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Lunes 12 de febrero. Yun Guang Lu Título: Hyperbolic Conservation Laws. Resumen: In this talk, I would like to introduce some basic behaviors of weak solutions for nonlinear hyperbolic conservation laws and the method of Compensated Compactness. Lunes 19 de febrero Lorenzo Acosta Título: Hausdorffización de espacios espectrales. Resumen: La "Hausdorffización" de un espacio topológico X es un espacio topológico H que es de Hausdorff y cuya topología es el mejor refinamiento posible de la topología de X. Se puede probar que en el contexto general de los espacios topológicos no siempre es posible construir una Hausdorffización de un espacio topológico dado. En esta charla mostraremos que en el contexto de los espacios espectrales, es decir aquellos que se obtienen como espectros primos de retículos distributivos acotados (o de anillos conmutativos con unidad), la construcción siempre es posible y corresponde a la construcción algebraica de la extensión booleana libre de un retículo distributivo. Lunes 12 de marzo Agustín Moreno Título: Resolviendo las ecuaciones x³+y³+z³=n, x³+y³+2z³=n. Resumen: Fermat's theorem of polygonal numbers (proved by Cauchy in 1813) states that every number is a sum of at most three triangular numbers, four squares, five pentagonal numbers and so on, where the k-th polygonal number of order n is given by the formula p^n_k = 1/2 [(n-2)k^2-(n-4)k]. Concerning the Fermat's theorem of polygonal numbers and sums of cubes R. Guy (in [1,2,3]) proposes several unsolved problems in number theory. The first one asks what theorems are there, stating that all numbers of a suitable shape are expressible as the sum of three (say) squares of numbers of a given shape? Particularly R. Guy (in [1,3]) asks if every sufficiently large number 8n + 3 can be expressible as the sum of three squares of numbers of the form 4r-1 with r positive or if every sufficiently large number of shape 24n + 3 is expressible as the sum of three squares of numbers of shape 6r-1. Another question of this type asks if every number of shape 40n + 27 can be expressible as a sum of three squares of the form (10r ± 3)2 , (problems C20,D3 in [3]). With respect to cubes R. Guy asks, if all numbers which are not of the form 9n ± 4 are the sum of three cubes or if every number is the sum of four cubes with two of them equal [2,3], (problem D5 in [3]). In the last decades algorithms of differentiation of posets have been the most useful tool in the theory of representation of posets. They have been used to obtain criteria of representation type finite, tame, and finite growth of posets both ordinary and with additional structures [4-9]. In this talk we will describe how we can use the ideas of differentiation algorithms of posets in order to obtain solutions of the problems men- tioned above about figurate numbers. Additionally we will give some advances to the problem concerning the number of representations of a number n as the sum of polygonal numbers [1]. Referencias [1] R. Guy, Every number is expressible as a sum of how many polygonal numbers, A.M.M 101 (1994), 169--172. [2]--, Nothing's new in number theory?, A.M.M 105 (1998), no. 10, 951--954. [3]--, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag, New York, 2004. [4] M.M. Kleiner, Partially ordered sets of finite type, Zap. Nauchn. Semin. LOMI 28 (1972), 32--41 (in Russian); English transl., J. Sov. Math 3 (1975), no. 5, 607--615. [5] L.A. Nazarova and A.V. Roiter, Partially ordered sets of infinite type, Izv. AN SSSR, Ser. Mat. 39 (1975), no. 5, 963--991 (in Russian); English transl., Math. USSR Izvestia 9 (1975), 911--938. [6] L.A. Nazarova and A.G. Zavadskij, Partially ordered sets of tame type, Akad. Nauk Ukrain. SSR Inst. Mat., Kiev (1977), 122--143 (Russian). [7] --, Partially ordered sets of finite growth, Function. Anal. i Prilozhen., 19 (1982), no. 2, 72--73 (in Russian); English transl., Functional. Anal. Appl., 16 (1982), 135--137. [8] A.G. Zavadskij, Tame equipped posets, Linear Algebra Appl. 365 (2003), 389-- 465. [9] --, Equipped posets of finite growth, Representations of Algebras and Re- lated Topics, AMS, Fields Inst. Comm. Ser. 45 (2005). Lunes 26 de marzo Alexandre Sinitsyn Título: Vlasov equation: existence and stability of solutions. Resumen: Lunes 9 de abril Benjin Xuan Título: Best constants for Sobolev's inequalities. Resumen: Lunes 23 de abril Victor Albis Título: Matemáticas y antropología. Resumen: Lunes 30 de abril Leonardo Rendón Título: Ecuaciones diferenciales y leyes de conservación Resumen: Lunes 7 de mayo Rodrigo de Castro Título: Estructuras ordenadas y semántica denotacional. Resumen: Lunes 14 de mayo Fernando Zalamea Título: Un panorama de la lógica categórica. Resumen: