Title: «No turning back: growth theory and sustainable development»
Abstract: According to the UN definition, sustainable development should meet the needs of the present generation without compromising the needs of future generations. Following an idea of Chichilnisky, I modify the classical Ramsey criterion of economic growth in order to model sustainable development. I then show that, in the absence of a commitment mechanism, optimal policies cannot be implemented. The only implementable policies are one-way: it is possible to limit the destruction of the environment, but one can never restore it to previous levels.
Title: “On the order of the automorphism group of foliations”
Given a holomorphic foliation $\mathcal F$ with ample canonical bundle on a smooth projective surface $X$, we obtain an upper bound on the order of its automorphism group in terms of Chern classes associated to $\mathcal F$ and $X$. This is joint work with M. Correa Jr.
Title: “Single-directional property of quasimonotone operators”
In this talk we will discuss some results about the single-directional property of set valued maps, specifically how this property is implied by different continuity properties on the class on quasimonotone operators.
Title: “Rotation sets and unbounded behavior for toral homeomorphisms”
The aim of this talk is to describe some dynamical consequences of the geometry of the rotation set of a homeomorphism of the two- dimensional torus in the homotopy class of the identity. The rotation set is a generalization of Poincar_e’s rotation number for homeomorphisms of the circle, and it is a dynamical invariant which contains information on the asymptotic mean velocity of rotation of orbits.
It turns out that this set also carries other interesting information on the dynamics. We will discuss the known results in this direction, and we will talk about some recent results which show that the rotation set in many cases also detects asymptotic rotation with mean velocity 0 (i.e. sub-linear rotation).
Title: “Explicit solutions for singular in_nite horizon calculus of variations”
We consider a one dimensional infinite horizon calculus of variations problem (P), where the integrand is linear with respect to the velocity. The Euler-Lagrange equation, when defined, is not a differential equation as usual, but reduces to an algebraic (or transcendental) equation $C(x)=0 $. Thus this first order optimality condition is not informative for optimal solutions with initial condition $x_0 $ such that $C(x_0) \neq 0 $. To problem (P) we associate an auxiliary calculus of variations problem whose solutions connect as quickly as possible the initial conditions to some constant solutions. Then we deduce the optimality of these curves, called MRAPs (Most Rapid Approach Path), for (P). According to the optimality criterium we consider, we have to assume a classical transversality condition. We observe that (P) possesses the Turnpike property, the Turnpike set being given by the preceding particular constant solutions of the auxiliary problem.
Title: “Models for neural networks; analysis, simulations and qualitative behavior”
Neurons exchange informations via membrane potential discharges which trigger firing of the many connected neurons. How to describe large networks of such neurons? How can such a network generate a collective activity? How does the variety of neurons act on the modeling?
Such questions can be tackled using nonlinear integro-differential equations. These are are now classically used to describe neuronal networks or neural assemblies. Among them, the Wilson-Cowan equations are the most wellknown and describe spiking rates in different locations. Another classical model is the integrate-and-fire equation that describes neurons through their voltage using a particular type of Fokker-Planck equations. It has also been proposed to describe directly the spike time distribution which seems to encode more directly the neuronal information. This leads to a structured population equation that describes at time $t$ the probability to find a neuron with time $s$ elapsed since its last discharge.
We will compare these models and perform some mathematical analysis. A striking observation is that solutions to the I\&F can blow-up in finite time, a form of synchronization. We can also show that for small or large connectivity the ‘elapsed time model’ leads to desynchronization. For intermediate regimes, sustained periodic activity occurs which profile is compatible with observations. A common tool is the use of the relative entropy method.
This talk is based on works with K. Pakdaman and D. Salort, M. Caceres and J. A. Carrillo.
Title: “Regularization of inverse ill-posed problems with generalized Tikhonov-Phillips methods.”
Several generalizations of the traditional Tikhonov-Phillips regularization method have been proposed during the last two decades. Many of these generalizations are based upon inducing stability throughout the use of different penalizers which allow the capturing of diverse properties of the exact solution (e.g. edges, discontinuities, borders, etc.). However, in some problems in which it is known that the regularity of the exact solution is heterogeneous or anisotropic, it is reasonable to think that a much better option could be the simultaneous use of two or more penalizers of different nature. Such is the case, for instance, in some image restoration problems in which preservation of edges, borders or discontinuities is an important matter. We will show some results on existence, uniqueness and stability of minimizers for arbitrary penalizers in generalized Tikhonov-Phillips functionals. Also, results on the simultaneous use of penalizers of $L^2$ and of bounded variation (BV) type will be shown. Open problems will be discussed and results to image restoration problems will be presented.
Title: “Igusa – Todorov function: a new homological measure.”
During the late 1950s Rosenberg and Zelinsky conjectured that the finitistic dimension1 of any finite dimensional algebra is finite, and it still remains as an open problem. K. Igusa and G. Todorov defined their functions in On the nitistic global dimension conjecture for Artin algebras. Representations of algebras and related topics., Fields Inst. Commun., 45, Amer. Math. Soc., 2005, getting a very powerful result: the Finitistic conjecture is in fact true for a big family of Artin algebras.
Since this work we promote Igua-Todorov function(s) (Φ, Ψ) as a new homological measure, with very nice properties. This tool, generalises the concept of projective dimension (or injective dimension) and refine fin.dim and global dimensions (in fact: fin.dim(A) ≤ Φ.dim(A) ≤ Ψ.dm(A) ≤ gl.dim(A), for any Artin algebra A).
We characterise the Φ.dimension of an Artin algebra A in terms of bifunctors ExtiA(-,-).
Furthermore, by using this characterisation of the Φ.dimension, we show that the finiteness of the Φ.dimension of an Artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz’s result as follows: for an Artin algebra A, a tilting A-module T and the endomorphism algebra B = EndA(T)op, we have that Φ.dim(A) – pd(T)≤ Φ.dim(B) ≤ Φ.dim(A)+ pd(T).
1 Finitistic dimension: supremum of the projective dimension of finite projective dimension modules.
Title: “Geodesic PCA in the Wasserstein space”
We introduce the method of Geodesic Principal Component Analysis (GPCA) analysis on the space of probability measures on the line, with finite second moments, endowed with the Wasserstein metric. We discuss the advantages of this approach over a standard
functional PCA of probability densities in the Hilbert space of square-integrable functions.
We establish the consistency of the method by showing that the empirical GPCA converges to its population counterpart as the sample size tends to infinity. We also give illustrative examples on simple statistical models to show the benefits of this approach for data analysis.
Title: “A Logistic Multi-Unit Auction Applied to Internet Service Provision in Buenos Aires Public Schools”
The multi-unit class of auctions, typically applied in tenders of multiple identical items, has been extensively studied. This article introduces a new subclass we call logistic multi-unit, whose principal characteristic is the presence of strong supplier-cost heterogeneity due to logistical factors that results in cost advantages or disadvantages for some potential bidders relative to others on the units being tendered.
Also presented is a practical application of this tender subclass that was designed by the authors and implemented in 2008 by the city of Buenos Aires in Argentina to optimize the cost of providing Internet services to its 709 public schools. A single round first-price sealed-bid auction, this tender required each participating firm to bid a single price for monthly service to each school, identify the individual schools it would supply, and specify volume discounts on predetermined graduated volume intervals. The format thus defined can be interpreted as a combinatorial auction. An integer linear programming model determined the optimal set of bids in a matter of seconds. Implementation of the model resulted in savings of about 20% over the estimated result that would have been obtained with the format the city had originally planned.