Algebra and Geometry Seminar
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Abstracts of talks 2013/2014
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Loïc Foissy (Université de Calais)
Feynman graphs, trees and combinatorial
Dyson-Schwinger equations (parte
I, la parte II avrà luogo il 30 ottobre)
Feynman graphs are
used in Quantum Field Theory to represent the possible interactions between the
particles studied by the theory. They are used to compute certain physical
constants (mass or charge of the electron, for example). For this, several
infinite series of Feynman graphs are considered; they can be uniquely defined
as the solution of a certain system of equations, called the combinatorial
Dyson-Schwinger equations of the system.
Moreover, the combinatorial operations on Feynman graphs (insertion, extraction
contraction) give them a structure of a Hopf algebra. The solution of the
Dyson-Schwinger system should be compatible with this algebraic structure, and
this imposes strong conditions on the system itself.
We present a classification of the systems which satisfy these conditions, with
the help of the Connes-Kreimer Hopf algebra and its universal property, which
allows
to replace Feynman graphs by rooted trees.
Paolo
Antonini (Paris-Sud XI - Orsay)
Fibrati piatti, algebra di von Neumann e
K-teoria a coefficient in R/Z
A un fibrato piatto E su una
varietà M, Atiyah Patodi e
Singer hanno associato una classe di K-teoria di M a coefficienti in R/Z. Il pairing di questa classe con una classe
di K-omologia rappresentata da un operatore ellittico e il teorema dell'indice
per fibrati piatti ed è
espresso dall'invariante ρ di Atiyah Patodi e Singer e dal flusso
spettrale di un opportuno cammino di operatori. Il passaggio a coefficienti R/Z
è necessario per
trattare gli invariati secondari come ρ e η. Nello stesso lavoro gli
autori hanno suggerito la possibilità di costruire un modello della
K-teoria a coefficienti in R/Z basato sulla teoria dei fattori di algebre
di von Neumann. Verrà esposto un lavoro in collaborazione con Sara
Azzali e Georges Skandalis in cui tale modello è costruito in un modo semplice non solo per
la K-teoria degli spazi topologici ma anche per le C*-algebre
nella categoria Bootstrap. La classe di K-teoria a coefficienti in R/Z
di un fibrato piatto ha in questo modo una descrizione puramente in termini di
K-teoria ordinaria ed è
collegata all'algebra di von Neumann della foliazione definita dalla struttura
piatta sul fibrato dei frames di E.
Inoltre il pairing di APS si realizza in questo modo come un prodotto di
Kasparov.
[AAS] P. Antonini, S. Azzali, G. Skandalis. Flat bundles, von Neumann algebras
and K-theory with R/Z-coefficients. arXiv:1308.0218
[APS1] M.F. Atiyah, V.K. Patodi, I.M. Singer. Spectral asymmetry and Riemannian geometry. I. Math.
Proc. Cambridge Philos. Soc. 77 (1975), 43–49.
[APS2] M.F. Atiyah, V.K. Patodi, I.M. Singer. Spectral asymmetry and Riemannian
geometry. II. Math. Proc. Cambridge Philos. Soc. 78 (1975), 405–432.
[APS3] Atiyah, V.K. Patodi, I.M. Singer. Spectral asymmetry and Riemannian
geometry. III. Math. Proc. Cambridge Philos. Soc. 79
(1976), 71–99
[AAS] P. Antonini, S. Azzali, G. Skandalis. Flat bundles, von Neumann algebras and K-theory with R/Z coefficients. arXiv:1308.0218
Homepage: http://paoloanton.wix.com/
Frédéric
Patras (Université de Nice)
The fine structure of time-ordered products
Time ordered products
are familiar objects: they appear in the Picard expansion of the solution of
linear differential equations, almost everywhere in quantum physics and more
generally, in disguise, in many computations in algebra, analysis and
probability.
There have been many advances recently in their understanding from a pure
algebraic (Lie and group-theoretical) point of view. The keywords are:
Rota--Baxter algebras, pre-Lie algebras, Lie idempotents.
The talk will survey some of these developments, their background, and insist
on the underlying pre-Lie structures. (Based on joint work with K.
