## Notiziario Scientifico

**Notiziario dei seminari di carattere matematico**

a cura del Dipartimento di Matematica

*Guido Castelnuovo*, Sapienza Università di Roma

Settimana dal 11-11-2024 al 17-11-2024

**Lunedì 11 novembre 2024**

Ore 9:00, Aula Picone, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Gérard Besson (Institut Fourier, Université de Grenoble)
*Ricci flow and the 1/4-pinched differentiable sphere theorem (after Brendle and Schoen), I*

We shall show that a Riemanian manifold whose sectional curvature is strictly between 1 and 1/4 is diffeomorphic to the standard sphere. The proof uses the Ricci flow without surgery and a nice work on the algebra of the curvature tensor.

**Lunedì 11 novembre 2024**

Ore 11:00, Aula Picone, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Bernhard Hanke (Institut für Mathematik, Augsburg Universität)
*Introduction to spin geometry, I*

Spin geometry arises from the attempt to define a first-order differential operator whose square is equal to the Laplace operator. In Euclidean space this problem can be solved after moving from scalar valued functions to functions taking values in Clifford representations. This construction is carried over to Riemannian manifolds and leads to the definition of Dirac operators. We will explain the main steps of this construction, study analytic properties of Dirac operators, and outline the famous Atiyah-Singer index theorem. These lectures provide some background for the lectures of Christian Bär, who will study the implications in scalar curvature geometry.

**Lunedì 11 novembre 2024**

Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Gilles Carron (Nantes Université, Laboratoire de Mathématiques Jean Leray)
*The Kato condition for Ricci curvature and consequences, I*

I will first explain the Kato class on the Euclidean space and on Riemannian manifold. Then I will explain some consequences for complete Riemannian manifold whose Ricci curvature in the Kato class. The last lecture will be devoted to the study of limits of Riemannian manifold under such a Kato constraint on the Ricci curvature.

**Lunedì 11 novembre 2024**

Ore 14:00, Aula B, Dipartimento di Matematica, Sapienza Università di Roma

Seminario di Probabilità

Pablo Groisman (University of Buenos Aires)
*The Kuramoto Model in Random Geometric Graphs*

The Kuramoto model is a nonlinear system of ODEs that represents the behavior of coupled oscillators. The coupling is determined by a given graph and pushes the system towards synchronization. An important question is whether there is global synchronization (the system converges to a state in which all the phases coincide from almost every initial condition) or if the system supports other patterns. We will consider the Kuramoto model on random geometric graphs in the d dimensional torus and prove a scaling limit. The limiting object is given by the heat equation. On the one hand this shows that the nonlinearities of the system disappear under this scaling and on the other hand, provides evidence that stable equilibria of the Kuramoto model on these graphs are, as \( n \to \infty \), in correspondence with those of the heat equation, which are explicit and given by twisted states. In view of this, we conjecture the existence of twisted stable equilibria with high probability as \( n \to \infty \). We'll prove this conjecture in dimension \(d=1\).

Per informazioni, rivolgersi a: * silvestri@mat.uniroma1.it*

**Lunedì 11 novembre 2024**

Ore 14:30, Aula B, Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre

Seminario di Teoria dei Numeri

Sebastian Eterovic (University of Leeds)
*Multiplicative Relations Among Differences of Singular Moduli*

A singular modulus is the j-invariant of an elliptic curve with complex multiplication; as such the arithmetic and algebraic properties of these numbers are of great interest. In particular, there are important results concerning the behavior of differences of singular moduli, and also about the multiplicative dependencies that can arise among singular moduli. In joint work with Vahagn Aslanyan and Guy Fowler we show that for every positive integer n there are a finite set S and finitely many algebraic curves \(T_1,...,T_k\) with the following property: if \((x_1,...,x_n,y)\) is a tuple of pairwise distinct singular moduli so that the differences \( (x_1-y),...,(x_n-y)\) are multiplicatively dependent, then \( (x_1,..., x_n, y)\) belongs either to \(S\) or to one of the curves \(T_1,...,T_k\).

