This summer (2017) will feature two seminar courses: Seminar Course #1 Math 8910 - Introduction to Discrete Analysis Instructor: Neil Lyall and Akos Magyar Ten Lectures on July 10-14 in Boyd 326 from 11:00-1:30 (2 lectures per day) Course Description: We plan to cover two distinct, but related topics: Geometric measure theory (4-5 lectures) Fine structure of measurable sets, fractal sets, Hausdorff dimension. Analytic approaches to classical problems such as the Falconer and Kakeya conjectures. Geometric Ramsey Theory, specifically analogues of Falconer’s Conjecture in positive density subsets of R^d and Z^d. Additive Combinatorics (4-5 lectures) Roth’s theorem on 3-term arithmetic progressions, Szemeredi’s theorem and the Green-Tao theorem on arithmetic progressions in the primes. Emphasis will be placed on modern approaches to these problems and the use of (hypergraph) regularity lemmas. Seminar Course #2 Math 8930 - Emergent applications of parameter spaces Instructor: Patricio Gallardo Ten Lectures on June 6-9, 13-16, 20-21 in Boyd 304 from 1:40 pm to 2:40 p.m. Course Description: We explore the construction of parameters and moduli spaces arising from applications within dynamical systems. The style of this class is a sequence of 7 or 10 lectures, which are seminar style. During the first classes, we will study how to construct parameter spaces associated to matrices up to different kind of equivalence relations (see [1][2]). Afterward, we will use such techniques for building parameter spaces related to systems of differential equations arising from chemistry and epidemiological models (for example see [3][4] and [5]). In particular, we will focus in describing their geometry and defining equations. [1] Introduction to the theory of moduli. D Mumford and K. Suominen. [2] Mukai, Shigeru, and W. M. Oxbury. An introduction to invariants and moduli. Vol. 81. Cambridge University Press, 2003. [3] Craciun, Gheorghe, et al. "Toric dynamical systems." Journal of Symbolic Computation 44.11 (2009): 1551-1565. [4] Van den Driessche, Pauline, and James Watmough. "Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission." Mathematical biosciences 180.1 (2002): 29-48. [5] Bader, Markus. "Quivers, geometric invariant theory, and moduli of linear dynamical systems." Linear Algebra and its Applications 428.11-12 (2008): 2424-2454.
