Coloquio de Matemáticas

We'll take a look at geometry of angles in the tropical plane by means of the so-called tropical wave front evolution. Resulting caustic produces a subdivision of the tropical angle to the elementary angles. Surprisingly, it can be seen as a geometric manifestation of the continued fractions, both in the classical form (with plus signs), and in the Hirzebruch-Jung form (with minus signs). Joint work with Mikhail Shkolnikov.

Abstract: An elastic surface resists not only changes in curvature but also tangential stretches and shears. In classical plate and shell theories, e.g., due to von Karman, the latter two strain measures are approximated infinitesimally. We motivate our approach via the phenomenon of wrinkling in highly stretched elastomers. We postulate a new, physically reasonable class of stored-energy densities, and we prove various existence theorems based on the direct method of the calculus of variations.

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En teoría de la demostración, una proposición matemática se puede representar por un tipo, es decir una colección de datos que siguen unas reglas precisas de introducción y eliminación. Este punto de vista sobre fórmulas matemáticas se conoce como la correspondencia de Curry-Howard y resulta ser útil tanto para cuestiones de fundamentos de las matemáticas como para la elaboración de lenguajes de programación que permitan formalizar una demostración. En esta charla, daremos una introducción informal a esas ideas y a la manera como se utilizan los asistentes de prueba.

Lie algebras are non-associative algebras, very helpful for understanding Lie groups. The finite-dimensional, simple Lie algebras over the complex numbers were quickly classified. Now in positive characteristic their classification is more surprising, and more recent. This suggests that Lie algebras seen as abstract algebraic structures are of interest, which naturally brings model theory into play. We'll focus on a logical generalisation of `dimension' called Morley rank, and report on ongoing work. The talk will assume knowledge of neither Lie theory nor model theory.

This will be a friendly, accessible introduction to the history of moduli spaces. Moduli, the plural of modulus, is a term coined by Riemann to describe a space whose points afford an alternative description as certain classes of geometric objects. For instance, projective space is the moduli space of lines in affine space passing through the origin. By tracing the origins through the discoveries of Riemann, Hilbert, Grothendeick, Mumford, and Deligne, we will explain many of the key concepts and theorems in moduli theory. We will then explore how these ideas have further evolved over the last 50 years.

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In this talk we present a quantitative version of the first Borel-Cantelli lemma, and its immediate consequences. Then we present the Polya urn process and its main properties, in particular, we verify that it is a bounded martingale with bounded increments which is known to converge a.s. Finally, with the help of the Azuma-Hoeffding inequality, we show how to apply the previous Borel-Cantelli lemma, in order to obtain a.s. rates and almost optimal tail estimates on the (random) modulus of continuity.