Ebrahimi-Fard).
Wednesday
23 October 2013 - Aula Consiglio, 15:45
June Huh (University of Michigan)
Rota's conjecture,
and the nef cone of the permutohedral toric variety
Rota's conjecture predicts that the coefficients of
the characteristic polynomial of a matroid form a log-concave sequence. I will
outline a proof for representable matroids using Milnor numbers and the Bergman
fan. The same approach to the conjecture in the general case (for possibly
non-representable matroids) leads to several intriguing questions on higher
codimension algebraic cycles in the toric variety associated to the
permutohedron.
Pietro
Mercuri (“Sapienza” Università di Roma)
Non-split Cartan
modular curves and rational points
The case of modular curves associated to Γ0(p),
for p a prime number, is well known, but the case of modular curve
associated to a non-split Cartan congruence subgroup is still not well
understood. In particular, it is interesting, by a Serre conjecture, to know
the rational points on these curves. Some papers in last 30 years shed some
light on this topic. I will present an extension of a method that allows to
find explicit equations for a projective model of these modular curves.
Giuseppe
Pipoli (“Sapienza” Università di
Roma)
Title
[Abstract]
Friday 15 November 2013 – Aula Consiglio, 15:30
Nicolás Andruskiewitsch (Universidad Nacional
de Córdoba, Argentina)
Nichols algebras of
diagonal type
Nichols algebras are Hopf algebras in braided tensor categories with
very particular properties; for instance, the positive parts of quantized
enveloping algebras at a generic parameter, and their finite-dimensional
counterparts when the parameter is a root of one, are Nichols algebras. The
input datum to define a Nichols algebra is a braided vector space, that is a
solution of the braid equation or equivalently of the quantum Yang-Baxter
equation. Nichols algebras of diagonal type are by definition those
corresponding to solutions of the braid equation given by a perturbation of the
usual transposition given by a matrix of non-zero scalars. These Nichols
algebras appear in the classification of pointed Hopf algebras with abelian group,
within the method proposed by Hans-Jürgen Schneider and myself. The main questions concerning them are:
The first question was solved by István Heckenberger using
notably the Weyl groupoid introduced by himself; the second was answered by
Iván Angiono in his thesis at the University of Córdoba.
Currently Angiono and myself are working in clarifications of some aspects of
these results, including relations with contragredient Lie (super) algebras,
either in zero or in positive characteristic. In this talk I will survey from
scratch the notion of Nichols algebras of diagonal type and the main results
evoked above and will report on the recent work in progress with Angiono.
Loïc Foissy (Université de Calais)
Feynman graphs, trees and
combinatorial Dyson-Schwinger equations (part II)
Feynman graphs are
used in Quantum Field Theory to represent the possible interactions between the
particles studied by the theory. They are used to compute certain physical
constants (mass or charge of the electron, for example). For this, several
infinite series of Feynman graphs are considered; they can be uniquely defined
as the solution of a certain system of equations, called the combinatorial
Dyson-Schwinger equations of the system.
Moreover, the combinatorial operations on Feynman graphs (insertion, extraction
contraction) give them a structure of a Hopf algebra. The solution of the
Dyson-Schwinger system should be compatible with this algebraic structure, and
this imposes strong conditions on the system itself.
We present a classification of the systems which satisfy these conditions, with
the help of the Connes-Kreimer Hopf algebra and its universal property, which
allows
to replace Feynman graphs by rooted trees.
Tuesday
3 December 2013 – Aula Consiglio, 14:00
Nigel Higson (Penn State University)
Contractions of Lie
groups and representation theory
Let K be a closed subgroup of a Lie group G. The contraction of
G to K is a Lie group that approximates G to first order near K.
