## Weekly Bulletin (it)

**Notiziario dei seminari di carattere matematico**

a cura del Dipartimento di Matematica

*G. Castelnuovo*, Sapienza Università di Roma

Settimana dal 25-03-2019 al 31-03-2019

**Lunedì 25 marzo 2019**

Ore 14:15, Aula di Consiglio, Dipartimento di Matematica

Seminario di Analisi Matematica

Elisabetta Rocca (Università di Pavia)
*Long-time dynamics and optimal control for some tumor growth models*

We consider the problem of the long time dynamics and optimal control for two diffuse interface models for tumor growth. The models describe the growth of a tumor surrounded by host tissues in the presence of a nutrient and consist of a Cahn-Hilliard-type equation for the tumor phase coupled with a reaction-diffusion equation for the nutrient concentration under different choices of proliferation and death of cells terms. In one case we prove that, under physically motivated assumptions on parameters and data, the corresponding initial-boundary value problem generates a dissipative dynamical system that admits the global attractor in a proper phase space. In the second case we include a possible medication that serves to eliminate tumor cells is in terms of drugs and we consider the problem of ``long-time treatment'' under a suitable given mass source and prove the convergence of any global solution to a single equilibrium as time goes to infinity. Second, we consider the ``finite-time treatment'' that corresponds to an optimal control problem. Here we allow the objective cost functional to depend on a free time variable, which represents the unknown treatment time to be optimized. We prove the existence of an optimal control and obtain first order necessary optimality conditions for both the drug concentration and the treatment time. This is a joint project with Alain Miranville and Giulio Schimperna and with Cecilia Cavaterra and Hao Wu.

**Lunedì 25 marzo 2019**

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

Algebra and Representation Theory Seminar

Claudio Procesi (Sapienza Università di Roma e Accademia dei Lincei)
*Perpetuants a lost treasure*

Perpetuant is one of the several concepts invented (in 1882) by J. J. Sylvester in his investigations of covariants for binary forms. It appears in one of the first issues of the American Journal of Mathematics which he had founded a few years before. It is a name which will hardly appear in a mathematical paper of the last 70 years, due to the complex history of invariant theory which was at some time declared dead only to resurrect several decades later. I learned of this word from Gian-Carlo Rota who pronounced it with an enigmatic smile. In this talk I want to explain the concept, a Theorem of Stroh, and some new explicit description.

**Lunedì 25 marzo 2019**

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

Seminario

Claudio Landim (IMPA, Rio de Janeiro, Brazil)
*Homogenization for diffusion processes*

We present general tools to prove the central limit theorem for addive functionals of Markov processes and discuss in some detail the application to diffusions in periodic or random enevironment.

**Martedì 26 marzo 2019**

Ore 14:15, Aula B, Dipartimento di Matematica, Sapienza Università di Roma

Seminario di Modellistica Differenziale Numerica

Luigi Preziosi (Dipartimento di Scienze Matematiche, Politecnico di Torino)
*Degenerate Parabolic Models for Sand Slides*

Four phenomena contribute to wind-induced sand movement and eventually to the formation and evolution of dunes: erosion from the sand bed, transport by the wind, sedimentation due to gravity, and sand grain slides occurring when the slope of the accumulated sand exceeds a critical angle of repose. In particular, erosion, sedimentation, and the formation of such small avalanches determine the evolution of the free-boundary over which wind blows and transports the sand. The need to couple the multiphase turbulent fluid-dynamics with the dynamics occurring at the sand surface requires to deduce mathematical models for such phenomena that are able to describe the evolution of the surface in an accurate way, but that is at the same time computationally fast. Starting from this need, the aim of this talk is to propose a new mathematical model based on using classical continuum mechanics tools under the assumptions that the thickness of the creep layer is small and that the grains in it move in the direction of the steepest descent with a speed that is determined by several constitutive closures (Coulomb-like or pseudo-plastic fluids). The mathematical models deduced as degenerate parabolic equations for the height of the sand pile. In spite of their simplicity, all the models reply many well known behaviours characterizing the evolution of sand piles, such as the non-uniqueness of static configurations in subcritical conditions and the link between critical stationary configurations and the eikonal equation, and behave well in all tested set-ups. The only slight difference among them lies in the temporal evolution of the interface.

