Notiziario Scientifico

Notiziario dei seminari di carattere matematico
a cura del Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma

Settimana dal 10-02-2025 al 16-02-2025

Lunedì 10 febbraio 2025
Ore 16:00, Aula D'Antoni , Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata
DocTorV Seminar
Giacomo Greco (Università degli Studi di Roma Tor Vergata )
From Statistical Mechanics towards Generative models, passing thruough (Stochastic) Optimal Transport
In this talk I will introduce a statistical mechanics problem known as the Schrödinger problem, which aims at finding the most likely evolution of a cloud of particles conditionally to initial and final configurations. Starting from this problem I will guide you towards stochastic optimal control, entropic optimal transport and finally generative models. (No prior knowledge of statistical mechanics is required, it might be useful basic knowledge of SDEs, though you might enjoy the talk even without it and with a bit of faith).
Per informazioni, rivolgersi a: doctorv.uniroma2@gmail.com

Martedì 11 febbraio 2025
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Probabilità
Jacopo Borga (Massachusetts Institute of Technology)
Lattice Yang-Mills theory in the large N limit via sums over surfaces
Lattice Yang-Mills theories are important models in particle physics. They are defined on the d-dimensional lattice Z^d using a group of matrices of dimension N, and Wilson loop expectations are the fundamental observables of these theories. Recently, Cao, Park, and Sheffield showed that Wilson loop expectations can be expressed as sums over certain embedded bipartite maps of any genus. Building on this novel approach, we prove in the so-called strongly coupled regime: - A rigorous formula in terms of embedded bipartite planar maps of Wilson loop expectations in the large N limit, in any dimension d. - An exact computation of Wilson loop expectations in the large N limit, in dimension d=2, for a large family of (simple and non-simple) loops. Previous results to the two aforementioned points were previously established by Chatterjee (2019) and Basu & Ganguly (2016), respectively. Our results extend these previous results, offer simpler proofs and provide a new perspective on these significant quantities. This work is a collaboration with Sky Cao and Jasper Shogren-Knaak.
Per informazioni, rivolgersi a: silvestri@mat.uniroma1.it

Martedì 11 febbraio 2025
Ore 14:30, Aula Dal Passo, Dipartimento di Matematica, Università di Roma "Tor Vergata"
Seminario di Analisi Matematica
Jacopo Schino (Uniwersytet Warszawski)
Normalised solutions to poly-harmonic equations with Hardy-type potentials via a Nehari-Pohozaev approach
Schrödinger-type equations model a lot of natural phenomena and their solutions have interesting and important properties. This gives rise to the search for normalised solutions, i.e., when the mass is prescribed. In this talk, I will exploit a novel variational approach, introduced in the context of autonomous Schrödinger equations, to find a least-energy solution to a problem involving the m-Laplacian and a Hardy-type potential. The growth of the non-linearity is mass-supercritical at infinity and at least mass-critical at the origin. An important step in this approach is to show that all the solutions satisfy the Pohozaev identity, which in the presence of a Hardy-type potential was previously known only in the spherical case with m = 1. This talk is based on a joint article with Bartosz Bieganowski and Jaroslaw Mederski, about energy-subcritical non-linearities, and a joint preprint with Bartosz Bieganowski and Olímpio H. Miyagaki, concerning exponential critical non-linear terms in dimension N = 2m.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006
Per informazioni, rivolgersi a: molle@mat.uniroma2.it

Martedì 11 febbraio 2025
Ore 14:30, aula d'Antoni, Dipartimento di Matematica, Università di Roma Tor Vergata
seminario di Geometria e teoria dei Numeri
René Schoof (Università di Roma ``Tor Vergata'')
La Congettura di Greenberg
Il contesto naturale della congettura di Greenberg è la teoria di Iwasawa. In questo colloquio introdurrò la teoria di Iwasawa e la congettura di Greenberg, per poi presentare risultati recenti ottenuti in collaborazione con Pietro Mercuri.
Per informazioni, rivolgersi a: guidomaria.lido@gmail.com

Mercoledì 12 febbraio 2025
Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
Seminario di Algebra e Geometria
Luca Schaffler (Università di Roma Tre)
Fineness and smoothness of a KSBA moduli of marked cubic surfaces
The moduli space of cubic surfaces marked by their \(27\) lines admits multiple compactifications arising from different perspectives. By work of Gallardo-Kerr-Schaffler, it is known that Naruki’s cross-ratio compactification is isomorphic to the normalization of the Kollár, Shepherd-Barron, Alexeev (KSBA) compactification parametrizing pairs \(\left(S,\left(\frac{1}{9}+\epsilon\right)D\right)\), where \(D\) is the sum of the \(27\) marked lines on \(S\), along with their stable degenerations. In this talk, we show that the normalization assumption is unnecessary by proving that this KSBA compactification is smooth. Additionally, we show it is a fine moduli space. This is achieved by studying the automorphisms and the \(\mathbb{Q}\)-Gorenstein obstructions of the stable pairs it parametrizes. This is joint work with Hanlong Fang and Xian Wu. -- This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