The terminology is due to the mathematical physicists, who examined the
group of Galilean transformations as a contraction of the group of Lorentz
transformations. In geometry, the group of isometric motions of Euclidean
space may be viewed as a contraction of the group of isometric motions of
hyperbolic space. It is natural to guess that there is some sort of
limiting relationship between representations of the contraction group and
representations of the original group. But in the 1970’s George
Mackey made calculations that suggested an interesting rigidity phenomenon: if K is the maximal compact subgroup of a
semisimple group, then representation theory remains (in some sense) unchanged
after contraction. In particular the irreducible representations of the
contraction group parametrize the irreducible representations of G. I shall formulate a reasonably
precise conjecture that was inspired by subsequent developments in C*-algebra theory and noncommutative
geometry, and I shall describe the evidence in support of it, which is by now
substantial. However a conceptual explanation for Mackey’s rigidity
phenomenon remains elusive.
18
December 2013 – Aula B, 15:30
Enrico Le Donne (University of Jyvaskyla)
Title
[Abstract]
Lewis Topley (University of East Anglia)
Presentations and
Representations of Finite W-algebras
Since they were first defined in the early 20th
century, Lie algebras have found their place at the very core of abstract
algebra and theoretical physics. Their representation theory was developed
rapidly and is still an area of vibrant interdisciplinary research today -
combining algebraic techniques with those of geometry, combinatorics and
category theory. The themes which arose in this theory have been replicated
successfully for many other algebras and so this body of work may be seen as a guiding
paradigm in representation theory.
In the late 1970's some glimpses began to appear of deep relationships between
the representations of Lie algebras and nilpotent orbits. These were mostly
understood by a variety of sophisticated methods although, at first, there were
very few unifying themes. In 2002, Alexander Premet defined what is now known
as the finite W-algebra. This is an associative, filtered algebra
attached to a complex semisimple Lie algebra and a nilpotent orbit therein. It
has since become clear that the representation theory of these algebras may
explain many of the aforementioned connections between representations and
nilpotent orbits of Lie algebras. As a result, some of the most challenging
questions in the representation theory of Lie algebras are now being answered.
I intend to contribute to the theory by initiating two parallel investigations.
The second of these depends upon the first. In type A, the finite W-algebras
may be described by generators and relations, thanks to the work of Brundan and
Kleshchev. Finding such a presentation in other types is perhaps the most
fundamental and pressing problem for theorists in this area. I have conceived
of a method to obtain such a presentation, making use of the (geometric) theory
of sheets of adjoint orbits. My second investigation shall reduce this
presentation to the characteristic p realm in order to study the modular
representations of Lie algebras.
Victor Kac (MIT)
Did Ramanujan know
representation theory of infinite dimensional Lie superalgebras?
[Abstract]
Giovanni
Cerulli Irelli (Roma “Sapienza”)
Isotropic quiver
Grassmannians
We introduce a new class of projective varieties
called isotropic quiver Grassmannians, in order to provide a quiver framework
for the study of flag varieties of type B, C and D and their degenerations.
Giulio
Codogni (Roma III)
Satake
compactifications, lattices and Schottky problem
We prove some results about the singularities of
Satake compactifications of classical moduli spaces, this will give an insight
into the relation among solutions of the Schottky problem in different genera.
The moduli space Ag lies in
the boundary of ASg+m
for every m. We will show that the
intersection between MSg+m
and Ag
contains the m-th infinitesimal
neighbourhood of Mg in Ag, this implies that stable
equations for Mg do not
exist. In particular, given two inequivalent positive even unimodular quadratic
forms P and Q, there is a curve whose period matrix distinguishes between the
theta series of P and Q; we are able to compute its genus in
the rank 24 case. On the other hand, the intersection of Ag and HypSg+m is transverse: this enables us to write
down many new stable equations for Hypg in terms of theta series. Our work relies upon some
formulae for the first order part of the period matrix of some degenerations.
26
February 2014 – Room G, 14:30
Martina Lanini (FAU Erlangen-Nürnberg)
Moment graph combinatorics for semi-infinite flags
Semi-infinite flags
arise as an affine variant of flag varieties and provide a geometric approach
to the study of representations of quantum groups at a root of unity and Lie
algebras in positive characteristic.
In this talk we explain
how a certain graph encodes all the information needed to compute local
intersection cohomology stalks of these varieties.
