Abstracts of talks 2014/2015
Markus Reineke (Wuppertal)
Topology and arithmetic of matrix invariants
We consider the action of a general linear group on tuples of matrices via simultaneous conjugation. The quotient by this action is called a space of matrix invariants, and we consider global invariants of these spaces. Closed formulas for their numbers of rational points over finite fields, and for their intersection cohomology, are derived.
Salvatore Stella (NCSU)
d-vector fans per cluster algebra di tipo finito ed affine
Ad ogni algebra cluster, grazie al fenomeno di Laurent, si può associare in maniera naturale una famiglia di vettori a coordinate intere (i vettori d delle variabili cluster).
Sorprendentemente questa famiglia codifica molte delle proprietà combinatoriche dell'algebra di partenza.
In questo seminario, dopo aver ricordato le definizioni di base, spiegherò come costruire l'insieme dei vettori d nel caso di algebre cluster di tipo finito ed affine utilizzando l'azione di una particolare simmetria del corrispondente sistema di radici.
Matthieu Gendulphe (Roma « Sapienza »)
A theorem of Maryam Mirzakhani: the asymptotic of the growth of simple closed geodesics on hyperbolic surfaces
Last summer, Maryam Mirzakhani received the Fields medal for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."
We will describe one of her most famous results, which illustrates the sentence above.
This talk is intended for non-experts.
Dmitri Panov (King’s College of London)
Definite connections and real symplectic Fano manifolds
A definite connection is an SO(3)-connection over a 4-manifold, whose curvature is non-zero on every tangent 2-plane. Such geometric objects were first considered by Allan Weinstein and were called fat bundles. Given such a connection, the associated S2-bundle is naturally a symplectic manifold. Surprisingly the symplectic manifold is either a symplectic Fano or a symplectic Calabi-Yau.
Important examples of such connections come from differential geometry. Namely, the Levi-Cevita connection on the twistor bundle of a Riemannian 4-manifold with sufficiently pinched curvature is definite. In particular we have examples such as the unit S4 that give rise to positive definite connections and examples such as hyperbolic 4-manifolds that give rise to negative definite connections.
In this talk I will discuss the following sphere-type theorem: the only four manifolds that admit an S1-invariant definite connection are S4 and CP2. This work is joined with Joel Fine.
Gregory Sankaran (Bath)
Fundamental groups of toroidal compactifications
We give a precise unified description of the fundamental group of a toroidal compactification of a locally symmetric variety, and explain some consequences in special cases.
This is joint work with A.K. Kasparian (Sofia).
Michael Bulois (Saint-Étienne)
Slice induction and sheets
Let G be an algebraic group acting on a variety V. Sheets of V are defined as irreducible components of the sets of elements with fixed orbit dimension. Knowing the made up and geometry of sheets is important to understand moduli spaces and some other varieties. In the talk, I will focus on the adjoint action and its generalization to symmetric Lie algebras. In this setting, many information follow by considering the Slodowy slices of the sheet. This allows to rebuilt important part of the theory of induction of nilpotent orbits.
Giovanni Cerulli Irelli (Roma « Sapienza »)
Realizzazione, come sottomoduli di Demazure, dei moduli abelianizzati per algebre di Lie di tipo A e C
In questo seminario presenterò un progetto che sto portando avanti con M. Lanini e P. Littelmann: sia L un'algebra di Lie di tipo A o C e sia V una sua rappresentazione irriducibile di dimensione finita. La filtrazione PBW dell'algebra inviluppante di L, induce una filtrazione di V il cui graduato associato Va si dice l'abelianizzato associato a V. Si noti che Va è un modulo per l'algebra simmetrica S(n), dove n è la sottoalgebra delle matrici triangolari strettamente inferiori di L. Motivati da una congettura di Vinberg, i moduli abelianizzati hanno ricevuto un discreto interesse negli ultimi anni, grazie ai lavori di Littelmann, Fourier, Finkelberg e Feigin. In questo seminario dimostriamo che Va è isomorfo (come S(n)-modulo) ad un opportuno sottomodulo di Demazure di un opportuno modulo irriducibile per un'algebra di Lie dello stesso tipo di L, ma di rango doppio.
