Prof. Michael Herty (RWTH Aachen University)
Abstract: We present recent work on interacting multi-agent optimization problems using meanfield limits. In particular, we are interested in the case of game aspects of possibly infinitely many players.A strategy for an efficient computation of the optimum is presented as well as meanfield limits. Theoretical and numerical examples are given.
Gli interessati sono cordialmente invitati ad intervenire.
Luigi Pepe
]]>Gli interessati sono cordialmente invitati ad intervenire.
Luigi Pepe
]]>Title: Logarithmic bundles of multi-degree arrangements in $P^n$
Abstract: Let D = {D_{1},...,D_{l}} be a multi-degree arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let O^{1}_P^n(logD) be the logarithmic bundle attached to it. We show that O^{1}_P^n(logD) admits a resolution of length 1 which explicitly depends on the degrees and on the equations of D_{1},...,D_{l}. Then we prove a Torelli type theorem when D is made of a "sufficiently" large number of hypersurfaces: indeed, we recover the components of D as unstable smooth hypersurfaces of D_{1},...,D_{l}. Moreover we analyze the case of one quadric and a pair of quadrics, which yield examples of non-Torelli arrangements. In particular, through a duality argument, we prove that two pairs of quadrics have isomorphic logarithmic bundles if and only if they have the same tangent
hyperplanes. Finally we give a description of the conic case and of some line-conic cases.
The Gauss--Green formulas are of significant relevance in many areas of analysis and mathematical physics. This motivated several investigations to extend such formulas to more general classes of integration domains and weakly differentiable vector fields, and thus led to the definition of the divergence-measure fields. These are $L^{p}$-summable vector fields on $\mathbb{R}^{n}$ whose distributional divergence is a Radon measure, and so they form a new family of function spaces, which in a sense generalize the $BV$ fields. The divergence-measure fields were introduced at first by Anzellotti in 1983 for the case $p = \infty$, and then they have been rediscovered in the early 2000s by many authors interested in various applications.
In this talk, we shall present an overview of such researches, with a particular focus on the results concerning essentially bounded divergence-measure fields. For such a family of vector fields, Silhavy (2005), Chen, Torres and Ziemer (2009) and Comi and Payne (2017) showed that Gauss--Green formulas hold for sets of finite perimeter, by proving the existence of essentially bounded interior and exterior normal traces on the reduced boundary of the given set.
Subsequent extensions of these results and generalizations to non-Euclidean geometries will be also briefly discussed.
]]>Giovedi' 26 aprile, ore 11:00 Aula 3, Dipartimento di Matematica e Informatica via Machiavelli 35
speaker: prof. Giuseppe Coclite (Universita' di Bari)
title: "Un problema di pesca ottimale con soluzioni a valori misura"
abstract: In questo seminario consideriamo un modello di pesca costituito da un'equazione parabolica. Dal momento che i funzionali costo hanno crescita sublineare rispetto alla strategia di pesca, le soluzioni ottimali hanno in generale valori misura. Dimostriamo l'esistenza di strategie ottimali per il problema del controllo. I risultati sono stati ottenuti in collaborazione con il professor Mauro Garavello.
-- Michele Miranda
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Abstract: Efficient enumeration of small objects in an abstract class may require deep structure theory (for instance, for groups) or smart combinatorial tricks (for instance, for quasigroups). I will talk about two important classes of quandles, where both approaches find their place. Certain configurations in transitive groups are used to describe algebraically connected quandles (these are the ones used for coloring knots). Then the deep theory of transitive groups can be exploited to enumerate connected quandles, currently up to size 47 and of sizes p, p^2, p^3, 2p for a prime p. Another class, medial quandles, is described using a heterogeneous affine structure over the orbit decomposition. A few combinatorial tricks then allow for enumeration of small medial quandles. For example, we can calculate that the number of isomorphism classes of 15-element medial quadles is 563753074951.