Tuesday
4 March 2014 – Aula Consiglio, 13:45
Aaron Lauda (University of Southern California)
The odd cohomology of
Springer varieties and the Hecke algebra at q=-1
The cohomology rings of type A Springer varieties carry an action of the symmetric group (or
Hecke algebra at q=1).
The top degree cohomology is the Specht module
corresponding to the shape of the partition defining the Springer variety.
Work of De Concini-Procesi and Tanisaki provide an
elementary description of these rings and the action of the symmetric group.
These cohomology rings appear many places in higher
representation theory.
In particular, the symmetric group action can be
interpreted using the nilHecke algebra which plays a fundamental role in the
categorification of sl(2).
It turns out that the categorification of sl(2) is not unique.
There are two different categorifications, the even
and odd version, which agree modulo 2, but are otherwise distinct.
Reexamining the cohomology of Springer varieties from
this odd perspective leads to a new
"odd" analog of the cohomology ring of Springer varieties by
replacing the nilHecke algebra with the odd nilHecke algebra. The top degree
component of the resulting "odd cohomology" of Springer varieties
turns out to be isomorphic to a Specht module for the Hecke algebra at q=-1.
(Joint work with Heather Russell)
Indrava
Roy (Roma “Sapienza”)
ρ-invariants for
foliations and their stability properties
In this talk we will
discuss the stability properties of some spectral invariants for Dirac-type
operators on foliated manifolds.
ρ-invariants (also known as relative eta-invariants), which were
introduced by Atiyah, Patodi and Singer for closed manifolds, reflect deep
geometric properties of the underlying manifold. In the case of measured
foliations, we will study "foliated" ρ-invariants associated
with a leafwise signature operator, extending a classical result of Cheeger and
Gromov in the L2 case (a
covering being a special case of a foliation). We will also discuss the proof
of the homotopy invariance of such spectral invariants under suitable
conditions. In this context we will give an analogue of Atiyah's L2-index theorem and also a
new proof of the leafwise homotopy invariance of the index of leafwise
signature operators for foliations.
Samuel Grushevsky (SUNY Stony Brook)
Stable cohomology of
the compactifications of the moduli space of abelian varieties
Borel showed that the degree k cohomology of the moduli space Ag of (complex principally polarized) Abelian g-dimensional varieties stabilized as g grows, that is does not depend on g, for g>k. Similarly, Madsen and Weiss showed that the cohomology of
the moduli space of curves Mg
stabilizes.
In this talk we study the stabilization of the
cohomology of compactifications, observing that the cohomology of the Deligne-Mumford
compactification of Mg does
not stabilize, of the second Voronoi toroidal compactification of Ag likely does not stabilize,
while proving that the cohomology of the perfect cone toroidal compactification
of Ag does stabilize.
Our methods are algebro-geometric and not topological;
based on joint work with Klaus Hulek and Orsola Tommasi.
Lie Fu (ENS Paris)
Algebraic cycles on
hyperkähler varieties: the Beauville-Voisin conjecture
Hyperkähler varieties are higher-dimensional
generalizations of K3 surfaces, consisting of fundamental building blocks of
varieties with trivial canonical bundles. According to the work of Beauville
and Voisin, the intersection theory of projective K3 surfaces has some
particular degeneracy property. Its generalization to hyperkähler
varieties, namely the Beauville-Voisin conjecture, says that the cycle
class map restricted to the sub-algebra in the Chow
ring (with rational coefficients) generated by the Chern classes of line
bundles and the tangent bundle is injective. I will talk about my result
verifying this conjecture for the generalized Kummer varieties.
Niels Kowalzig (INdAM Marie Curie fellow – Roma
“Tor Vergata”)
Operads and
differential calculi
In this talk, we show under what additional ingredients a left module in
negative degrees over an operad with multiplication can be given the structure
of a cyclic k-module and how the
underlying simplicial homology gives rise to a calculus structure (or
Batalin-Vilkovisky module) over the cohomology of the operad, which is in some
sense a dual picture to the relationship between cyclic operads and
Batalin-Vilkovisky algebras. In particular, we obtain explicit expressions for
a generalised Lie derivative as well as a generalised (cyclic) cap product that
obey a Cartan-Rinehart homotopy formula. Examples include the calculi known for
the Hochschild theory of associative algebras, for Poisson structures, but
above all the calculus for a general left Hopf algebroid with respect to
general coefficients (in which the classical calculus of vector fields and
differential forms is contained). Time permitting, we will also dicuss how the
classical Koszul bracket is obtained in this framework.