Indrava Roy (Roma « Sapienza »)
Analytic surgery exact sequence and L2 spectral invariants
For a discrete group Γ, the analytic surgery exact sequence of Higson and Roe characterizes the failure of the Baum-Connes assembly map μ: KÛ(BΓ)¦KÛ(CÛΓ) to be an isomorphism. In particular, there are obstruction structure groups SÛ(Γ) which vanish precisely when the assembly map μ is bijective. Higson and Roe showed that the certain spectral invariants associated with finite-dimensional representations of Γ called relative η-invariants (a.k.a. ρ-invariants), defined in the seminal work of Atiyah, Patodi and Singer, appear naturally as morphisms from these structure groups to R. This allowed them to conceptualize and re-prove classical deep results of Mathai, Weinberger, Keswani, and Piazza-Schick on the vanishing of the Atiyah-Patodi-Singer rho-invariants for metrics of positive scalar curvature for spin Dirac operators and their homotopy invariance for signature operators, when Γ is torsion-free and the assembly map μ is an isomorphism. We shall extend the utility of the analytic surgery exact sequence to the semi-finite case associated with Galois Γ-coverings, by introducing L2-structure groups. These structure groups appear in an exact sequence which is compatible with the one studied by Higson and Roe. We thus prove the corresponding vanishing results for L2 ρ-invariants, first introduced and studied by Cheeger and Gromov.
(Joint work with M.-T. Benameur).
Henning Krause (Bielefeld)
Koszul, Ringel, and Serre duality for strict polynomial functors
Strict polynomial functors were introduced by Friedlander and Suslin in their work on the cohomology of finite group schemes. More recently, a Koszul duality for strict polynomial functors has been established by Chalupnik and Touze. In my talk I will give a gentle introduction to strict polynomial functors (via representations of divided powers) and will explain the Koszul duality, making explicit the underlying monoidal structure which seems to be of independent interest. Then I connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors.
Francesco Vaccarino (Politecnico di Torino)
Homological scaffold of the psychedelic brain
We will introduce our recent results on the use of persistent homology in the field of complex networks. In particular we will show how persistence homology can be used to discriminate the variation of the brain functional connectome under the influence of psilocybin extracted from magic mushrooms. We furthermore give account of our theorem which shows that a persistence module on a finite poset P can be obtained from the filtration of a graph weighted over P.
Silvana Bazzoni (Padova)
t-strutture e equivalenze indotte da moduli tilting classici e infinitamente generati
Ricorderò i risultati classici di Brenner-Butler, Happel-Ringel e Rickard riguardanti le equivalenze indotte da moduli tilting finitamente generati (e complessi tilting). Presenterò la generalizzazione naturale a moduli tilting infinitamente generati, le proprietà delle classi tilting e le equivalenze indotte. Accennerò ad un recente problema posto da Saorin riguardante le proprietà della categoria abeliana “cuore” della t-struttura associata a un modulo tilting.
Margherita Lelli-Chiesa (Centro « De Giorgi », Pisa)
Varietà di Severi e teoria di Brill-Noether di curve su superfici abeliane
Le varietà di Severi e la teoria di Brill-Noether di curve su superfici K3 sono ben comprese; al contrario, poco è noto riguardo le curve su superfici abeliane.
Data una superficie abeliana generale S con polarizzazione H di tipo (1,n), dimostreremo la non vuotezza e la regolarità della varietà di Severi che parametrizza curve δ-nodali C nel sistema lineare |H| per 0≤δ≤n-1. Studieremo poi le serie lineari di tipo g1d sulla normalizzazione di C; già nel caso liscio, la gonalità delle curve in |H| risulterà non costante. Tali risultati sono ottenuti per degenerazione ad una superficie semiabeliana. Nell'ultima parte del seminario, proporrò un approccio per studiare le serie lineari di tipo grd con r≥2.