Per informazioni, rivolgersi a: * laura.capuano@uniroma3.it*

**Lunedì 11 novembre 2024**

Ore 16:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Andrea Malchiodi (Scuola Normale Superiore di Pisa)
*Prescribing scalar curvature in conformal geometry, I*

We will consider the classical problem, proposed by Kazdan and Warner in the 70’s, of prescribing the scalar curvature of a Riemannian manifold via conformal deformations of the metric. This amounts to solving an elliptic nonlinear PDE with critical exponent, presenting difficulties due to lack of compactness. There are in general obstructions, but still several contributions to the existence theory have been given using different techniques. These include direct methods of the calculus of variations, blow-up analysis, Liouville theorems, gluing constructions and topological or Morse-theoretical tools. We will give a general presentation of the subject, describing the principal contributions in the literature and arriving to more recent developments.

**Martedì 12 novembre 2024**

Ore 9:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Bernhard Hanke (Institut für Mathematik, Augsburg Universität)
*Introduction to Spin Geometry, II*

Spin geometry arises from the attempt to define a first-order differential operator whose square is equal to the Laplace operator. In Euclidean space this problem can be solved after moving from scalar valued functions to functions taking values in Clifford representations. This construction is carried over to Riemannian manifolds and leads to the definition of Dirac operators. We will explain the main steps of this construction, study analytic properties of Dirac operators, and outline the famous Atiyah-Singer index theorem. These lectures provide some background for the lectures of Christian Bär, who will study the implications in scalar curvature geometry.

**Martedì 12 novembre 2024**

Ore 11:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Gérard Besson (Institut Fourier, Université de Grenoble)
*Ricci flow and the 1/4-pinched differentiable sphere theorem (after Brendle and Schoen), II*

We shall show that a Riemanian manifold whose sectional curvature is strictly between 1 and 1/4 is diffeomorphic to the standard sphere. The proof uses the Ricci flow without surgery and a nice work on the algebra of the curvature tensor.

**Martedì 12 novembre 2024**

Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Andrea Malchiodi (Scuola Normale Superiore di Pisa)
*Prescribing scalar curvature in conformal geometry, II*

We will consider the classical problem, proposed by Kazdan and Warner in the 70’s, of prescribing the scalar curvature of a Riemannian manifold via conformal deformations of the metric. This amounts to solving an elliptic nonlinear PDE with critical exponent, presenting difficulties due to lack of compactness. There are in general obstructions, but still several contributions to the existence theory have been given using different techniques. These include direct methods of the calculus of variations, blow-up analysis, Liouville theorems, gluing constructions and topological or Morse-theoretical tools. We will give a general presentation of the subject, describing the principal contributions in the literature and arriving to more recent developments.

**Martedì 12 novembre 2024**

Ore 14:30, aula d'Antoni, Dipartimento di Matematica, Università di Roma Tor Vergata

seminario di Geometria

Filippo Fagioli (Università di Roma Tor Vergata)
*On Griffiths' conjecture about the positivity of Chern–Weil forms*

In the last years there has been a renewed interest around a long-standing conjecture by Griffiths characterizing which should be the positive characteristic forms for any Griffiths positive vector bundle. This conjecture can be interpreted as the differential geometric counterpart of the celebrated Fulton–Lazarsfeld theorem on positive polynomials for ample vector bundles. In this talk, we present some results that confirm the above conjecture for several characteristic forms. The positivity of these forms is due to a theorem which provides the version at the level of representatives of the universal push-forward formula for flag bundles valid in cohomology.

Per informazioni, rivolgersi a: * guidomaria.lido@gmail.com*

**Martedì 12 novembre 2024**

Ore 16:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Gilles Carron (Nantes Université, Laboratoire de Mathématiques Jean Leray)
*The Kato condition for Ricci curvature and consequences, II*

I will first explain the Kato class on the Euclidean space and on Riemannian manifold. Then I will explain some consequences for complete Riemannian manifold whose Ricci curvature in the Kato class. The last lecture will be devoted to the study of limits of Riemannian manifold under such a Kato constraint on the Ricci curvature.

**Mercoledì 13 novembre 2024**

Ore 9:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Anlaysis

Andrea Malchiodi (Scuola Normale Superiore di Pisa)
*Prescribing scalar curvature in conformal geometry, III*

We will consider the classical problem, proposed by Kazdan and Warner in the 70’s, of prescribing the scalar curvature of a Riemannian manifold via conformal deformations of the metric. This amounts to solving an elliptic nonlinear PDE with critical exponent, presenting difficulties due to lack of compactness. There are in general obstructions, but still several contributions to the existence theory have been given using different techniques. These include direct methods of the calculus of variations, blow-up analysis, Liouville theorems, gluing constructions and topological or Morse-theoretical tools. We will give a general presentation of the subject, describing the principal contributions in the literature and arriving to more recent developments.

**Mercoledì 13 novembre 2024**

Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Gilles Carron (Nantes Université, Laboratoire de Mathématiques Jean Leray)
*The Kato condition for Ricci curvature and consequences, III*

I will first explain the Kato class on the Euclidean space and on Riemannian manifold. Then I will explain some consequences for complete Riemannian manifold whose Ricci curvature in the Kato class. The last lecture will be devoted to the study of limits of Riemannian manifold under such a Kato constraint on the Ricci curvature.

**Mercoledì 13 novembre 2024**

Ore 16:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Christian Bär (Institut für Mathematik, Potsdam Universität)
*Scalar curvature and the Dirac operator, I*

Building on the introduction to spin geometry taught by Bernhard Hanke, I will use spinorial methods to investigate manifolds whose scalar curvature satisfies restrictions. Scalar curvature contains much less information than sectional or Ricci curvature which is why relatively sophisticated methods are needed to understand its implications. The results to be covered include Geroch’s conjecture on the impossibility of positive scalar curvature on tori as well as Llarull’s famous rigidity theorem.

**Mercoledì 13 novembre 2024**

Ore 17:00, M4, Dipartimmento di Matematica e Fisica, Università degli Studi Roma Tre

Seminario di Analisi Matematica

Berardino Sciunzi (Università della Calabria)
*Symmetry and monotonicity results in the context of semilinear problems involving singular nonlinearities.*

I shall consider elliptic problems addressing the study of the geometric properties of the solutions. This issue is in general related to the classification of the solutions or to Liouville type theorems. During the talk, in particular, I will discuss a recent classification result for the solutions to semilinear elliptic problems in half spaces involving singular nonlinerarities (in collaboration with L. Montoro and L. Muglia).

Per informazioni, rivolgersi a: * luca.battaglia@uniroma3.it*

**Giovedì 14 novembre 2024**

Ore 9:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Gilles Carron (Nantes Université, Laboratoire de Mathématiques Jean Leray)
*The Kato condition for Ricci curvature and consequences, IV*

I will first explain the Kato class on the Euclidean space and on Riemannian manifold. Then I will explain some consequences for complete Riemannian manifold whose Ricci curvature in the Kato class. The last lecture will be devoted to the study of limits of Riemannian manifold under such a Kato constraint on the Ricci curvature.

**Giovedì 14 novembre 2024**

Ore 11:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Andrea Malchiodi (Scuola Normale Superiore di Pisa)
*Prescribing scalar curvature in conformal geometry, IV*

We will consider the classical problem, proposed by Kazdan and Warner in the 70’s, of prescribing the scalar curvature of a Riemannian manifold via conformal deformations of the metric. This amounts to solving an elliptic nonlinear PDE with critical exponent, presenting difficulties due to lack of compactness. There are in general obstructions, but still several contributions to the existence theory have been given using different techniques. These include direct methods of the calculus of variations, blow-up analysis, Liouville theorems, gluing constructions and topological or Morse-theoretical tools. We will give a general presentation of the subject, describing the principal contributions in the literature and arriving to more recent developments.