Giovedì 13 febbraio 2025
Ore 14:15, Aula M4, Dipartimento di Matematica e Fisica, Università Roma Tre
Seminario di Geometria
Felix Röhrle (Tübingen)
Tropical enriched count for plane curves with conjugate point conditions
Consider the classical problem in enumerative geometry of counting rational plane curves through a fixed configuration of points. The problem may be considered over any base field and the point conditions might be scheme theoretic points. Recently, Kass--Levine--Solomon--Wickelgren have used techniques from \(\mathbb{A}^1\)-homotopy theory to define an enumerative invariant for this problem which is defined over a large class of possible base fields. This new theory generalizes Gromov-Witten invariants (base field = complex numbers) and Welschinger invariants (base field = real numbers) simultaneously. In this talk I will report on work in progress which explores the tropical approach to computing these new invariants. More specifically, for point conditions which are defined over (at most) quadratic extensions of the base field, we are developing a tropical correspondence theorem which expresses the KLSW-invariant as a tropical count. This result generalizes earlier correspondence theorems by Mikhalkin and Shustin and allows us to effectively compute some quadratically enriched invariants which were not known before. This is joint work in progress with Andrès Jaramillo-Puentes, Hannah Markwig, and Sabrina Pauli.
Per informazioni, rivolgersi a: amos.turchet@uniroma3.it

Giovedì 13 febbraio 2025
Ore 14:30, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma
P(n)/N(p) : Problemi differenziali nonlineari/Nonlinear differential problems
Carlo Alberto Antonini (Università di Parma)
Second-order estimates in anisotropic elliptic problems
In recent years, various results showed that second-order regularity of solutions to the p-Laplace equation can be properly formulated in terms of the expression under the divergence, the so-called stress field, see [3]. I will discuss the extension of these results to the anisotropic p-Laplace problem, namely equations of the kind \(-\mathrm{div}\,\big(\mathcal{A}(\nabla u)\big)=f\,, \) in which the stress field is given by \(\mathcal{A}(\nabla u)=H^{p-1}(\nabla u)\,\nabla_\xi H(\nabla u)\), where \(H(\xi)\) is a norm on \(\mathbb{R}^n\) satisfying suitable ellipticity assumptions. \(W^{1,2}\)-Sobolev regularity of \(\mathcal{A}(\nabla u)\) is established when \(f\) is square integrable, and both local and global estimates are obtained. The latter apply to solutions to homogeneous Dirichlet problems on sufficiently regular domains. A key point in our proof is an extension of Reilly's identity to the anisotropic setting. This is joint work with A. Cianchi, G. Ciraolo, A. Farina and V.G. Maz'ya. References [1] C.A. Antonini, G. Ciraolo, A. Farina, \textit{Interior regularity results for inhomogeneous anisotropic quasilinear equations}, Math. Ann. (2023). [2] C.A. Antonini, A. Cianchi, G. Ciraolo, A. Farina, V.G. Maz'ya, \textit{Global second-order estimates in anisotropic elliptic problems}, arXiv preprint (2023) arXiv:2307.03052. [3] A. Cianchi, V.G. Maz'ya, \textit{Second-order two-sided estimates in nonlinear elliptic problems}, Arch. Ration. Mech. Anal. 229 (2018), no. 2, 569-599.
Per informazioni, rivolgersi a: galise@mat.uniroma1.it

Venerdì 14 febbraio 2025
Ore 14:30, Aula Dal Passo, Dipartimento di Matematica, U Roma Tor Vergata
Algebra and Representation Theory Seminar (ARTS)
Alessandro Carotenuto (U Parma)
Complex geometry of the full quantum flag manifold of quantum SU(3)
The noncommutative differential geometry of quantum flag manifolds has seen rapid growth in recent years, following the remarkable finding of a complex structure for flag manifolds of irreducible type by Heckenberger and Kolb. With a large part of the theory for the irreducible cases already figured out, it is now time to tackle the question of how to obtain the same structure for other types of flag manifolds. In this work in collaboration with R. Ó Buachalla and J. Razzaq, we give a complex structure for the full flag manifold of quantum SU(3) that includes the differential calculus discovered by Ó Buachalla and Somberg as its holomorphic sub-complex. I shall review this construction that makes use of Lusztig quantum root vectors, while at the same time giving a general overview of the theory of noncommutative differential calculi for quantum homogeneous spaces.

Venerdì 14 febbraio 2025
Ore 16:00, Aula Dal Passo, Dipartimento di Matematica, U Roma Tor Vergata
Joint Topology & Algebra and Representation Theory Seminar (T-ARTS)
Andrea Guidolin (U Southampton)
Algebraic Wasserstein distances and stable homological invariants of data
Persistent homology, a popular method in Topological Data Analysis, encodes geometric information of data into algebraic objects called persistence modules. Invoking a decomposition theorem, these algebraic objects are usually represented as multisets of points in the plane, called persistence diagrams, which can be fruitfully used in data analysis in combination with statistical or machine learning methods. Wasserstein distances between persistence diagrams are a common way to compare the outputs of the persistent homology pipeline. In this talk, I will explain how a notion of p-norm for persistence modules leads to an algebraic version of Wasserstein distances which fit into a general framework for producing distances between persistence modules. I will then present stable invariants of persistence modules which depend on Wasserstein distances and can be computed efficiently. The use of these invariants in a supervised learning context will be illustrated with some examples.

Venerdì 14 febbraio 2025
Ore 16:00, Aula Enriques, Dipartimento di Matematica, Sapienza Università di Roma
Seminari PLS per docenti
Sergio Caprara (Sapienza Università di Roma)
Atomi e molecole: i mattoni della materia condensata

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