Questo lavoro è in collaborazione con A. L. Knutsen e G. Mongardi.
Giovanna Carnovale (Padova)
Un analogo del teorema di Katsylo per sheet di classi di coniugio sferiche
Le sheet per l'azione di un gruppo algebrico G su una varietà X sono le componenti irriducibili degli insiemi di punti di X la cui orbita ha dimensione fissata. Nel caso dell'azione di G su se stesso per coniugio, esse sono le componenti irriducibili delle parti di una partizione introdotta recentemente da G. Lusztig in termini di corrispondenza di Springer. Mostreremo come descrivere le parti e le sheet nel caso in cui esse contengano una classe di coniugio sferica e dimostreremo che per queste sheet e parti esiste un quoziente geometrico, isomorfo ad una sottovarietà affine di G modulo l'azione di un gruppo abeliano finito.
Martina Balagovic (Newcastle)
Irreducible modules for degenerate double affine Hecke algebras of type A
Double affine Hecke algebras and their degenerate versions have been an active area of study since their original definition in 1993, when they were used by Ivan Cherednik to solve Macdonald conjectures. When studying their representation theory, one usually focuses on Category O. Here one defines Verma modules M as certain induced modules, shows that with appropriate assumptions these modules always have an irreducible quotient L=M/K, and that all irreducible modules in Category O can be uniquely realized in this way as quotients of Verma modules. However, the description of this maximal submodule K, and consequently of the irreducible module L, are not known in general. This is why alternative characterizations of irreducible modules are interesting.
In this talk, I will focus on the question “Can irreducible Category O modules for degenerate double affine Hecke algebras of type A always be realized as submodules of Verma modules?”
The corresponding answer for affine Hecke algebras, as discovered by Guzzi, Nazarov and Papi, is always yes. The answer for double affine Hecke algebras is more involved.
I will describe both the modules which do not allow such a realization and the imbedding of the modules which do allow it in terms of the combinatorics of the affine symmetric group and periodic skew Young diagrams.
Shifra Reif (University of Michigan)
The Kac-Wakimoto character formula
Finding a character formula for finite dimensional representations has been a central problem in the theory of Lie superalgebras in the past few decades. In 1994, Kac and Wakimoto conjectured an elegant character formula for modules admitting certain choices of simple roots. We shall give several characterizations for these modules and show how they can be used to prove the formula for the general linear superalgebra. Joint with Crystal Hoyt and Michael Chmutov.
14 January 2015
Marco Zambon (Leuven)
Multisymplectic manifolds, homotopy moment maps, and conserved quantities
Multisymplectic structures are higher generalizations of symplectic structures, where forms of higher degree are considered. We introduce a notion of moment map for such structures, based on Stasheff’s notion of L∞-algebra, and by relating it to equivariant cohomology we are able to produce examples. Even though moment maps are quite involved, they admit a simple cohomological description. Finally, we discuss some geometric implications of our moment maps (the existence of conserved quantities).
Fosco Loregian (SISSA)
A characterization of quasicategorical t-structures
After having introduced the basic definition of ∞-category and stable ∞-category, we characterize t-structures in stable ∞-categories as suitable quasicategorical factorization systems. More precisely we show that a t-structure t on a stable ∞-category C is equivalent to a normal torsion theory F on C, i.e. to a factorization system F=(E,M) where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts.
Luca Vitagliano (Salerno)
Homotopy Algebras and PDEs
The jet space approach is a powerfool tool to study invariant properties of PDEs, i.e. properties independent of the choice of coordinates (symmetries, conservation laws, recursion operators, etc.). The highest outcome of this approach is that the space of solutions of a PDE is a "derived manifold", which roughly means that vector fields, differential forms, etc. on it form "algebras up to homotopy". The aim of the talk is providing a smooth introduction to the homotopical algebra of PDEs.