**Giovedì 14 novembre 2024**

Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Christian Bär (Institut für Mathematik, Potsdam Universität)
*Scalar curvature and the Dirac operator, II*

Building on the introduction to spin geometry taught by Bernhard Hanke, I will use spinorial methods to investigate manifolds whose scalar curvature satisfies restrictions. Scalar curvature contains much less information than sectional or Ricci curvature which is why relatively sophisticated methods are needed to understand its implications. The results to be covered include Geroch’s conjecture on the impossibility of positive scalar curvature on tori as well as Llarull’s famous rigidity theorem.

**Giovedì 14 novembre 2024**

Ore 14:15, Aula M1, Dipartimento di Matematica e Fisica, Università Roma Tre

Seminario di Geometria

Angelo Felice Lopez (Roma tre)
*Ulrich subvarieties and a lower bound on the Ulrich complexity of complete intersections*

Let \(X \subset \mathbb{P}^N\) be a smooth irreducible n-dimensional variety. A well-known conjecture predicts that \(X\) always carries an Ulrich vector bundle, that is a bundle \(\mathcal{E}\) such that \(H^i(\mathcal{E}(−p)) = 0\) for \(i \geq 0\) and \(1 \leq p \leq n\). In the talk we will report on three recent results in collaboration with D. Raychaudhury. The first one is that any given \(X\) carries an Ulrich bundle if and only if it contains a subvariety satisfying certain conditions. The second one is an application of this result to low rank Ulrich bundles on complete intersections of dimension \(n ≥ 5\), or on general complete intersections of dimension \(n = 4\). The third one is an application to rank 2 Ulrich bundles on general hypersurfaces of dimension \(n\) with \(2 \leq n \leq 3\).

Per informazioni, rivolgersi a: * amos.turchet@uniroma3.it*

**Giovedì 14 novembre 2024**

Ore 16:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Gérard Besson (Institut Fourier, Université de Grenoble)
*Ricci flow and the 1/4-pinched differentiable sphere theorem (after Brendle and Schoen), III*

We shall show that a Riemanian manifold whose sectional curvature is strictly between 1 and 1/4 is diffeomorphic to the standard sphere. The proof uses the Ricci flow without surgery and a nice work on the algebra of the curvature tensor.

**Venerdì 15 novembre 2024**

Ore 9:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Christian Bär (Institut für Mathematik, Potsdam Universität)
*Scalar curvature and the Dirac operator, III*

Building on the introduction to spin geometry taught by Bernhard Hanke, I will use spinorial methods to investigate manifolds whose scalar curvature satisfies restrictions. Scalar curvature contains much less information than sectional or Ricci curvature which is why relatively sophisticated methods are needed to understand its implications. The results to be covered include Geroch’s conjecture on the impossibility of positive scalar curvature on tori as well as Llarull’s famous rigidity theorem.

**Venerdì 15 novembre 2024**

Ore 11:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Christian Bär (Instituts für Mathematik, Potsdam Universität)
*Scalar curvature and the Dirac operator, IV*

Building on the introduction to spin geometry taught by Bernhard Hanke, I will use spinorial methods to investigate manifolds whose scalar curvature satisfies restrictions. Scalar curvature contains much less information than sectional or Ricci curvature which is why relatively sophisticated methods are needed to understand its implications. The results to be covered include Geroch’s conjecture on the impossibility of positive scalar curvature on tori as well as Llarull’s famous rigidity theorem.

**Venerdì 15 novembre 2024**

Ore 11:30, Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata

DocTorV Seminar

Alessio Oliviero (Sapienza Università di Roma)
*A modified-Patankar semi-Lagrangian scheme for the control of production-destruction systems*

We present a numerical scheme for the solution of optimal control problems associated with production-destruction systems (PDS). We start by introducing these differential systems and their properties, such as positivity and conservativity. Then, we define a class of controlled PDS in a way that preserves the properties mentioned earlier and we formulate the related general finite horizon optimal control problem. Following the Dynamic Programming approach, the problem is addressed through the solution of an evolutive partial differential equation in the state-space of the PDS, called the Hamilton--Jacobi--Bellman (HJB) equation. We devise a parallel-in-space semi-Lagrangian (SL) scheme for the solution of the HJB equation based on modified-Patankar methods, which preserve positivity and conservativity also at the numerical level, thus obtaining the optimal feedback controls and trajectories. Finally, we show with two examples, enzyme-catalyzed biochemical reactions and infectious diseases, that the proposed scheme provides better solutions compared with a classical SL scheme.

