Dipartimento di Matematica -UTV-

Date	Type	Start	End	Room	Speaker	From	Title
25/10/24	Seminario	14:30	15:30	1201 Dal Passo	Linda HOYER	RWTH Aachen University	Algebra & Representation Theory Seminar (ARTS) "Orthogonal Determinants of Finite Groups" N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006) Abstract Let G be a finite group. It is not hard to see that for any representation ρ : G ⟶ GL(V) for V a real vector space, there exists a G-invariant bilinear form β on V, i.e., a non-degenerate bilinear form such that β(ρ(gv,ρ(g)w) = β(v,w) for all g ∈ G, v, w ∈ V. If ρ is "orthogonally stable" (so it is a sum of even-dimensional irreducible real representations) then the square class of the determinant of the Gram matrix for any basis (the "orthogonal determinant") does not depend on the choice of β, giving us interesting invariants of our group G. Richard Parker conjectured that these orthogonal determinants are always "odd", for any finite group. We will see that the conjecture holds for the symmetric groups, as well as the general linear groups GL(q) for q a power of an odd prime. In the discussion, important concepts like (standard) Young tableaux and Iwahori-Hecke algebras will come up. This talk has the additional purpose of giving a small introduction (with many examples) into the representation theory of finite groups. As such, no previous knowledge in that area will be assumed. N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)
22/10/24	Seminario	14:30	16:00	1101 D'Antoni	Taro Sano	Kobe University	Geometry Seminar Delta invariants of Fano weighted hypersurfaces Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027) Abstract K-stability (or existence of Kähler-Einstein metrics) of explicit Fano varieties has been studied for a long time. Delta invariants (stability thresholds) detect the K-stability of Fano varieties. Moreover, Abban--Zhuang developed a powerful method to compute the delta invariants by adjunctions. In this talk, I will explain our recent results on the K-stability of some Fano weighted hypersurfaces via the Abban--Zhuang method. This is based on joint work with Luca Tasin.
22/10/24	Seminario	14:30	15:30	1201 Dal Passo	Valerio Assenza	IMPA, Rio de Janeiro	Seminario di Equazioni Differenziali Magnetic curvature and one application to the existence of closed magnetic geodesic Abstract Magnetic systems are the natural toy model for the motion of a charged particle moving on a Riemannian manifold under the influence of a (static) magnetic force. In this talk we introduce a curvature operator called magnetic curvature which encodes the information of the classical Riemannian curvature together with terms of perturbation due to the magnetic interaction. In a variational setting, we use this new notion of curvature to approach the problem of finding closed trajectories for small energy levels. Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)
16/10/24	Colloquium	15:00	16:00	1201 Dal Passo	Silvia Pappalardi	University of Cologne	Colloquium Levi-Civita Free probability approaches to quantum many-body dynamics Note: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006). Abstract Understanding how to characterize quantum chaotic dynamics is a longstanding question. The universality of chaotic many-body dynamics has long been identified by random matrix theory, which led to the well-established framework of the Eigenstate Thermalization Hypothesis. In this talk, I will discuss recent developments that identify Free Probability -- a generalization of probability theory to non-commuting objects -- as a unifying mathematical framework to describe correlations of chaotic many-body systems. I will show how the full version of the Eigenstate Thermalization Hypothesis, which encompasses all the correlations, can be rationalized and simplified using the language of Free Probability. This approach uncovers unexpected connections between quantum chaos and concepts in quantum information theory, such as unitary designs.
11/10/24	Seminario	14:30	15:30	1101 D'Antoni	Thibault JUILLARD	Université Paris-Saclay	Algebra & Representation Theory Seminar (ARTS) "Reduction by stages for affine W-algebras" N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006) Abstract Affine W-algebras form a family of vertex algebras indexed by the nilpotent orbits of a simple finite dimensional complex Lie algebra. Each of them is built as a noncommutative Hamiltonian reduction of the corresponding affine Kac-Moody algebra. In this talk, I will present a joint work with Naoki Genra about the problem of reduction by stages for these affine W-algebras: given a suitable pair of nilpotent orbits in the simple Lie algebra, it is possible to reconstruct one of the two affine W-algebras associated to these orbits as the Hamiltonian reduction of the other one. I will insist on how this problem relates to our previous work about reduction by stages between Slodowy slices, which are Poisson varieties associated with affine W-algebras. I will also mention some applications and motivations coming from Kraft-Procesi rule for nilpotent Slodowy slices, and isomorphisms between simple affine admissible W-algebras. N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).
