Notiziario Scientifico
a cura del Dipartimento di Matematica G. Castelnuovo, Sapienza Università di Roma
Settimana dal 06-01-2020 al 12-01-2020
Martedì 07 gennaio 2020
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Lionel Levine (Cornell University)
Random walks with local memory
The theme of this talk is walks in a random environment of "signposts" altered by the walker. I'll focus on three related examples:
1. Rotor walk on Z^2. Your initial signposts are independent with the uniform distribution on {North,East,South,West}. At each step you rotate the signpost at your current location clockwise 90 degrees and then follow it to a nearest neighbor. Priezzhev et al. conjectured that in n such steps you will visit order n^{2/3} distinct sites. I'll outline an elementary proof of a lower bound of this order. The upper bound, which is still open, is related to a famous question about the path of a light ray in a grid of randomly oriented mirrors. This part is joint work with Laura Florescu and Yuval Peres.
2. p-rotor walk on Z. In this walk you flip the signpost at your current location with probability 1-p and then follow it. I'll explain why your scaling limit will be a Brownian motion perturbed at its extrema. This part is joint work with Wilfried Huss and Ecaterina Sava-Huss.
3. p-rotor walk on Z^2. Rotate the signpost at your current location clockwise with probability p and counterclockwise with probability 1-p, and then follow it. This walk “organizes” its environment by destroying cycles of signposts. A native environment -- stationary in time, from your perspective as the walker -- is an orientation of the uniform spanning forest, plus one additional edge. This part is joint work with Swee Hong Chan, Lila Greco, and Peter Li: https://arxiv.org/abs/1809.04710
Martedì 07 gennaio 2020
Ore 15:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Modellistica Differenziale Numerica
Andrea Aspri (Ricam)
Data driven regularization
In this talk I will present a data-driven iteratively regularized Landweber iteration for solving linear and nonlinear ill-posed inverse problems. The method takes into account training data, which are used to estimate the interior of a black box, which is used to define the iteration process. I will show convergence and stability results for the scheme in the infinite dimensional Hilbert spaces and then I will discuss some numerical experiments.
Mercoledì 08 gennaio 2020
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Algebra e Geometria
Daniel Labardini-Fragoso (UNAM (Mexico))
Algebraic and combinatorial decompositions of Fuchsian groups
The discrete subgroups of \(PSL_2(R)\) are often called 'Fuchsian groups'. For Fuchsian groups \(\Gamma\) whose action on the hyperbolic plane \(H\) is free, the orbit space \(H/\Gamma\) has a canonical structure of Riemann surface with a hyperbolic metric, whereas if the action of \(\Gamma\) is not free, then \(H/\Gamma\) has a structure of 'orbifold'. In the former case, there is a direct and very clear relation between \(\Gamma\) and the fundamental group \(\pi_1(H/\Gamma,x)\): a theorem of the theory of covering spaces states that they are isomorphic. When the action of \(\Gamma\) is not free, the relation between \(\Gamma\) and \(\pi_1(H/\Gamma,x)\) is subtler. A 1968 theorem of Armstrong states that there is a short exact sequence \(1\rightarrow E\rightarrow \Gamma\rightarrow \pi_1(H/\Gamma,x)\rightarrow 1\), where \(E\) is the subgroup of \(\Gamma\) generated by the elliptic elements. For \(\Gamma\) finitely generated, non-elementary and with at least one parabolic element, I will present full algebraic and combinatorial decompositions of \(\Gamma\) in terms of \(\pi_1(H/\Gamma,x)\) and a specific finitely generated subgroup of \(E\), thus improving Armstrong's theorem.This talk is based on an ongoing joint project with Sibylle Schroll and Yadira Valdivieso-Díaz that aims at describing the bounded derived categories of skew-gentle algebras in terms of curves on surfaces with orbifold points of order 2.
Mercoledì 08 gennaio 2020
Ore 16:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Fisica Matematica
Giacomo De Palma (MIT)
Quantum optimal transport with quantum channels
We propose a new generalization to quantum states of the Wasserstein distance, which is a fundamental distance between probability distributions given by the minimization of a transport cost. Our proposal is the first where the transport plans between quantum states are in natural correspondence with quantum channels, such that the transport can be interpreted as a physical operation on the system. Our main result is the proof of a triangle inequality for our transport distance. We then specialize to quantum Gaussian systems, which provide the mathematical model for the electromagnetic radiation in the quantum regime. We prove that the noiseless quantum Gaussian attenuators and amplifiers are the optimal transport plans between thermal quantum Gaussian states, and that our distance recovers the classical Wasserstein distance in the semiclassical limit. Finally, we prove that the distance between a quantum state and itself is intimately connected with the Wigner-Yanase metric on the manifold of quantum states.
