Notiziario Scientifico
a cura del Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma
Settimana dal 07-11-2022 al 13-11-2022
Lunedì 07 novembre 2022
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Analisi Matematica
Giacomo Canevari (Università di Verona)
Gamma-convergence for the Ginzburg-Landau functional on complex line bundles
The Ginzburg-Landau functional was originally proposed as a model for superconductivity in Euclidean domains. However, invariance with respect to gauge transformations - which is one of the most prominent features of the model - suggests that the functional can be naturally defined in the setting of complex line bundles, where it can be regarded as an Abelian Yang-Mills-Higgs theory. In this talk, we shall consider the Ginzburg-Landau functional on an Hermitian line bundle over a closed Riemannian manifold, in the so-called "non-self dual scaling" (which is closer to the original motivation from superconductivity theory). We shall focus on the variational aspects of the problem; more precisely, we will discuss a Gamma-convergence result for sequences whose energy grows at most logarithmically in the Ginzburg-Landau coupling parameter. As we shall see, the London equation for superconductivity plays a significant role in our analysis. The talk is based on a joint work with Federico Dipasquale (Università Federico II, Napoli) and Giandomenico Orlandi (Verona).
Per informazioni, rivolgersi a: spadaro@mat.uniroma1.it
Martedì 08 novembre 2022
Ore 10:30, Aula B, Dipartimento di Matematica, Sapienza Università di Roma
Corso di Dottorato, Prima lezione
Siye Wu (National Tsing Hua University)
Symplectic Geometry
Abstract. Symplectic geometry is a branch of differential geometry that studies symplectic manifolds, which are smooth manifolds equipped with closed non-degenerate two-forms. The course begins with basic concepts such as Hamiltonian vector fields, Poisson brackets, Lagrangian submanifolds and the Darboux theorem. Examples include cotangent bundles, K ̈ahler man- ifolds, coadjoint orbits and fibrations of Lagrangian tori. Comparisons will be made with contact and Poisson manifolds. The second part is about symmetries of symplectic man- ifolds. Important notions to be introduced are Hamiltonian group actions, moment maps and their images, and symplectic quotients. Interesting examples are toric manifolds and moduli space of flat connections on surfaces. The last part of the course is to be on the applications of symplectic geometry to classical mechanics (Lagrangian and Hamiltonian mechanics), solving problems on the motion of rigid bodies and integrable systems. The course is suitable for students who have already taken an introductory course on man- ifolds (with calculus of differential forms) and who wish to engage their knowledge in a constructive and useful setting. Il programma dettagliato del corso è disponibile nella pagina web del dottorato in Matematica a Roma Sapienza. Il corso di riunisce 2 volte a settimana, ogni volta per due ore, per un totale di 16 ore. Il calendario dopo l'8 Novembre è : Giovedi' 10/11, ore 14, Aula B; Martedi' 15/11, 22/11, 29/11, ore 14, Aula B; Giovedi' 17/11, 23/11, 1/12, ore 14, Aula B. (Il calendario dopo l'8 Novembre è da confermare.)
Per informazioni, rivolgersi a: piazza@mat.uniroma1.it
Martedì 08 novembre 2022
Ore 13:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario
Jesus Juyumaya (University of Valparaiso)
Polinomi Homfly-pt e di Kauffman con lacci
Inizieremo dando una panoramica degli invarianti polinomiali di Homflypt e Kauffman, concentrandosi sulle loro costruzioni algebriche utilizzando le algebre di Hecke e BMW, rispettivamente. Generalizziamo quindi queste algebre introducendo "ties" (lacci) nei generatori che le definiscono. Vedremo che queste generalizzazioni definiscono nuove invarianti di "links".
