
Mathematics
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October 24, 10:00 a.m., Science conference room.
Title: Growth of balls in the universal cover of graphs & surfaces
Speaker: Dr. Steve Karam, University of Lille, France
Abstract: We prove that if the area of a closed Riemannian surface M of genus at least two is sufficiently small with respect to its hyperbolic area, then for every radius R ≥ 0 the universal cover of M contains an Rball with area at least the area of a cRball in the hyperbolic plane, where c is a universal positive constant in (0; 1). In particular (taking the area of M smaller if needed), we prove that for every radius R ≥ 1, the universal cover of M contains an Rball with area at least the area of a ball with the same radius in the hyperbolic plane.
This result answers positively a question of L. Guth for surfaces. We also prove an analog result for graphs. Specifically, we prove that if G is a connected metric graph of first Betti number b ≥ 2 and of length suffciently small with respect to the length of a connected trivalent graph Gof the same Betti number where the length of each edge is 1, then for every radius R ≥ 0 the universal cover of G_{b} contains an Rball with length at least c times the length of an Rball in the universal cover of G_{b} where c is a universal constant in (0.5, 1).
November 18th, 2014, 12:30 p.m., Science Conference Room.
Title: Particle representations, field equations and supersymmet
Speaker: Dr. Michel Egeileh (AUB, Lebanon)
Abstract: In the setting of relativistic quantum mechanics, 1particle states of an elementary particle constitute a Hilbert space which carries an irreducible unitary representation of the Poincaré group. We start by recalling Wigner's classification of these representations, which shows how the various types of elementary particles are labeled by a nonnegative real number m (the mass) and a halfinteger s (the spin). Then, we describe a general mechanism to construct invariant differential operators that realize the massive representations of the Poincaré group with arbitrary spin as solutions of field equations in fourdimensional spacetime, using a Fourier transform between sections of equivariantly trivialized vector bundles. Finally, time permitting; we will present a recent generalization of the above construction to supermanifolds, which allows us to obtain supersymmetric field equations from momentum space.
December 17th, 2014, 10:00 a.m., Science Conference Room.
Speaker: Dr. Raafat Talhouk, Lebanese University, Hadath.
Title: A coupled anisotropic chemotaxisfluid model: Mathematical and numerical analysis.
Abstract: In this lecture, we present a chemotaxisfluid model arising from biology, consisting of parabolicparabolic chemotaxis equations coupled to viscous incompressible NavierStokes equations by transport and gravitational forcing. The motivation of this model is the study of the behavior of many cell organisms towards a chemoattractant in a fluid. For example, bacteria often swim towards higher concentration of oxygen to survive. The unknowns are the density of cells, the concentration of chemoattractant, the velocity and the pressure in the fluid. In space dimensions less or equal than three. We first present a global existence result of weak solutions for the chemotaxisNavierStokes system. Second, we propose a new convergent numerical scheme to analyze and simulate numerically our model; many numerical tests will be presented to show the effectiveness of this numerical scheme. This work is a joint work with Georges Chamoun And Mazen Saad.
January 19th, 2015, 10:00 a.m., Science Conference Room.
Title: Poletsky theory of discs in almost complex manifold.
Speaker: Dr. Florian Bertrand (AUB, Lebanon)
Abstract: In this talk, I will present Poletsky’s construction of plurisubharmonic functions in the framework of almost complex manifolds. The main consequence of that construction is the nice characterization of polynomial hulls of compact sets in almost complex manifolds in term of pseudoholomorphic discs; roughly speaking, a point p belongs to the hull of a relatively compact set K if and only if there are holomorphic discs centered at p and with most of their boundary arbitrarily close to K. This is joint work with Uros Kuzman.
March 18th, 2015, 10:00 a.m., Science Conference Room.
Speaker: Dr. Wael Mahboub
Title: Valuations centred in the local ring k[x,y]_{(x,y)}
Abstract: Consider the field extension k(x) →k(x,y) and let v be a valuation of k(x). We can classify the valuations of k(x,y) that extends v by constructing sequences of key polynomials associated to each valuation. Those constructions give a parametrization for the valuative tree associated to this field extension. In my talk, I will explain explicitely this processus and I will give examples about the application of the theory of key polynomials to the resolution of singularities problem.
April 15th, 2015, 10:00 a.m., Science Conference Room.
Title:: Random walks on linear groups and applications.
Speaker: Dr. Richard Aoun (USJ, Lebanon)
Abstract:Random walks on groups have recently gained great interest due to their applications in group theory, or more precisely geometric group theory. Some geometric/algebraic properties of a given group G can be deduced from probabilistic properties of a random walk generated on G. We present results obtained recently by the author concerning linear groups, i.e. subgroups of the general linear group 〖GL〗_d (K,) K being a field. In particular, we give a probabilistic version of the Tits alternative (1972), which states that any finitely generated linear group which is not virtually solvable contains a free subgroup on two generators. We also discuss the "size" of algebraic subvarieties in such groups using probabilistic tools, namely random matrix products theory.
May 20th, 2015, 10:00 a.m., Science Conference Room.
Title: Covariance Matrix Estimation in Statistics
Speaker: Dr. Stefano Monni (AUBLebanon)
Abstract: In this talk, I will first give an overview of the problem of estimation in statistics. I will then zero in on the estimation of covariance matrices, detailing the limitations of the usual estimator (the sample covariance matrix) and describing some approaches that lead to better estimators in terms of risk.
