeffreyKirwan
reduction formula. 
04.10.2011 
from 13:30 at Villa
Battelle 
Tuesday 
Minicourse
on Thom polynomials 
Lecture
2: How to ask a question in eumerative geometry? 

by András
Szenes (UniGe) 

Principal
and associated bundles, introduction to Thom polynomials. 
30.09.2011 
from 14:00 at Villa
Battelle 
Friday 
JeffreyKirwan reduction formula I. 
by Zsolt
Szilágyi (UniGe) 

Introduction
to equivariant cohomology,
following the book of GuilleminSternberg: Supersymmetry
and equivariant de Rham theory.
This talk is part of a series of talks, aiming to explain the JeffreyKirwan reduction formula. 
27.09.2011 
from 13:30 at Villa
Battelle 
Tuesday 
Minicourse
on Thom polynomials 
Lecture
1: Enumerative Geometry and Cohomology 

by András
Szenes (UniGe) 

Vector
bundles, Chern classes, Bott localization formula. 
Contact:
András Szenes Webmaster: Zsolt Szilágyi