Angela Ortega (Humboldt Universität Berlin)
The Brill-Noether curve and Prym-Tyurin varieties
It is a result of L. Masiewicki that the Jacobian of a non hyperelliptic curve of genus 5 is isomorphic to the Prym variety associated to the natural involution on the singular locus of the Θ-divisor of the curve. We will show that this result can be generalized to a curve C of arbitary odd genus g=2a+1. More precisely, the Jacobian of C can be realized as the Prym-Tyurin variety for the Brill-Noether curve W1a+2(C) of pencils of minimal degree on C. As a consequence we obtain an application to the enumerative geometry of the secants to C.
Steve Karam (Lille)
Growth of balls in the universal cover of graphs and surfaces
We prove that if the area of a closed Riemannian surface M of genus at least two is sufficiently small with respect to its hyperbolic area, then for every radius R≥0 the universal cover of M contains an R-ball with area at least the area of a cR-ball in the hyperbolic plane, where c is a universal positive constant in (0,1). In particular (taking the area of M smaller if needed), we prove that for every radius R≥1, the universal cover of M contains an R-ball with area at least the area of a ball with the same radius in the hyperbolic plane.
This result answers positively a question of L. Guth for surfaces. We also prove an analog result for graphs. Specifically, we prove that if G is a connected metric graph of first Betti number b≥2 and of length sufficiently small with respect to the length of a connected trivalent graph Gb of the same Betti number where the length of each edge is 1, then for every radius R≥0 the universal cover of Gb contains an R-ball with length at least c times the length of an R-ball in the universal cover of Gb where c is a universal constant in (½,1).
Gianluca Pacienza (Strasbourg)
Curve razionali e gruppi di Chow di varietà irriducibili simplettiche
Mori e Mukai hanno dimostrato che ogni sistema lineare ampio su di una superficie K3 contiene un elemento somma di curve razionali. Beauville e Voisin hanno osservato come da ciò discenda l’esistenza di uno 0-ciclo «canonico» sulla superficie K3, dato dalla classe di un punto qualsiasi su una qualunque curva razionale sulla K3.
Nel seminario parlerò di un lavoro in corso e in collaborazione con François Charles nel quale generalizziamo i risultati di Mori-Mukai e di Beauville-Voisin alle varietà irriducibili simplettiche di dimensione arbitraria che sono deformazioni di schemi di Hilbert di punti su una K3.
Lidia Angeleri Hügel (Verona)
I moduli silting generalizzano una nozione introdotta recentemente da Adachi, Iyama e Reiten nello studio di mutazioni in teoria cluster. Sono i moduli che corrispondono a complessi silting non necessariamente compatti formati da due termini, e sono strettamente legati a certe t-strutture nella categoria derivata. Ci soffermeremo in particolare sul ruolo della teoria silting nello studio di epimorfismi di anelli e localizzazioni, mostrando che in alcuni casi le localizzazioni di un anello sono parametrizzate da moduli silting. Questo lavoro è in collaborazione con Frederik Marks e Jorge Vitória.
Valentina Di Proietto (Strasbourg)
On the homotopy exact sequence for the algebraic fundamental group
There is a strong link between the fundamental group of a variety and the linear differential equations we can define on it. The definition of the fundamental group given in terms of homotopy classes of loops does not generalize easily to algebraic varieties defined over an arbitrary field. But exploiting this link we can give another definition that makes sense in very general contexts: it is called the algebraic fundamental group. We prove the homotopy exact sequence for the algebraic fundamental group for a fibration with singularities with normal crossing. This is a joint work with Atsushi Shiho.
Jesus Juyumaya (Valparaiso)
Knot invariants from the Yokonuma-Hecke algebras
In this talk we define invariants for classical knots, singular knots and framed knots. These invariant are constructed by using the Yokonuma-Hecke algebras in the Jones recipe.