Sarah Scherotzke (Universität Bonn)
Graded quiver
varieties and derived categories
Nakajima's quiver varieties are important geometric
objects in representation theory that can be used to give geometric
constructions of quantum groups. Very recently, graded quiver varieties also
found application to monoidal categorification of cluster algebras. Nakajima's
original construction uses geometric invariant theory. In my talk, I will give
an alternative representation theoretical definition of graded quiver
varieties. I will show that the geometry of graded quiver varieties is governed
by the derived category of the quiver Q.
This approach brings about many new and surprising results: for instance, it
turns out that a large class of quiver varieties has a very simple geometry,
indeed they are isomorphic to affine spaces. Also, I will explain that familiar
geometric constructions in the theory of quiver varieties, such as
stratifications and degeneration orders, admit a simple conceptual formulation
in terms of the homological algebra of the derived category of Q. If time permits, I will also explain
interesting applications of our work to desingularization of quiver
Grassmannians.
Domenico
Fiorenza (Roma “Sapienza”)
Abel, Jacobi and the
double homotopy fibers
Let X be a
Kähler complex manifold and let Z
be a complex submanifold.
Within the framework of derived deformation theory, the Abel-Jacobi map for the
pair (X,Z) has a natural
interpretation as a morphism from a homotopy fiber to a double homotopy fiber.
Using this fact and the dictionary between formal moduli problems and
differential graded Lie algebras up to homotopy, it is easy to describe a
linear L∞-morphism encoding the Abel-Jacobi map, and in terms
of this L∞-morphism it is immediate to reobtain the classical
description of the differential of the Abel-Jacobi map as well as the result
that Bloch's semiregularity map annihilates the obstructions to deforming Z inside X.
This provides a geometrical interpretation of recent results by Iacono-Manetti
and Pridham.
Tuesday
22 April 2014 – Aula Consiglio, 15:00
Siye Wu (University of Hong Kong)
Hitchin's equations
over a non-orientable manifold
We study Hitchin's equations and Higgs bundles over a
non-orientable
manifold whose oriented cover is compact Kähler.
Using the involution induced
by the deck transformation, we show that Hitchin's
moduli space
is Langrangian/complex with respect to the
hyper-Kähler structure on Hitchin's
moduli space associated to the oriented cover. We then
establish a
Donaldson-Corlette type correspondence and relate
Hitchin's moduli space to
representation varieties. This is a joint work with
N.-K. Ho and G. Wilkin.
Gavril Farkas (Humboldt Universität - Berlin)
What is the principally
polarized Abelian variety of dimension six?
The general principally polarized abelian variety of
dimension at most five is known to be a Prym variety. This reduces the study of
abelian varieties of small dimension to the beautifully concrete theory of
algebraic curves. I will discuss recent progress on finding a structure theorem
for principally polarized abelian varieties of dimension six, and the
implications this uniformization result has on the geometry of their moduli
space.
Corrado
De Concini (“Sapienza” Università di Roma)
Covariants in the
exterior algebra of a simple Lie algebra
For a simple complex Lie algebra g we
study the space of invariants
A=(Λ g *Ä g *)g (which describes the isotypical component
of type g in Λg* as a module over the
algebra of invariants (Λg*)g.
As main result we prove that A is a
free module, of rank twice the rank of g,
over the exterior algebra generated by all primitive invariants in (Λg*)g,
with the exception of the one of highest degree. Joint with P. Papi and C.
Procesi.
Wednesday
14 May 2014 – Aula di Consiglio, 14:30
Charles Vial (University of Cambridge)
On multiplicative
Chow-Kunneth decompositions
This is joint work with Mingmin Shen. I will introduce
the notion of multiplicative Chow-Kunneth decomposition and give first examples
of varieties endowed with such a decomposition. I will then explain why we
expect hyperKähler varieties to be endowed with such a decomposition,
sketch a proof that the Hilbert scheme of length-2 subschemes on a K3 surface
is endowed with such a decomposition and relate this to the Fourier
decomposition.
