In a series of recent papers, W.M. Schmidt and L. Summerer develop a remarkable theory of parametric geometry of numbers which enables them to recover many results about simultaneous rational approximation to families of Q-linearly independent real numbers, or about the dual problem of forming small linear integer combinations of such numbers. They recover classical results of Khintchine and Jarnik as well as more recent results by Bugeaud and Laurent. They also find many new results of Diophantine approximation. Their theory provides constraints on the behavior of the successive minima of a natural family of one parameter convex bodies attached to a given n-tuple of real numbers, in terms of this varying parameter. In this talk, we are interested in the converse problem of constructing n-tuples of numbers for which the corresponding successive minima obey given behavior. We will present the general theory of Schmidt and Summerer, mention some applications, and report on recent progress concerning the above problem.
On Schmidt and Summerer parametric geometry of numbers (NOTE: THURS)
Apr. 10, 2014 3pm (Webber 20…
Algebraic Geometry
Dusty Ross (Michigan)
X
Given a quasi-homogeneous polynomial of degree d, Landau-Ginzburg theory studies certain intersection numbers on the moduli space of d-spin curves (parametrizing curves with d-th roots of the canonical bundle). I will describe a generalization of these intersection numbers obtained by allowing some of the points on the curves to be weighted in the sense of Hassett. As one changes the weights, the invariants thus obtained can be related by a wall-crossing formula. I will explain how the wall-crossing formula generalizes the mirror theorem of Chiodo-Iritani-Ruan, and in particular how it gives a completely enumerative (A-model) interpretation of the mirror phenomenon.
On Schmidt and Summerer parametric geometry of numbers (NOTE: THURS)