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for events the day of Thursday, September 18, 2014.

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Thursday, September 18, 2014

11:00 am in 241 Altgeld Hall,Thursday, September 18, 2014

Deformations of Galois representations

Patrick Allen   [email] (Northwestern University)

Abstract: In the theory of representations of Galois groups, a central object of study is Mazur's universal deformation space. It is an object that parametrizes Galois representations in a continuous way. There are many unknown questions and conjectures about this space. For example: What is its dimension? Can we describe the smooth points? Is the subset constructed via modular forms a dense subspace? In this talk we will give an overview of the theory of deformations of Galois representations, and discuss some progress on the above questions.

1:00 pm in Altgeld Hall 243,Thursday, September 18, 2014

Counting closed geodesics on flat surfaces

Elise Goujard (Rennes)

Abstract: Counting periodic trajectories in polygonal billiards is related in some cases to counting closed geodesics on corresponding flat surfaces. The asymptotic of the number of closed geodesics on a flat surface is given by a constant called Siegel-Veech constant. For a flat surface defined by a quadratic differential we explain how this constant is related to the volumes of moduli spaces of quadratic differentials, extending the work of Masur-Zorich and Athreya-Eskin-Zorich. We illustrate this correspondence with an example of small complexity, for which we compute the volume explicitly. A video recording of this talk can be found at

2:00 pm in 007 Illini Hall,Thursday, September 18, 2014

Sum of Dilates

George Shakan (UIUC Math)

Abstract: Let A be a finite subset of the integers and q be an integer. Set $A+q \cdot A = \{a + qb : a,b \in A\}$. We discuss that $|A+q \cdot A| \geq (|q|+1)|A| - C_q,$ where $C_q$ is a constant that only depends on $q$. This is seen to be the best possible, up to the additive constant, by allowing $A$ to be $\{1 , ... , |A|\}$. We also discuss other related work and open problems in the area.

3:00 pm in 243 Altgeld Hall,Thursday, September 18, 2014

Bounds for the Hilbert-Samuel Multiplicity

Javid Validashti (UIUC Math)

Abstract: A classical inequality due to Lech states that the Hilbert-Samuel multiplicity of a zero-dimensional ideal in a regular local ring is bounded above by the normalized colength of that ideal. Simple examples show that this inequality is sharp asymptotically, but it gives a very weak bound for the Hilbert-Samuel multiplicity in general. In this talk, I will discuss improvements of Lech's inequality, which also yield stronger inequalities on the Hilbert coefficients of an ideal. Joint work with Craig Huneke (University of Virginia) and Ananth Hariharan (Indian Institute of Technology).

4:00 pm in 245 Altgeld Hall,Thursday, September 18, 2014

The mathematics behind biological invasion processes

Mark Lewis (University of Alberta)

Abstract: Models for invasions track the front of an expanding wave of population density. They take the form of parabolic partial differential equations and related integral formulations. These models can be used to address questions ranging from the rate of spread of introduced invaders and diseases to the ability of vegetation to shift in response to climate change. In this talk I will focus on scientific questions that have led to new mathematics and on mathematics that have led to new biological insights. I will investigate the mathematical and empirical basis for multispecies invasions, for accelerating invasion waves, and for nonlinear stochastic interactions that can determine spread rates.