## Notiziario Scientifico

### Notiziario dei seminari di carattere matematico a cura del Dipartimento di Matematica G. CastelnuovoSapienza Università di Roma Settimana dal 9 al 15 aprile 2018

Lunedì 9 aprile 2018
Ore 14:15, aula di Consiglio
seminario di Analisi Matematica
Carlo Mantegazza (Università di Napoli Federico II)
Evolution by curvature of networks in the plane
We will present the state-of-the-art of the problem of the motion by curvature of a network of curves in the plane, discussing existence, uniqueness, singularity formation and asymptotic behavior of the flow.

Martedì 10 aprile 2018
Ore 10:00, aula 2E, Dipartimento di Scienze di Base e Applicate per Ingegneria (SBAI), via A. Scarpa 14
corso di dottorato
Giovanni Cerulli Irelli (Sapienza Università di Roma)
Quiver representations and cluster algebras, I
We will give an introduction to the representation theory of quivers and finite dimensional algebras. We will then apply the theory to study the geometry of quiver Grassmannians. Time permitting we will see applications to the Fomin-Zelevinsky theory of cluster algebras. The course is intended for non--expert with standard background in mathematics.

Martedì 10 aprile 2018
Ore 14:30, aula Dal Passo, dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1
seminario di Equazioni Differenziali
Massimo Grossi (Sapienza Università di Roma)
Radial nodal solution for Moser-Trudinger problems
We study the asymptotic behavior of least-energy nodal solutions for suitable Moser-Trudinger problems. We will show that appear different phenomena with respect to other nonlinearities (for example power or sinh-type nonlinearites).

Martedì 10 aprile 2018
Ore 14:30, aula 311, Università di Roma Tre, largo san Leonardo Murialdo 1
seminario di Probabilità
Hubert Lacoin (IMPA, Rio de Janeiro)
Wetting and Layering for Solid-and-Solid
Solid-on-Solid (SOS) was introduced in the early 50s as a simplified model for lattice interfaces. It is believed to display the same low temperature behavior as three-dimensiona systems with phase coexistence while being considerably easier to analyze. The objective of this talk is to present the result we recently obtained for SOS interacting with a solid substrate, which is the problem associated with the following energy functional
$V(\phi)= ta \sum_{x\sim y} |\phi(x)-\phi(y)| -\sum_{x} \left(h{f 1}_{{\phi(x)=0}}-\infty{f1}_{{\phi(x)<0}} ight)$ for $$(\phi(x))_{x\in\mathbf{Z}^2}$$ (the graph of $$\phi$$ representing the interface). We prove that for $$ta$$ sufficiently large, there exists a decreasing sequence $$(h^*_n(ta))_{n\ge 0}$$, satisfying $$\lim_{n o\infty}h^*_n(ta)=h_w(ta)$$, and such that:
(A) The free energy associated with the system is infinitely differentiable on $$\mathbf{R} \setminus \left( {h^*_n}_{n\ge 1}\cup h_w( ta ) ight)$$, and not differentiable on $${h^*_n}_{n\ge 1}$$.
(B) For each $$n\ge0$$ within the interval $$(h^*_{n+1},h^*_n)$$ (with the convention $$h^*_0=\infty$$), there exists a unique translation invariant Gibbs state which is localized around height $$n$$, while at a point of non-differentiability, at least two ergodic Gibbs states coexist. The respective typical heights of these two Gibbs states are $$n-1$$ and $$n$$. The value $$h^*_n$$ corresponds thus to a first order layering transition from level $$n$$ to level $$n-1$$.

Mercoledì 11 aprile 2018
Ore 14:00, aula di Consiglio
seminario di Algebra e Geometria
Thomas Schick (Georg-August-Universität Göttingen)
Minimal hypersurfaces and positive scalar curvature
There is a long history to find relations between the topology of a smooth manifold and its (Riemannian) geometry. The first such is the Gauss-Bonnet theorem which says that the Euler characterestic of a compact 2-dimensional surface without boundary is (upto a positive constant) the scalar curvature of that manifold. Particular conclusion: if the Euler characterestic is not positive (i.e. if the surface is not a sphere or a real projective plane) then there is no metric such that the scalar curvature is everywhere positive. The use of the Dirac operator allows to obtain similar obstructions to the existence of positive scalar curvature in higher dimension; but only for spin manifolds (as otherwise this operator doesn't exist). There is one further approach -invented by Schoen and Yau, which does not rely on the spin condition, but rather uses minimal hypersurfaces. We will present this approach and its main implications. There are two crucial problems with this approach:
* in its initial incarnation, it requires regularity results on minimial hypersurfaces which are available only in dimension less than 8,
* it needs a large integral first homology.
We will report on current work which aims to overcome part of these problems, due to Schoen-Yau for the first problem, and developped in joint work with Simone Cecchini for some aspects of the second problem. Specifically, we will introduce and discuss the case of 'enlargeable manfolds' (as introduced by Gromov and Lawson).