Per informazioni, rivolgersi a: * doctorv.uniroma2@gmail.com*

**Venerdì 15 novembre 2024**

Ore 12:00, Aula INdAM, INdAM, Dipartimento di Matematica Sapienza Università di Roma

Number Theory Seminar

Bianca Gariboldi (Università degli Studi di Bergamo)
*Irregularities of distribution and discrepancy theory: from Weyl up to nowadays*

Starting from the Weyl criterion for uniformly distrubuted sets of points, we introduce the discrepancy theory, focusing on some classical results by Roth, Davenport, Cassels and Montgomery. We conclude the seminar with some new results on manifolds strictly related to the classical ones.

Per informazioni, rivolgersi a: * cherubini@altamatematica.it*

**Venerdì 15 novembre 2024**

Ore 14:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

School on Curvature and Geometric Analysis

Gérard Besson (Institut Fourier, Université de Grenoble)
*Ricci flow and the 1/4-pinched differentiable sphere theorem (after Brendle and Schoen), IV*

We shall show that a Riemanian manifold whose sectional curvature is strictly between 1 and 1/4 is diffeomorphic to the standard sphere. The proof uses the Ricci flow without surgery and a nice work on the algebra of the curvature tensor.

**Venerdì 15 novembre 2024**

Ore 14:30, Aula Dal Passo, Dipartimento di Matematica, U Roma Tor Vergata

Algebra and Representation Theory Seminar

Alfonso Tortorella (U Salerno)
*Deformations of Symplectic Foliations via Dirac Geometry and \(L_\infty\) Algebras*

In this talk, based on joint work with Stephane Geudens and Marco Zambon, we develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leafwise symplectic form. The main result is that each symplectic foliation is attached with a cubic L∞ algebra controlling its deformation problem. Indeed, we establish a one-to-one correspondence between the small deformations of a given symplectic foliation and the Maurer-Cartan elements of the associated L∞ algebra. Further, we prove that, under this one-to-one correspondence, the equivalence by isotopies of symplectic foliations agrees with the gauge equivalence of Maurer-Cartan elements. Finally, we show that the infinitesimal deformations of symplectic foliations can be obstructed.

**Venerdì 15 novembre 2024**

Ore 16:00, Aula Dal Passo, Dipartimento di Matematica, U Roma Tor Vergata

Joint Topology & Algebra and Representation Theory Seminar (T-ARTS)

Marco Moraschini (U Bologna)
*An introduction to amenability in bounded cohomology*

Bounded cohomology of groups is a variant of ordinary group cohomology introduced by Johnson in the 70s in the context of Banach algebras and then intensively studied by Gromov in his seminal paper "Volume and bounded cohomology" in relation to geometry and topology of manifolds. Since the 80s bounded cohomology has then grown up as an independent and active research field. On the other hand, it is notoriously hard to compute bounded cohomology. For this reason it is natural to first investigate groups with trivial bounded cohomology groups. During this talk we survey recent advances around "acyclicity" in bounded cohomology and we will introduce a new algebraic criterion for the vanishing of bounded cohomology. This is part of a joint work with Caterina Campagnolo, Francesco Fournier-Facio, and Yash Lodha.

**Venerdì 15 novembre 2024**

Ore 16:00, Sala di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

Seminari PLS per docenti

Alessandro Foschi (Convitto Nazionale), Marta Menghini (Sapienza Università di Roma) & Graziano Surace (Sapienza Università di Roma)
*Punti di vista sulle coniche*

Le comunicazioni relative a seminari da includere in questo notiziario devono pervenire esclusivamente
mediante apposita form da compilare online, entro le ore 24 del giovedì precedente la settimana
interessata. Le comunicazioni pervenute in ritardo saranno ignorate.
Per informazioni, rivolgersi all'indirizzo di posta elettronica
*seminari@mat.uniroma1.it*.

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