Program

Schedule

Addresses:

See also the interactive campus map to locate the buildings (JFB, JWB and LCB)

Abstracts

Liz Vivas

Title: Introduction to one and several variables complex dynamics

Abstract: 

Charles Favre

Title: Holomorphic dynamics: the global point of view

Abstract: These introductory lectures are intended to give an overview in holomorphic dynamics. In recent years, new techniques from algebraic geometry and arithmetic intersection theory have been brought into this very active area of research. We shall present how Berkovich theory can be used to understand degenerations of complex dynamical systems and analyze special hyperbolic components.

complex dynamics1-Fatou Julia theory.pdf
complex dynamics2-moduli space.pdf

Rebecca Winarski

Title: Polynomials, branched covers, and trees

Abstract: Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction.  We use topological techniques – adapting tools used to study mapping class groups – to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a topological branched cover is equivalent to.  This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.

John Lesieutre

Title: Fibrations in algebraic dynamics

Abstract: A birational map of an algebraic surface for which the degrees of the iterates grow subexponentially always admits an invariant fibration.  A question of Cantat asks whether the same might hold true in higher dimensions as well.  I will introduce the basic ideas surrounding this problem, and then describe an unsuccessful attempt to find a counterexample.  Although it didn't work, it will allow us to touch on a variety of techniques and open questions in the area, and still yield some interesting geometry in the end.

Sayantika Mondal

Title: Length infima and self-intersection number of filling curves 

Abstract: We look at filling closed curves on hyperbolic surfaces and consider its length infima in the moduli space of the surface. In this talk I will focus on relations between the length infimum of curves and their self-intersection number. In particular, construct families of filling curves with same intersection number but different length infimum on any surface.

Emil Geilser

Title: Representations of the symmetric group from geometry

Abstract: Representation stability was introduced to study mathematical structures which stabilize when viewed from a representation theoretic framework. The instance of representation stability studied in this project is that of ordered complex configuration space, denoted PConf_n(C):

PConf_n(C) := {(x1, x2, . . . , xn) ∈ C^n | xi ̸= xj}

PConf_n(C) has a natural Sn action by permuting its coordinates which gives the cohomology groups H^i(PConf_n(C); Q) the structure of an Sn representation. The cohomology of PConf_n(C) stabilizes as n tends toward infinity when viewed as a family of S_n representations. From previous work, there is an explicit description for H^i(PConf_n(C); Q) as a direct sum of induced representations for any i, n, but this description does not explain the behavior of families of irreducible representations as n → ∞. We implement an algorithm which, given a Young Tableau, computes the cohomological degrees where the corresponding family of irreducible representations appears stably as n → ∞. Previously, these values were known for only a few Young Tableaus and cohomological degrees. Using this algorithm, results have been found for all Young Tableau with up to 8 boxes and certain Tableau with more, which has led us to conjectures based on the data collected.