**Martedì 26 marzo 2019**

Ore 14:30, Aula 311, Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo S.L. Murialdo, 1

Seminario di Fisica Matematica

Sebastien Ott (Università di Ginevra)
*Entropic repulsion in low temperature 2D Potts model and critical pre-wetting in 2D Ising model.*

Consider a Potts (or Ising) model in a square box of side N with boundary conditions 2 (-1 for Ising) on the lower side and 1 on the three others at low temperature. In this talk, I will present two problems about the interface generated by the boundary conditions. 1) convergence (under diffusive scaling) of the interface in the homogeneous Potts model to a Brownian excursion, 2) convergence (under suitable scaling) of the (bulk) Ising interface to a Ferrari-Spohn iffusion when a field of magnitude a/N is applied. I will start by introducing the two problems and then discuss the general strategy used in both proofs.

**Martedì 26 marzo 2019**

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

Seminario di Equazioni Differenziali

Alessandro Fortunati (University of Bristol (UK))
*Arnold’s diffusion and variational methods: a constructive proof*

The talk is focused on the well known phenomenon of topological instability that can occour in a class of quasi-integrable systems, as pointed out by V.I. Arnold in 1964. Right after the publication of the comprehensive work by Chierchia and Gallavotti in 1994 (based on geometric methods), a pioneering paper by U. Bessi proposed a variational formulation for the very same model studied in the original paper by Arnold. However, the constructivity of this approach is not as manifest as in techniques of geometric nature such as the Chierchia-Gallavotti or the Windows methods. The work presented is based on a revisitation of the tools proposed by Bessi then developed by Berti, Biasco and Bolle, in order to obtain a rigorous and constructive proof in the case of the Arnold example. New tools are introduced in order to formulate the quantitative estimates necessary for a machine implementation apt to construct the diffusing trajectories.

**Martedì 26 marzo 2019**

Ore 14:30, Aula Picone, Dipartimento di Matematica, Sapienza Università di Roma

Jean-Eric Pin (CNRS and IRIF, University of Paris 7)
*A Mahler's theorem for word functions*

Let p be a prime number. We prove a noncommutative generalization of Mahler's theorem on interpolation series, a celebrated result of p-adic analysis. Mahler's original result states that a function from N to Z is uniformly continuous for the p-adic metric if and only if it can be uniformly approximated by polynomial functions. We prove an analogous result for functions from a free monoid to a free group, where the p-adic metric is replaced by the pro-p metric. This result relies on a noncommutative extension of Newton's forward difference formula, which states that for each function from N to Z, there is a unique sequence of integers \(\delta_k f\) such that, for all n in N, \(f(n)= \sum_{k=0} ^\infty {n\choose k} \delta_k f.\) The value of these coefficients \(\delta_k f\) is given by the formula \(\delta_k f = (\Delta ^k f)(0)\), where \(\Delta ^k\) is the k-th iteration of the difference operator \(\Delta\), defined by \((\Delta f)(n)=f(n+ 1) - f(n).\) The noncommutative extension involves a noncommutative difference operator which applies to any function from a free monoid to a group. This lecture is for the most part of combinatorial nature. In particular, it does not require any prior knowledge in p-adic analysis. This is a joint work with Christophe Reutenauer (UQAM, Montréal).

**Martedì 26 marzo 2019**

Ore 15:00, Aula 1B1, Dipartimento di Scienze di Base ed Applicate per L'Ingegneria, Sapienza Università di Roma, Pal. RM002

Incontri di Algebra e Geometria allo SBAI

Stefano Capparelli (Sapienza Università di Roma)
*Rogers-Ramanujan identities, Lie algebras and Vertex Operator Algebras: a short historical introduction.*

There are close connections between combinatorial objects such as integer partitions, as in the Rogers-Ramanujan identities, and classical algebraic objects such as Lie algebras, and a modern evolution of these algebras which is Vertex Operator Algebra (VOA) theory. This circle of ideas also involves finite group theory, modular forms, and various physics concepts. In this talk I want to sketch an elementary introduction to these ideas via a short historical overview.