Frédéric Patras (Université de Sophia-Antipolis)
B∞ structures and finite topologies
Models (simplicial, cellular...) of topological spaces are usually equipped with various operations: products, dualities, cohomological operations, and so on. One can also consider topological spaces from another point of view: instead of looking at the internal structure of each space, one can consider their collective structure. In the particular case of finite spaces, they are in bijection with quasi-orders, and it is natural to investigate their linear span using techniques of algebraic combinatorics. New operations and structure theorems arise from this point of view, as well as connexions with Hopf algebras such as the one of quasi-symmetric functions.
Joint work with L. Foissy and C. Malvenuto.
Gabriele Mondello (« Sapienza » Roma)
On the existence of spherical metrics with conical singularities on a 2-sphere
A celebrated result by Poincaré states that a compact connected Riemann surface has a conformal metric of constant curvature, unique up to rescaling and biholomorphisms. Clearly, the case of genus 0 is not so exciting, being trivially solved by any Fubini-Study metric on the complex projective line.
The problem becomes more interesting if we require such metrics to have conical singularities of prescribed angles at a finite subset of n marked points. The case of negative and zero curvature was settled by McOwen and Troyanov in 1989-1991: they established the existence and uniqueness of such a metric in each conformal class.
The case of positive curvature is more delicate: existence and uniqueness results are known for small angles (Troyanov), whereas non-uniqueness results are known in positive genus (Bartolucci-De Marchis-Malchiodi).
In a joint work with D.Panov (still in progress), we determine for which angle assignment there exists a surface of genus 0 with a metric of curvature 1 and conical singularities of such prescribed angles (and non-coaxial holonomy).
Stéphane Launois (Kent)
On the quadratic Poisson Gel’fand-Kirillov problem
We study the field of fractions of certain Poisson polynomial algebras.
For a large class of Poisson polynomial algebras, we prove that their field of fractions is isomorphic to a field of rational functions endowed with a so-called quadratic Poisson brackets. The result extends to homogeneous Poisson prime quotients. Examples include Poisson matrix varieties and Poisson determinantal varieties. This is joint work with Cesar Lecoutre.
Elena Martinengo (Hannover)
Singolarità di spazi di moduli di fasci su K3
Negli anni Ottanta Mukai dimostra che le singolarità dello spazio dei moduli di fasci su una superficie K3 sono contenute nel luogo dei fasci strettamente semistabili. Se la polarizzazione rispetto a cui si considera la stabilità è generica e il vettore di Mukai è primitivo tutti i fasci semistabili sono stabili e lo spazio dei moduli è liscio e ha una struttura simplettica. Nel caso in cui il vettore di Mukai è non primitivo, Kaledin, Lehn e Sorger congetturano che la dg-algebra che controlla le deformazioni di fasci su una K3 sia formale, che implicherebbe una completa descrizione delle possibili singolarità dello spazio di moduli. La congettura è stata dimostrata in alcuni casi da Kaledin-Lehn e successivamente da Zhang. Le tecniche utilizzate sono simili e consistono nell'analizzare i sollevamenti dei fasci sulla K3 alla famiglia twistor e poi applicare il teorema di formalità in famiglie di Kaledin.
In un lavoro in progress in collaborazione con Manfred Lehn vogliamo completare la dimostrazione della congettura. Nel seminario parlerò della nostra dimostrazione per un caso mancante e del tentativo di estendere la nostra costruzione ad hoc a una dimostrazione generale.
Olivier Debarre (ENS Paris + Paris 7)
Fano varieties and EPW sextics
We explore a connection between smooth projective varieties X of dimension n with an ample divisor H such that
Hn=10 and KX=-(n-2)H
and a class of sextic hypersurfaces of dimension 4 considered by Eisenbud, Popescu, and Walter (EPW sextics). This connection makes possible the construction of moduli spaces for these varieties and opens the way to the study of their period maps. This is work in progress in collaboration with Alexander Kuznetsov.
