Georgia Institute of Technology College of Sciences

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects which we call pastures; they generalize both hyperfields in the sense of Krasner and partial fields in the sense of Semple and Whittle. We then define matroids over pastures; in fact, there are at least two natural notions of matroid in this general context, which we call weak and strong matroids. We present ``cryptomorphic'’ descriptions of each kind of matroid. To a (classical) rank-$r$ matroid $M$ on $E$, we can associate a universal pasture (resp. weak universal pasture) $k_M$ (resp. $k_M^w$). We show that morphisms from the universal pasture (resp. weak universal pasture) of $M$ to a pasture $F$ are canonically in bijection with strong (resp. weak) representations of $M$ over $F$. Similarly, the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios'', which we call the foundation of $M$, parametrizes rescaling classes of weak $F$-matroid structures on $M$. As a sample application of these considerations, we give a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.

In these lectures we discuss some statistical problems with an interesting combinatorial structure behind. We start by reviewing the "hidden clique" problem, a simple prototypical example with a surprisingly rich structure. We also discuss various "combinatorial" testing problems and their connections to high-dimensional random geometric graphs. Time permitting, we study the problem of estimating the mean of a random variable.

We consider a class of singular ordinary differential equations, describing systems subject to a quasi-periodic forcing term and in the presence of large dissipation, and study the existence of quasi-periodic solutions with the same frequency vector as the forcing term. Let A be the inverse of the dissipation coefficient. More or less strong non-resonance conditions on the frequency assure different regularity in the dependence on the parameter A: by requiring a non-degeneracy condition on the forcing term, smoothness and analyticity, and even Borel-summability, follow if suitable Diophantine conditions are assumed, while, without assuming any condition, in general no more than a continuous dependence on A is obtained. We investigate the possibility of weakening the non-degeneracy condition and still obtaining a solution for arbitrary frequencies.

In 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved.Their fifth problem asks the following.Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let C(K,x)=vol(K\cap H_x)dist (0, G). (proportional to the volume of the cone spanned by the secion and the support point). If there exists a constant C such that for all directions x we have C(K,x)=C, does it follow that K is an ellipsoid?We give an affirmative answer to this problem for bodies sufficiently close to the Euclidean ball in the Banach Mazur distance.This is a joint work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.