**Martedì 26 marzo 2019**

Ore 15:15, Aula B, Dipartimento di Matematica, Sapienza Università di Roma

Seminario di Modellistica Differenziale Numerica

Athena Picarelli (Dipartimento di Scienze Economiche, Università di Verona)
*Some high order filtered schemes for parabolic Hamilton-Jacobi-Bellman equations*

We consider high order numerical schemes for second order Hamilton-Jacobi-Bellman (HJB) equations. For high order approximation schemes (where “high” stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. We present a class of ``filtered'' schemes: a suitable local modification of the high order scheme is introduced by ``filtering'' it with a monotone one. The resulting scheme can be proven to converge and it still shows an overall high order behavior for smooth enough solutions. We give theoretical proofs of these claims and validate the results with numerical tests.

**Mercoledì 27 marzo 2019**

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

Seminario di Algebra e Geometria

François Bergeron (Université du Québec à Montréal)
*Rectangular Multivariate Modules of Harmonic Polynomials*

Much of the work of the last 25 years on (modified) Macdonald symmetric polynomials, and operators for which they are joint eigenfunctions, has recently be nicely synthesized via an operator realization of the elliptic Hall algebra (introduced by Burban-Vasserot-Schifmann). This has opened the way to generalizations of the interaction between interesting questions in many areas including: Algebraic Combinatorics (rectangular Catalan combinatorics), Symmetric Functions (compositional shuffle conjecture/theorem, nabla operator), Knot Theory (Khovanov-Rozansky homology of (m,n)-torus knots), and Theoretical Physics (boson-fermion supersymmetry). We will present a new link between all these subjects and representation theory, by describing modules of bivariate diagonal polynomials that generalize to the (m,n)-rectangular context the Garsia-Haiman module of diagonal harmonic polynomials (in 2 sets of n variables). This makes it natural to extend most of the previous work to the multivariate context (k sets of n variables). If time allows, we will finally explain how such an extension unifies in a surprising manner many questions of the domain.

**Mercoledì 27 marzo 2019**

Ore 14:30, Aula M3, Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo S.L. Murialdo, 1, nuovo edificio

Final exam Ph.D in Mathematics

Edmond Koudjinan
*Quantitative KAM normal forms and sharp measure estimates*

**Giovedì 28 marzo 2019**

Ore 11:00, Aula 211, Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo S.L. Murialdo, 1

Mini-corso

Alex Küronya (Goethe-Universität Frankfurt am Main)
*Syzygies of Algebraic Varieties*

The topic of the course is the study of embeddings of varieties into projective spaces. Given a very ample line bundle L on a projective variety \( X \), the Kodaira map associated to L gives rise to an embedding of \( X \) into some projective space, thus realizing \( X \) as a common zero set of equations in a polynomial ring. Syzygies of the pair \( (X;L) \) are algebraic invariants of this embedding which describe the higher order relations among the arising system of equations. Following a quick introduction to the basics of the subject, in particular reviewing the necessary material from commutative algebra, we will start focusing on pairs \( (X;L) \) where the associated syzygy modules are as simple as possible. This is made more precise by the so-called 'property (Np)', which was first considered by Green and Lazarsfeld, and which means that the first \( p + 1\) syzygies are linear. We will study how verifying property (Np) can be reduced to checking the vanishing of higher cohomology of vector and line bundles, and look at the case of abelian varieties where one obtains a surprisingly uniform answer. In the last part of the course we consider the case of surfaces in more detail and see how one can characterize property (Np) in terms of forbidden subvari- eties. The necessary prerequisites are the basics of graded rings and a working knowledge of positivity, cohomology, and vanishing theorems for projective varieties.

**Giovedì 28 marzo 2019**

Ore 14:30, Aula 211, Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo S. L. Murialdo, 1

Seminario di Geometria

Marco D'Ambra (Università di Roma Tor Vergata)
*Partially ample line bundles and base loci*

By the well-known Cartan-Grothendieck-Serre’s result, the ampleness of a line bundle on a projective variety is characterized in terms of the vanishing of the higher cohomology groups. By weakening this condition, we obtain a notion of 'partial ampleness', that intuitively measures how much a line bundle is far from being ample and that shares many important properties with the usual one. In the first part of this talk, I will give an introduction to the theory of partially ample line bundles and discuss some of their main features. In the second part of the talk, I will present some original results that help us to interpret geometrically the partial ampleness of a line bundle.

