# Early Research Directions

## Home

This course is intended to broaden students' perspective of mathematical research areas, while simultaneously introducing important mathematical concepts. More than a survey course, faculty volunteers plan to introduce their research topics at a beginning graduate student level to the point that students can grasp some of the central ideas and objectives of the area.

Spring 2012 Course

Organizers: Alla Borisyuk & Andrejs Treibergs
ERD course will meet Tuesdays at 10:45 am in AEB 320.

## Calendar

• 1/17: Lajos Horvath, "Statistical Analysis of Dependent Functional Data"
I'll show some examples when the data are curves or due to the large number of observations they can be modeled as curves. The data sets are coming from applications, including finance, biology, space physics. I'll discuss how to reduce the dimension of the data into finite dimension using empirical projections. I give a brief outline how Hilbert space theory, probability and statistics can be used to provide some answers to real life questions based on these data sets.

• 1/24: Andrejs Treibergs, "Heat Equation and Geometry"
Recent progress in geometry has come from by exploiting geometric flows such as the Ricci Flow, which is a solution of a parabolic partial differential equation on a manifold. I'll mention results in higher dimensions and discuss two simple flows in detail: the heat equation on the circle and curvature flow for closed curves.

• 1/31: Ken Golden, "Mathematics and the Melting Polar Ice Caps"
Sea ice is a leading indicator of climate change. It also hosts extensive microbial communities which support life in the polar oceans. The precipitous decline of the summer Arctic sea ice pack is probably the most visible, large scale change on Earth's surface in recent years. Most global climate models, however, have significantly underestimated these losses. We will discuss how mathematical models of composite materials and statistical physics are being used to study key sea ice processes such as melt pond evolution, snow-ice formation, and nutrient replenishment for algal communities. These processes must be better understood to improve projections of the fate of sea ice, and the response of polar ecosystems. Video from recent Antarctic expeditions where we measured sea ice properties will be shown.

• 2/7: Andrej Cherkaev, "How Structures Break"
The talk discusses modeling the process of damage propagation through a structure. The consideration requires Differential Eequations, geometry, wave theory, algebra, stability theory, probability, quasiconvexity, phase transition, etc.

• 2/14: Firas Rassoul Agha, "Drunken Sailors in Disordered Cities", -notes-.
I will do a short presentation on the history of Brownian motion, culminating with diffusions in random media leading to current research topics. And then I'll present a *very* simple model of random walk in random environment and compute the asymptotic velocity. The formula immediately shows a few interesting effects of random media. This computation shows how things are different for a random walk in random environment from what they are in a classical random walk.

• 2/21: Christel Hohenegger, "Swimming Near a Wall"
The dynamics of active suspensions - bacterial baths and artificial swimmers are important examples - has been the focus of much experimental and numerical work in recent years. While most of the experiments are in a confined geometry, simulations are usually performed in an infinite or periodic domain. In this talk, we will discuss the two classical suspensions models (discrete or continuum) and the difficulties of imposing boundary conditions in the continuum case.

• 2/28: Kevin Wortman, "An Example in Geometric Group Theory"
Let F be the field with two elements, and let F[t] be the one-variable polynomial ring over F. With addition, F[t] is an infinite Abelian group.
In this talk I'll prove the following:
Proposition: F[t] surjects onto an infinite Abelian group.
One proof is to let the surjection be the identity, but I'll give a much longer proof that illustrates some techniques of proofs used in geometric group theory. I'll also indicate how these same techniques have applications to more interesting problems.

• 3/6: Dan Ciubotaru, "Unitary Representations of Reductive Lie Groups: Some Examples"
In this talk, I will attempt to give a brief introduction to the theory of unitary representations for reductive groups (e.g., GL(n,R), GL(n,Q_p)). I will define all the mathematical terms involved (such as "representations") and give lots of examples, then try to convince you that this is an interesting (and difficult) problem.

• 3/20: Alla Borisyuk, "Mathematics in Neuroscience"
I will discuss how mathematical and computational tools contribute to neuron and brain studies. On math side we will talk about dynamical systems, stochastics and numerics. I will show applications in hearing studies, and in studies of brain oscillations. We will also talk about how approaches in mathematical biology can cover a spectrum from quite mathematical to much more experiment-oriented.

• 3/27: David Dobson, "Analysis and Optimization of Structures Admitting Surface Plasmons."
Surface plasmons are electromagnetic fields, highly localized near interfaces between dielectric materials and certain metals. Possible devices utilizing surface plasmons, including sensors and improved photovoltaics, are receiving a lot of attention and study in engineering. Unfortunately, mathematical analysis of these structures is lagging behind a little bit. We will outline some of the issues in this area, and opportunities for future research.

• 4/3: Peter Alfeld, "Multivariate Splines"
Splines are piecewise polynomial functions. They come in two flavors: univariate (one independent variable), and multivariate (several independent variables). Univariate splines are used ubiquitously and routinely in numerical analysis for many purposes, including approximation of data and functions, numerical integration, solution of differential equations, designs of curves, interpolation and approximation. Multivariate splines are much more complicated, their applicability is more limited, and they exhibit more diversity than univariate splines. I will focus on piecewise polynomial splines defined on triangulations which has been my main area of research for many years. I'll discuss issues, techniques, and open problems.

• 4/10: Aaron Bertram, "Hurwitz Numbers and Tropical Moduli Spaces"
If you want to count branched covers of the Riemann sphere as an old-school complex algebraic geometer, you'd set two of the branch points at 0 and infinity and ram the rest of them into these. This involves the construction of a configuration space of points. The new-school tropical algebraic geometer is much more relaxed, and simply counts trees. The combinatorics of the two approaches is, mysteriously, the same, though the geometry is radically different, with analytic geometry replaced by piece-wise linear geometry. Pretty cool.

• 4/17: Y. P. Lee, "Gromov-Witten Theory in Algebraic Geometry"