Wednesday
14 May 2014 – Aula di Consiglio, 15:45
Mingmin Shen (Universiteit van Amsterdam)
On Fourier
decomposition of the Chow ring of certain hyperKähler fourfolds
This is joint work with Charles Vial. I will explain
how to define a Fourier transform using a conjecturally canonical codimension-2
cycle on the self product of certain hyperKähler fourfolds. Under certain
conditions, the Fourier transform induces a decomposition of the Chow ring. I
will explain how this can be carried out for the variety of lines on a cubic
fourfold and the Hilbert scheme of two points on a K3 surface.
Christophe Hohlweg (Université du Québec
à Montréal)
Weak order and
imaginary cone in infinite Coxeter groups
The weak order is a nice combinatorial tool intimately
related to the study of reduced words in Coxeter groups. In finite Coxeter
groups, it is a lattice and an orientation of the 1-skeleton of the permutahedron
that provides a nice framework to construct generalized associahedra (via N.
Reading's Cambrian lattices).
In this talk, we will discuss a conjecture of Matthew
Dyer that proposes a generalization of the framework weak order/reduced words
to infinite Coxeter groups. On the way, we will talk of the relationships
between limits of roots and tilings of their convex hull, imaginary cones,
biclosed sets and inversion sets of reduced infinite words (partially based on
joint works with M. Dyer, J.P. Labbé and V. Ripoll).
Tuesday
27 May 2014 – Room C, 16:00
Xiaonan Ma (Université de Paris VII)
Geometric
quantization for proper moment maps: the Vergne conjecture
We establish an analytic interpretation for the index
of certain transversally elliptic symbols on non-compact manifolds. By using
this interpretation, we establish a geometric quantization formula for a
Hamiltonian action of a compact Lie group acting on a non-compact symplectic
manifold with proper moment map. In
particular, we present a solution
to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture. Joint
with Weiping Zhang.
Emanuele Macrí (Ohio State University)
Geometria birazionale
degli spazi di moduli di fasci su superfici K3
In questo seminario, basato su alcuni lavori in
collaborazione con Arend Bayer, presenterò tecniche di categoria
derivata di superfici K3 - in particolare, condizioni di stabilità di
Bridgeland - per studiare la geometria birazionale di spazi di moduli di fasci
stabili.
Wednesday
4 June 2014 – Aula di Consiglio, 14:30
Marco
Franciosi (Università di
Pisa)
Il teorema di
Clifford e la congettura di Green per curve m-connesse
In questo seminario verranno esposti alcuni risultati
ottenuti in collaborazione con Elisa Tenni relativi ad alcune proprietà
algebro-geometriche di curve stabili m-connesse. In generale una curva C Gorenstein si dice m-connessa se per ogni sottocurva B il grado del fascio dualizzante
ristretto a B è maggiore o
uguale di 2pa(B)-2+m. Per curve stabili m-connesse è possibile definire l'indice di Clifford ed è
possibile studiare i gruppi di Koszul del divisore canonico.
Verranno descritti esempi di curve di genere alto
rispetto alla connessione numerica m
e verranno forniti alcuni risultati parziali per curve binarie (curve ottenute
dall'unione di due curve
razionali).
Wednesday
4 June 2014 – Aula di Consiglio, 15:45
Thomas Strobl (Université de Lyon 1)
20 Years of the
Poisson Sigma Model: lessons we learnt, lessons to learn
The Poisson sigma model has been found first so as to
unify gravitational and Yang-Mills forces in physical toy models for
two-dimensional space-times. Already in the beginning it was noted that its
quantization is related to the quantization of the target Poisson manifold.
However, it was in particular Kontsevich, who showed that it leads to a
solution of the by then longstanding open problem of deformation quantization.