Alessio Fiorentino (Roma « Sapienza »)
La matrice delle bitangenti di una quartica piana non singolare da un punto di vista modulare
È noto che una quartica piana ammette 28 bitangenti e può essere ricostruita a partire da uno qualunque dei suoi 288 sistemi di Aronhold. Sfruttando questo teorema classico e un risultato di Hesse sull'esistenza di 36 rappresentazioni determinantali non equivalenti per la quartica piana non singolare, Plaumann, Sturmfels e Vinzant hanno recentemente definito una matrice 8x8 simmetrica di rango 4 che parametrizza le bitangenti di una quartica piana non singolare. Nel seminario verrà presentato un lavoro scritto in collaborazione con Francesco Dalla Piazza e Riccardo Salvati Manni, in cui si è determinata l'espressione della matrice in termini di funzioni Θ di Riemann; tale approccio permette di considerare la matrice delle bitangenti da un punto di vista modulare, come funzione della curva.
6 May 2015
Claus Michael Ringel (Bielefeld)
Representations of quivers over the algebra of dual numbers
The representations of a quiver Q over a field k (the kQ-modules, where kQ is the path algebra of Q over k) have been studied for a long time, and one knows quite well the structure of the module category Mod(kQ). It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A.
The lecture will draw the attention to the case when A=k[ε] is the algebra of dual numbers (the factor algebra of the polynomial ring k[T] in one variable T modulo the ideal generated by T2), thus to the A-modules, where A=kQ[ε]=kQ[T]/<T2>.
The algebra Λ is a 1-Gorenstein algebra, thus the torsionless -modules are known to be of special interest (as the Gorenstein-projective or maximal Cohen-Macaulay modules). They form a Frobenius category L, thus the corresponding stable category L is a triangulated category. Actually, this category L is the category of perfect differential kQ-modules and L is the corresponding homotopy category. As we will see, the homology functor H: Mod → Mod(kQ) yields a bijection between the indecomposables in L and those in Mod(kQ) and the kernel of H is a finitely generated ideal of the category L which will be described explicitly.
This is a report on joint investigations with Zhang Pu (Shanghai).
Ulrike Rieß (Bonn)
On Beauville's conjectural weak splitting property
We present a recent result on the Chow ring of irreducible symplectic varieties.
The main object of interest is Beauville's conjectural weak splitting property, which predicts the injectivity of the cycle class map restricted to a certain subalgebra of the rational Chow ring (the subalgebra generated by divisor classes).
For special irreducible symplectic varieties we relate it to a conjecture on the existence of rational Lagrangian fibrations.
After deducing that this implies the weak splitting property in many new cases, we present parts of the proof.
Alessandro Valentino (Bonn)
Central extensions of mapping class groups from characteristic classes
I will discuss a functorial construction of extensions of mapping class groups of smooth manifolds which are induced by extensions of (higher) diffeomorphisms groups via the group stack of automorphisms of manifolds equipped with higher degree topological structures. The problem of constructing such extensions arises naturally in the study of topological quantum field theories, in particular in 3d Chern-Simons theory. Joint work with Domenico Fiorenza and Urs Schreiber.
Ghislain Fourier (Glasgow)
Quantum PBW filtrations and monomial ideals
Our aim is to define a PBW-type filtration for quantum groups, following the framework from recent years in the non-quantum setup. For this we recall main properties of this classical filtration which guideline us towards a quantum PBW filtration. In type An we obtain an approriate degree function via dimensions of homomorphism space of the corresponding oriented quiver.
By specializing to q=1, we obtain a new filtration on the universal enveloping algebra and hence on any simple module. We can give a basis for the associated graded module and see that the annihilating ideal is monomial, in contrast to the standard PBW filtration. We finish with exploring the relations to flag varieties, Schubert varieties, quiver Grassmanian and their degenerations to toric varieties.
This is joint work with X.Fang and M.Reineke.
Simone Cecchini (Northeastern University)
Von Neumann algebra valued differential operators over non-compact manifolds
We provide criteria for self-adjointness and τ-Fredholmness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra A endowed with a trace τ. We extend the Callias-type index to operators acting on sections of such bundles and show that this index is stable under compact perturbations. (Joint work with Maxim Braverman).
