# Summer Mini-Courses 2017

This summer (2017) will feature two mini-courses:

**Mini-Course #1****Math 8910 - Introduction to Discrete Analysis**

Instructor: Neil Lyall and Akos Magyar (Provisionally scheduled to take place from July 3 - July 14)

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.

**Mini-Course #2**__Math 8930 - Emergent applications of parameter spaces__

Instructor: Patricio Gallardo (Provisionally scheduled to take place from June 5 - June 16)

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.