Mercoledì 11 aprile 2018
Ore 14:30, aula F, Università di Roma Tre, largo san Leonardo Murialdo 1
colloquium di Matematica
Massimiliano Sala (Università di Trento)
Optimal non-linear Boolean functions as multivariable polynomials: the even case
To guarantee security w.r.t. known attacks (especially differential cryptanalysis) it is necessary to design a block cypher very carefully. One type of component which is often used is the so-called S-Box (Substitution Box). It turns out that the ideal situation would be to have an APN (Almost Perfect Nonlinear) permutation of dimension d even, possible 4, 8 or another power of 2. The experimental results show that: there is no APN permutation for d=4, there is one APN permutation for d=6 (but there could be more) and none has been found so far for d=8, being any higher dimension intractable with a computer nowadays. In recent papers, we have investigated the situation by considering APN permutations as multivariable polynomials (vectorial Boolean functions). In other words, they are polynomial maps from a binary space of dimension d to itself. We have proved several theoretical results (that partially explain the computational findings): no component can have degree less than three (for any d even), no APN permutations exist for d=4, no pure cubic APN permutations exist for d=6. This is joint work with M. Calderini, I. Villa and M. Zaninelli.

Mercoledì 11 aprile 2018
Ore 17:00, aula di Consiglio
seminario di Fisica Matematica
Vojkan Jaksic (McGill University, Montreal)
Time and Entropy
This talk concerns mathematical theory of the so-called Fluctuation Relation (FR) and Fluctuation Theorem (FT) in context of dynamical systems relevant to physics. The FR refers to a certain universal identity linked to statistics of entropy production generated by a reversal operation and FT to the related mathematical large deviations result. The discovery of FR goes back to numerical experiments and Evans, Cohen and Morris (1993) and theoretical works of Evans and Searles (1994), Gallavotti and Cohen (1995). These discoveries generated an enormous body of numerical, theoretical and experimental works which have fundamentally altered our understanding of non-equilibrium physics, with applications extending to chemistry and biology. In this talk I will introduce modern theory of FR and FT on an example and comment on a current research program on this topic.

Giovedì 12 aprile 2018
Ore 14:30, aula 311, Università di Roma Tre, largo san Leonardo Murialdo 1
seminario di Fisica Matematica
A. Sorrentino (Università di Roma Tor Vergata)
On Birkhoff conjecture for convex billiards
A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. This simple model has been first proposed by G.D. Birkhoff as a mathematical playground where 'it the formal side, usually so formidable in dynamics, almost completely disappears and only the interesting qualitative questions need to be considered'. Since then billiards have captured much attention in many different contexts, becoming a very popular subject of investigation. Despite their apparently simple (local) dynamics, their qualitative dynamical properties are extremely non-local. This global influence on the dynamics translates into several intriguing rigidity phenomena, which are at the basis of several unanswered questions and conjectures. In this talk I shall focus on some of these questions. In particular, I shall describe some recent results related to the classification of integrable billiards (also known as Birkhoff conjecture).

Giovedì 12 aprile 2018
Ore 15:00, aula 7, Università UNINT, via Cristoforo Colombo 200
seminario
Liviu Ornea (Università di Bucarest)
Potenziali positivi su varietà localmente conformemente kaehleriane compatte
Verrà mostrato che, se una varietà LCK compatta ammette un potenziale sul rivestimento universale, sul quale il gruppo delle trasformazioni del rivestimento agisce in modo naturale (automorfo), allora la varietà ammette anche un potenziale positivo, sempre automorfo. Questo è un risultato ottenuto assieme a Misha Verbitsky.

Venerdì 13 aprile 2018
Ore 14:30, aula D'Antoni, dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1
seminario
Velleda Baldoni (Università di Roma Tor Vergata)
Multiplicities & Kronecker coefficients
Multiplicities of representations appear naturally in different contexts and as such their description could use different languages. The computation of Kronecker coefficients is in particular a very interesting problem which has many applications. I will describe an approach based on methods from symplectic geometry and residue calculus (joint work with M. Vergne and M. Walter). I will state the general formula for computing Kronecker coefficients and then give many examples computed using an algorithm that implements the formula. The algorithm does not only compute individual Kronecker coefficients, but also symbolic formulas that are valid on an entire polyhedral chamber. As a byproduct, it is possible to compute several Hilbert series.

Venerdì 13 aprile 2018
Ore 16:00, aula Picone
seminario per insegnanti (Piano Lauree Scientifiche)
Fabio Spizzichino (Sapienza Università di Roma)
Il calcolo combinatorio nel Liceo matematico

Venerdì 13 aprile 2018
Ore 16:00, aula D'Antoni, dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1
seminario
René Schoof (Università di Roma Tor Vergata)
Il teorema di Lagrange per schemi in gruppi piatti e finiti
Il teorema di Lagrange dice che in un gruppo di cardinalità n la potenza n-esima di ogni elemento è uguale all'elemento neutro. Una congettura classica afferma che un risultato simile vale per schemi in gruppi piatti e finiti. Spiegherò la dimostrazione di un caso speciale della congettura.

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