**Giovedì 28 marzo 2019**

Ore 14:30, Aula di Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

Seminario P(n)/N(p)

Wilfrid Gangbo (UCLA)
*On intrinsic differentiability in the Wasserstein space \( {\mathcal P}_2(\mathbb{R}^d) \)*

We elucidate the connection between different notions of differentiability in \( {\mathcal P}_2(\mathbb{R}^d) \) : some have been introduced intrinsically by Ambrosio-Gigli-Savarè, the other notion due to Lions, is extrinsic and arises from the identification of \( {\mathcal P}_2(\mathbb{R}^d) \) with the Hilbert space of square-integrable random variables. We mention potential applications such as uniqueness of viscosity solutions for Hamilton-Jacobi equations in \( {\mathcal P}_2(\mathbb{R}^d) \), the latter not known to satisfy the Radon-Nikodym property. (This talk is based on a joint work with A. Tudorascu).

**Giovedì 28 marzo 2019**

Ore 15:00, Sala di Fisica, Palazzo Corsini, Accademia Nazionale dei Lincei

Tavola rotonda

M. Cattaneo (1 moderatore), G. Corbellini (2), A. Sgamellotti (3), G. Di Battista (4), F. Scoppola (5), A. Tesei (3), L. Toniolo (6), R. Natalini (7) (1 Direttore "Le Scienze", 2 Dipartimento Scienze Umane, CNR, 3 Accademia Nazionale dei Lincei, 4 Università di Roma Tre e DTC Lazio, 5 MIBAC, 6 Politecnico di Milano, 7 Istituto per la Applicazioni del Calcolo "M. Picone", CNR)
*Pietre Matematiche: presente e futuro della matematica per la salvaguardia dei Beni Culturali*

**Venerdì 29 marzo 2019**

Ore 11:00, Aula 211, Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo S.L. Murialdo, 1

Mini-corso

Alex Küronya (Goethe-Universität Frankfurt am Main)
*Syzygies of Algebraic Varieties*

The topic of the course is the study of embeddings of varieties into projective spaces. Given a very ample line bundle L on a projective variety \( X \), the Kodaira map associated to L gives rise to an embedding of \( X \) into some projective space, thus realizing \( X \) as a common zero set of equations in a polynomial ring. Syzygies of the pair \( (X;L) \) are algebraic invariants of this embedding which describe the higher order relations among the arising system of equations. Following a quick introduction to the basics of the subject, in particular reviewing the necessary material from commutative algebra, we will start focusing on pairs \( (X;L) \) where the associated syzygy modules are as simple as possible. This is made more precise by the so-called 'property (Np)', which was first considered by Green and Lazarsfeld, and which means that the first \( p + 1\) syzygies are linear. We will study how verifying property (Np) can be reduced to checking the vanishing of higher cohomology of vector and line bundles, and look at the case of abelian varieties where one obtains a surprisingly uniform answer. In the last part of the course we consider the case of surfaces in more detail and see how one can characterize property (Np) in terms of forbidden subvari- eties. The necessary prerequisites are the basics of graded rings and a working knowledge of positivity, cohomology, and vanishing theorems for projective varieties.

**Venerdì 29 marzo 2019**

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

Seminario

Claudio Landim (IMPA, Rio de Janeiro, Brazil)
*Homogenization for diffusion processes, part 2*

Central limit theorems for martigales. Hoogenization for random walks in random environment.

**Venerdì 29 marzo 2019**

Ore 16:00, Aula Picone, Dipartimento di Matematica, Sapienza Università di Roma

Seminario per Insegnanti (Piano Lauree Scientifiche)

Fabio Spizzichino*, Giuliana Massotti**, Marina Mayer*** , Donatella Ricalzone***, Lucilla Galterio §, Erminia Izzo §§, Francesca Ruzzi §§ (* Sapienza Università di Roma, ** Liceo Avogadro (Roma), *** Liceo Pascal (Roma), § Liceo Grassi (Latina), §§ Liceo Lucrezio Caro (Roma))
*\( \pi \)*

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