Giovedì 09 gennaio 2020
Ore 14:00, Aula B, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di geometria differenziale
Pier Paolo La Pastina (Sapienza Università di Roma)
Deformations of vector bundles in the categories of Lie algebroids and groupoids
VB-algebroids and VB-groupoids can be considered as vector bundles in the categories of Lie algebroids and groupoids and encompass several classical objects, including Lie algebra and Lie group representations, 2-vector spaces and the tangent and the cotangent algebroid (groupoid) to a Lie algebroid (groupoid). Moreover, they are geometric models for some kind of representations of Lie algebroids (groupoids), namely 2-term representations up to homotopy. Finally, it is well known that Lie groupoids are “concrete” incarnations of differentiable stacks, hence VB-groupoids can be considered as representatives of vector bundles over differentiable stacks, and VB-algebroids their infinitesimal versions. We attach to every VB-algebroid and VB-groupoid a cochain complex controlling its deformations. Moreover, the deformation complex of a VB-algebroid is equipped with a DGLA structure. The basic properties of these complexes are discussed: their relationship with the deformation complexes of the total spaces and the base spaces, particular cases and generalizations. The main theoretical results are a linear van Est theorem, that gives conditions for the linear deformation cohomology of a VB-groupoid to be isomorphic to that of the corresponding VB-algebroid, and a Morita invariance theorem, that implies that the linear deformation cohomology of a VB-groupoid is really an algebraic invariant of the associated vector bundle of differentiable stacks. Finally, several examples are discussed, showing how the linear deformation cohomologies are related to other well-known cohomologies.
Giovedì 09 gennaio 2020
Ore 14:50, Aula B, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di algebra
Francesco Allegra (Sapienza Università di Roma)
Affine W-algebras in type A and the Arakawa-Moreau conjecture
W-algebras are an important class of vertex algebras associated with a reductive Lie algebra g, a nilpotent element f ∈ g and a scalar k ∈ C, which are closely related with various area of mathematics such as integrable systems, two-dimensional conformal field theories, modular representation theory, four dimensional gauge theory, and geometric Langlands program. Moreover, there has been a renewed interest in W-algebras since they appear as invariants of Argyres-Douglas theory via the 4D/2D correspondence recently discovered in physics. However, despite of the importance of W-algebras the problem of finding all the generators for every affine W-algebra remains unsolved. The only results known so far are from Kac-Wakimoto for minimal nilpotent elements, and from Arakawa-Molev for rectangular nilpotent elements, with the restriction of g = gl_N. In this thesis we obtained an explicit list of generators of W-algebras of type A associated with quasi-rectangular nilpotent elements. This is a nice generalization of the aforementioned results, since both are quasi-rectangular. Furthermore, as an application we were able to confirm a conjecture of Anne Moreau and Tomoyuki Arakawa in some cases on the isomorphism of simple quotients of W-algebras. This is a promising result since it confirms also some expectations by physicists that arose in the recent study of the 4D/2D correspondence.
Venerdì 10 gennaio 2020
Ore 14:30, Aula "Claudio D'Antoni", Dipartimento di Matematica dell'Università degli Studi di Roma "Tor Vergata"
Markus Reineke (University of Bochum)
Cohomological Hall algebras of quivers
Cohomological Hall algebras form a class of graded algebras
which are defined by a convolution operation on representation spaces of
quivers. In the talk, we will motivate their definition, construct them,
and review basic properties and known structural results. Then we turn
to the special case of the Kronecker quiver and derive a description by
generators and relations of the corresponding cohomological Hall
algebra, which is related to Yangians.
N.B.: this talk is part of the activity of the MIUR Excellence
Department Project CUP E83C18000100006
Venerdì 10 gennaio 2020
Ore 15:45, Aula "Claudio D'Antoni", Dipartimento di Matematica dell'Università degli Studi di Roma "Tor Vergata"
Peter McNamara (University of Melbourne)
Geometric Extension Algebras
A number of algebras that we study in Lie theory have
geometric interpretations, appearing as a convolution algebra in
Borel-Moore homology or equivalently as the Ext-algebra of a pushforward
sheaf. We will discuss how information on the representation theoretic
side (like being quasihereditary) is related to information on the
geometric side (like odd cohomology vanishing). The primary application
is to KLR and related algebras.
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