Algebra and Geometry Seminar



Abstracts of talks 2013/2014




9 October 2013

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.




16 October 2013

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




23 October 2013

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.





6 November 2013

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.



13 November 2013 – Time 15:30

Giuseppe Pipoli (“Sapienza” Università di Roma)







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.



27 November 2013

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)






15 January 2014

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.



22 January 2014

Victor Kac (MIT)

Did Ramanujan know representation theory of infinite dimensional Lie superalgebras?





12 February 2014

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.



19 February 2014

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)



5 March 2014 – Room E, 14:30

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.




19 March 2014

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.




26 March 2014

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.




2 April 2014

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.



9 April 2014

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.




16 April 2014

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.






30 April 2014

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.



7 May 2014

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= g *Ä g *)g  (which describes the isotypical component of type g in Λg* as a module over the algebra of invariants g*)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 g*)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.




21 May 2014

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.



28 May 2014

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.





18 June 2014

Jerzy Weyman (University of Connecticut)














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.