Martedì 08 novembre 2022
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Probabilità
Matteo Quattropani (Sapienza Università di Roma)
Mixing of the Averaging process on graphs
Abstract: The Averaging process (a.k.a. repeated averages) is a mass redistribution model over the vertex set of a graph. Given a graph G, the process starts with a non-negative mass associated to each vertex. The edges of G are equipped with Poissonian clocks: when an edge rings, the masses at the two extremes of the edge are equally redistributed on these two vertices. Clearly, as time grows to infinity the state of the system will converge (in some sense) to a flat configuration in which all the vertices have the same mass. The process has been introduced to the probabilistic community by Aldous and Lanoue [1] in 2012, and recently received some attention thanks to the work of Chatterjee, Diaconis, Sly and Zhang [2], where the authors show an abrupt convergence to equilibrium (measured in L^1 distance) in the case in which the underlying graph is complete (and of diverging size). In this talk, I will present some recent results obtained in collaboration with F. Sau (IST Austria) [3,4] and P. Caputo (Roma Tre) [4]. In [3] we show that if the underlying graph is “finite dimensional” (e.g., a finite box of Z^d), then the convergence to equilibrium is smooth (i.e., without cutoff) when measured in L^p with p in [1,2]. On the other hand, in [4] we show that a cutoff phenomenon (for the L^1 and L^2 distance to equilibrium) takes place when the underlying graph is the hypercube or the complete bipartite graph. [1] David Aldous, and Daniel Lanoue. A lecture on the averaging process. Probab. Surv., 9:90–102, 2012. [2] Sourav Chatterjee, Persi Diaconis, Allan Sly, and Lingfu Zhang. A phase transition for repeated averages. Ann. Probab. 50(1):1–17, 2022. [3] Matteo Quattropani and Federico Sau. Mixing of the Averaging process and its discrete dual on finite-dimensional geometries. Ann. Appl. Probab. (to appear). [4] Pietro Caputo, Matteo Quattropani and Federico Sau. Cutoff for the Averaging process on the hypercube and complete bipartite graphs. (to appear).
Per informazioni, rivolgersi a: matteo.quattropani@gmail.com
Martedì 08 novembre 2022
Ore 14:30, Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata
Seminario di Geometria
Andrea Petracci (Università di Bologna)
Toric geometry and K-moduli of Fano varieties
Roughly speaking, Fano varieties are the algebraic varieties with positive curvature. Very recently K-stability (i.e. the existence of Kähler-Einstein metrics) has been applied to construct moduli spaces of Fano varieties, called K-moduli. Toric varieties are very special algebraic varieties which are constructed in a combinatorial flavour, starting from discrete objects such as polytopes. In this talk, I would like to explain a couple of applications of the deformation theory of toric Fano varieties to the study of K-moduli spaces of Fano varieties: a) K-moduli spaces of Fano varieties can be quite singular; b) (joint work with H.Abban, I.Cheltsov, A.Kasprzyk) the K-moduli space of quartic 3-folds contains points corresponding to K-polystable varieties which are not quartic 3-folds.
Per informazioni, rivolgersi a: onorati@mat.uniroma2.it
Martedì 08 novembre 2022
Ore 15:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Modellistica Differenziale Numerica
Chiara Segala (RWTH Aachen University)
Predictive control for non-linear collective dynamics and their mean-field limit
In this talk I will present the synthesis of control laws for interacting agent-based dynamics and their mean-field limit. In particular, a linearization-based approach is used for the computation of suboptimal feedback laws obtained from the solution of differential matrix Riccati equations. Quantification of dynamic performance of such control laws leads to theoretical estimates on suitable linearization points of the nonlinear dynamics. Subsequently, the feedback laws are embedded into a nonlinear model predictive control framework where the control is updated adaptively in time according to dynamic information on moments of linear mean-field dynamics.
Per informazioni, rivolgersi a: giuseppe.visconti@uniroma1.it
Mercoledì 09 novembre 2022
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
seminario di Algebra e Geometria
Siarhei Finski (CMLS - École polytechnique & CNRS)
On the metric structure of section rings and holomorphic extension theorem
We study the relationship between metric and algebraic structures on section rings of polarized projective manifolds. More precisely, we prove that once the kernel is factored out, the multiplication operator of the section ring becomes an approximate isometry (up to normalization) with respect to the \(L^2\)-norm. We then show that this algebraic property characterizes \(L^2\)-norms and describe some applications of this characterisation. The semiclassical version of Ohsawa-Takegoshi theorem, describing holomorphic extensions from submanifolds to global manifolds of holomorphic sections of sufficiently large tensor powers, lies at the heart of our approach.