But this is not all: the reduced phase space of the model carries the structure
of a symplectic groupoid as shown by Cattaneo and Felder and are thus
intimately related to the integration problem of Lie algebroids to Lie
groupoids. Finally, the Poisson sigma model serves ongoing attempts for a
deepend understanding gauge theories in an essential way, paving the way to
higher characteristic classes as well as to gauge theories governed by
structural Lie n-groupoids.
Tuesday
10 June 2014 – Aula di Consiglio, 14:30
Eli Aljadeff (Technion – Israel Institute of
Technology)
On a G-graded version of Jordan’s
theorem
A well known theorem of Camille Jordan (1878) says
that if G is a finite group which may
be embedded in GLn(C)
then it is ``almost abelian'' in the sense that it contains an abelian subgroup
H whose index [G:H] is bounded by a function of n.
The main object of this lecture is to present an
analogous result for G-graded
algebras where G is arbitrary (in
particular it may be infinite). The main tools are Kemer's representability
theorem for PI-algebras and Giambruno-Zaicev theorem on PI-asymptotics.
In the lecture I will recall the necessary concepts
and terminology.
Joint work with Ofir David.
Tuesday
10 June 2014 – Aula di Consiglio, 16:00
Ngò Viet Trung (Institute of Mathematics, VAST)
Cohen-Macaulayness of
monomial ideals
A combinatorial criterion for the Cohen-Macaulayness
of monomial ideals will be presented. This criterion helps to explain all
previous results on this topics. Moreover, it helps to establish a striking
relationship between the Cohen-Macaulayness of powers of Stanley-Reisner ideals
and matroid complexes.
11
June 2014 – Aula Consiglio, 14:30
Giulia Saccà (SUNY Stony Brook)
Spazi di moduli di
fasci su K3 e varietà di Nakajima
Lo scopo del seminario è quello di studiare le
singolarità di alcuni spazi di moduli di fasci su una superficie K3,
usando della varietà quiver nel senso di Nakajima.
Le singolarità di cui ci occuperemo sono
causate della scelta di una polarizzazione non generica, rispetto a cui si considera la
stabilità.
Con la verifica della stabilità del fibrato di
Lazarsfeld-Mukai, per una classe di fasci puri di dimensione uno su una
superficie K3, si forniscono esempi di di spazi di moduli che sono, localmente
attorno ad un punto singolare, isomorfi ad una varietà quiver nel senso
di Nakajima.
Usando questo isomorfismo, si dimostra che le naturali
risoluzioni simplettiche di questi spazi di moduli, date dal cambiamento delle
condizione di stabilità,
corrispondono alle naturali risoluzioni simplettiche delle
varietà di Nakajima che provengono da variazioni del quoziente nel senso
delle teoria geometrica degli invarianti.
Questo è un lavoro svolto in collaborazione con
Enrico Arbarello.
11 June 2014 – Aula Consiglio, 15:45
Ezra
Getzler (Northwestern University)
Courant algebroids on
derived manifolds, Deligne cohomology, and derived Poisson brackets
This talk presents an approach to the following
question: what is the extension of Noether's theorem in classical field theory
to manifolds with corners? (Equivalently, how does one keep track of all of the
integrations by parts performed in the course of the application of Noether's
theorem?)
Our answer involves derived manifolds (via the
Batalin-Vilkovisky approach to classical field theory), Deligne cohomology on
derived manifolds, and the work of Fiorenza and Manetti on L-infinity
structures on cones. Assembling these ingredients, I obtain a replacement for
the Poisson complex for classical field theory which provides the desired
extension of Noether's theorem.
Jerzy Weyman (University of Connecticut)
Title
[Abstract]
Pierre Albin (UIUC)
A Cheeger-Muller
theorem on manifolds with cusps
Ever since the proof by Cheeger and Muller of the
equality of the analytic torsion and Reidemeister torsion on smooth compact
manifolds, there has been a search for a corresponding theorem on singular
spaces. In 1985 Dar proposed a topological torsion on singular spaces using
intersection homology chains and conjectured that it will have an analytic description.
In joint work with Frederic Rochon and David Sher, we prove that Dar's torsion
is the analytic torsion of a manifold with cusps. Our proof is through an
analytic surgery that starts with a smooth manifold and deforms it to a
manifold with cusps.