Pierre Mathieu (Aix-Marseille)
Random walks on (hyperbolic) groups
Fabio Trova (Roma « Tor Vergata »)
Quantization of local systems over finite homotopy types
In Topological Quantum Field Theories from Compact Lie Groups D.S. Freed, M.J. Hopkins, J. Lurie, and C. Teleman have suggested the existence of an n-dimensional Extended Topological Quantum Field Theory arising from representations of finite homotopy types. Unfortunately, no real indication is given on how this TQFT should be defined: only a few hints concerning the need of equivalences between certain limits and colimits and a sketchy account of the case of trivial representations can be found in the article.
I will make this conjecture rigorous in dimension 1, under the sole assumption of the existence of duals, and explain how the construction could be promoted to that of an n-dimensional theory. As an aside result, I will show how certain notions from the representation theory of finite groups can be stated in the general framework of categories with duals; in particular, the Nakayama isomorphism will be recovered.
Time permitting, related problems will be discussed.
Jan Schröer (Bonn)
Enveloping algebras of Kac-Moody algebras as convolution algebras
This is joint work with Christof Geiss and Bernard Leclerc. The positive part of a symmetric Kac-Moody Lie algebra can be realized as a convolution algebra of constructible functions on representation varieties of acyclic quivers.
We will explain how to generalize this from the symmetric to the symmetrizable case. In this more general setup we need quivers with loops and relations.
Ryan Kinser (Iowa)
Combinatorial formulas for type A orbit closures
This talk is based on joint work with Allen Knutson and Jenna Rajchgot. Orbit closures of quivers come up in several of areas of mathematics, for example: representations of finite-dimensional algebras, Lusztig's construction of the canonical basis, generalizations of determinantal varieties, and degeneracy loci for maps of vector bundles. This talk is most related to the last one, where "formulas" for orbit closures means their equivariant K-classes and cohomology classes.
Previous work of many people (e.g. Buch, Fulton, Feher, Rimanyi, Knutson, Miller, Shimozono) produced such formulas in the case where all arrows of the quiver point the same direction, often having positive structure constants in some particular basis. We generalize some of these formulas to Type A quivers of arbitrary orientation. The main ingredient is the bipartite Zelevinsky map constructed in previous work of Rajchgot and mine, which identifies orbit closures with intersections of Schubert varieties and opposite Schubert cells in a partial flag variety.
Fabio Gavarini (Roma « Tor Vergata »)
Affine supergroups and super Harish-Chandra pairs
Together with any supergroup, one can naturally associate the pair made of its classical (i.e. non super) underlying group and its tangent Lie superalgebra, two objects which obey some obvious mutual compatibility constraints; any similar pair is called "super Harish-Chandra pair" (=sHCp). This construction leading from supergroups to sHCp's is functorial, and actually an equivalence, as an explicit quasi-inverse functor is known.
In this talk I present a new, totally different recipe for such a quasi-inverse: indeed, it extends to a much larger setup, with a more geometrical method. I shall mainly adopt the point of view of algebraic (super)geometry, but the bunch of ideas and results we shall be dealing with actually applies to the real differential, the real analytic and the complex analytic case as well.
F. Gavarini, "Global splittings and super Harish-Chandra pairs for affine supergroups", Transactions of the American Mathematical Society (to appear), 56 pages – see http://arxiv.org/abs/1308.0462
Kieran O’Grady (Roma « Sapienza »)
Paul Arne Østvær (Oslo)
Milnor's conjecture on quadratic forms
John Milnor proposed two influential conjectures on Galois cohomology and quadratic forms in his 1969 landmark paper "Algebraic K-theory and quadratic forms" (Inv. Math.). Both conjectures were solved in the affirmative by Vladimir Voevodsky and collaborators in papers published in 2003 (IHES journal) and 2007 (Annals of Math). We propose an alternate solution to Milnor's conjecture on quadratic forms. The proof proceeds by an explicit spectral sequence computation. This is joint work with Oliver Rondigs (arXiv:1311.5833, to appear in: Geometry and Topology).