Per informazioni, rivolgersi a: diverio@mat.uniroma1.it
Mercoledì 09 novembre 2022
Ore 14:15, online (zoom), disponibile alla pagina https://indico.gssi.it/event/410/
ciclo Mathematical Challenges in Quantum Mechanics
Dirk Hundertmark (Kalsruher Institut für Technologie)
Cwikel's bound reloaded
There are several proofs by now for the famous Cwikel–Lieb–Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel’s approach yields bad constants, is not set in stone: We give a substantial refinement of Cwikel’s original approach which highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis. Moreover, it gives an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies. My talk is based on a paper which was recently published in Inventiones mathematicae and is available online at https://doi.org/10.1007/s00222-022-01144-7 or as a PDF from https://link.springer.com/content/pdf/10.1007/s00222-022-01144-7.pdf. In my talk I will explain the background and the proof in the case of a usual Schrödinger operator. I will keep the discussion free from technicalities, the main tool in the proof is the Cauchy Schwarz inequality.
Per informazioni, rivolgersi a: monaco@mat.uniroma1.it
Mercoledì 09 novembre 2022
Ore 15:00, Aula Picone, Dipartimento di Matematica, Sapienza Università di Roma
Pomeriggio matematico al Castelnuovo
Claudio Procesi (Sapienza Università di Roma)
Riemann ed il suo tempo
Giovedì 10 novembre 2022
Ore 14:50, Aula dei Seminari, Edificio RM004, Facoltà di Ingegneria, Dipartimento SBAI
Seminario
Manuel V. Gnann ( TU Delft, Netherland)
Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise
We prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grün, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise. The talk is based on joint work with Konstantinos Dareiotis (University of Leeds), Benjamin Gess (Bielefeld University/MPI Leipzig), and Günther Grün (University of Erlangen-Nuremberg).
Giovedì 10 novembre 2022
Ore 13:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario
Jesus Juyumaya (University of Valparaiso)
Nodi, monoidi e algebre
Discuteremo relazioni tra invarianti di nodi classici e nodi singolari con algebre e monoidi che intervengono nelle loro definizioni. In particolare, vedremo come queste relazioni portano a considerare alcune algebre che possono essere viste come deformazioni dei cosiddetti "monoidi ramificati".
Venerdì 11 novembre 2022
Ore 14:30, Aula "Roberta Dal Passo", ipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"
Algebra and Representation Theory Seminar
Azzurra Ciliberti ("Sapienza" Università di Roma)
Categorification of skew-symmetrizable cluster algebras through symmetric quivers
I will present my attempt to categorify cluster algebras of type B and C using the theory of symmetric quivers in the sense of Derksen and Weyman
Venerdì 11 novembre 2022
Ore 16:00, Aula "Roberta Dal Passo", Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"
Algebra and Representation Theory Seminar
Sabino Di Trani (Università degli Studi di Trento )
Smoothness Criteria for T-Fixed Points in Flat Linear Degenerations of the Flag Variety
Linear Degenerations of the Flag Variety arise as very natural generalizations of the Complete Flag Variety and their geometrical properties very often appear to be linked with interesting combinatorial patterns. The talk will focus on a special class of linear degenerations, the Flat Degenerations, that have the remarkable property of being equidimensional algebraic varieties of the same dimension as the Complete Flag Variety. In some very recent works of M.Lanini and A.Pütz it is proved that Linear Degenerations of the Flag Variety can be endowed with a structure of GKM variety, under the action of a suitable algebraic torus T. The aim of the talk is to show how GKM Theory can be applied in this setting to prove some new results about the smooth locus in Flat Degenerations, generalizing a smoothness criterion proved by G.Cerulli Irelli, E. Feigin and M.Reineke for Feigin Degeneration. Finally, we provide a different combinatorial criterion, linking the smoothness property of a T-fixed point to the complete graph and to its orientations.
Le comunicazioni relative a seminari da includere in questo notiziario devono pervenire esclusivamente
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