09/10/24	Seminario	16:00	17:00	1201 Dal Passo	Fausto Di Biase	Università "G. D'Annunzio" di Chieti-Pescara	Operator Algebras Seminar On the Differentiation of Integrals in Measure Spaces Along Filters: II Abstract Let X be a complete measure space of finite measure. The Lebesgue transform of an integrable function f on X encodes the collection of all the mean-values of f on all measurable subsets of X of (finite and) positive measure. In the problem of the differentiation of integrals, one seeks to recapture f from its Lebesgue transform. In previous work we showed that, in all known results, f may be recaptured from its Lebesgue transform by means of a limiting process associated to an appropriate family of filters defined on the collection A of all measurable subsets of X of (finite and) positive measure. The first result of the present joint work with Steven G. Krantz, is a precise proof of a result announced in a previous work: the existence of such a limiting process is equivalent to the existence of a Von Neumann-Maharam lifting of X. In the second result of this work, we provide an independent argument that shows that the recourse to filters is a necessary consequence of the requirement that the process of recapturing from its mean-values may be extended to a natural transformation, in the sense of category theory. This result essentially follows from the Yoneda lemma. As far as we know, this is the first instance of a significant interaction between category theory and the problem of the differentiation of integrals. In a third result, we have proved, in a precise sense, that natural transformations fall within the general concept of homomorphism. As far as we know, this is a novel conclusion: Although it is often said that natural transformations are homomorphisms of functors, this statement appears to be presented as a mere analogy, not in a precise technical sense. In order to achieve this result, we had to bring to the foreground a notion that is implicit in the subject but has remained hidden in the background, i.e., that of partial magma.
08/10/24	Seminario	15:00	16:00	1200 Biblioteca Storica	Mark Hughes	Brigham Young University Utah	Topology Seminar Representations of knots for applications in machine learning Abstract Knots form an infinite and complex data set, with topological invariants that are often intertwined in ways not yet fully understood. Many foundational challenges in knot theory and low-dimensional topology can be recast as problems in reinforcement learning and generative machine learning. A key decision in approaching knot theory through an ML lens is determining how to represent knots in a machine-readable format, which can be thought of as selecting a suitable prior distribution over the space of all knots. In this talk, I will explore the challenges of representing knots for ML applications and showcase recent examples where machine learning has been successfully applied to problems in low-dimensional topology.
08/10/24	Seminario	14:30	16:00	1101 D'Antoni	Emanuele Macrì	Université Paris-Saclay	Geometry Seminar Modelli di Mukai per varietà di Fano Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027) Abstract La classificazione delle varietà di Fano di dimensione 3 e indice 1 è uno dei risultati fondamentali in geometria algebrica, completata da Iskovskikh e Mukai più di trent'anni fa. In questo seminario, basato su un progetto in collaborazione con Arend Bayer e Alexander Kuznetsov, presenterò una nuova dimostrazione, basata sempre sulle idee di Mukai, che si estende anche al caso singolare e in dimensione superiore. I will present an approach to the problem using logarithmic geometry which allows us to extend the theory of linear series to arbitrary nodal curves. A prominent role is played by vector bundles on Olsson fans, which I will introduce. This is joint work in progress with Luca Battistella and Jonathan Wise.
01/10/24	Seminario	14:30	16:00	1101 D'Antoni	Federico Caucci	Università di Roma Tor Vergata	Geometry Seminar On syzygies of abelian and Kummer varieties Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027) Abstract Equations defining projective varieties and their syzygies have been classically studied. In this talk, starting from the case of curves, I will recall several results about syzygies of projective varieties, especially focusing on some recent ones about abelian and Kummer varieties.
27/09/24	Seminario	16:00	17:00	1201 Dal Passo	Gastón Andrés GARCÍA	Universidad Nacional de La Plata / CONICET	Algebra & Representation Theory Seminar (ARTS) "Hopf algebras and finite simple groups of Lie type" N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006) Abstract Hopf algebras (and variations of them) are the algebraic counterpart of (strict, rigid) tensor categories. As such, they appear as symmetries of different categorial, geometrical and physical objects. In particular, several applications may be found in diverse areas of mathematics, physics and theoretical computer sciences. Hopf algebras were already studied in the 60's and had a big impulse in the 80's after the work of Drinfeld on quantum groups. Despite more than 60 years of study, not much is known about them: general results are sparse and the classification is only known for (quite) small dimensions or for families with different properties. This talk will be about a joint project with N. Andruskiewitsch and G. Carnovale in our attempt to determine finite-dimensional pointed Hopf algebras over finite-simple groups of Lie type. The main idea we exploit is the reduction of the problem to group-theoretical criteria to determine the finite-dimensionality of our objects, which boils down to the use of different properties of conjugacy classes, root systems and computational tools.