Equivalence relations show up at all levels in mathematics from kindergarten to graduate school: regrouping in addition and subtraction; equivalent fractions; equivalent algebraic expressions and equations; vectors; modular arithmetic; row reduction in matrices; cardinality; etc. I hypothesize that one reason students have difficulties with these topics is the subtle difference between "equivalent" and "equal" in these settings. In spite of the centrality of equivalence relations to understanding so many math topics, we don't explicitly talk about this to students, even to math majors and prospective math teachers, until late in their education, if at all.

I will talk about the mathematics that underlies the various uses of equivalence relations in these diverse settings. There will be something here for everyone who teaches math, or who prepares math teachers, at all grade levels.

Thermoacoustic tomography (TAT) is an emerging modality of medical imaging which produces high resolution, high contrast tomographic images without exposing patients to dangerous ionizing radiation. In a TAT scan, pulsed radio frequency energy applied to a tissue sample causes rapid thermoelastic expansion and contraction, propagating a pressure wave through the sample. Transducers arranged around the patient's body record this ultrasonic pressure data and there are several reconstruction regimes available to recover a tomographic image from this information. In this talk, I will describe the standard wave equation model for thermoacoustic signal generation and highlight several of the image reconstruction techniques discovered in the past twenty years. I will then present a new Neumann series exact solution due to P. Stefanov and G. Uhlmann. This very elegant method motivates and theoretically justifies wave equation time reversal methods for TAT reconstruction and I will outline some promising numerical algorithms of this type.

In this talk I will discuss the friendly index set of certain classes of trees called starlike trees.

Building upon the work of Kalai and Adin, we extend the concept of a spanning tree from graphs to simplicial complexes, which are just higher-dimensional analogs of graphs. For any complex satisfying a mild technical condition, we show that its simplicial spanning trees can be enumerated using reduced Laplacian matrices, generalizing the Matrix-Tree Theorem.

We use these higher-dimensional spanning trees to extend the concept of a critical group of a graph (related to the sandpile model and the chip-firing game) to simplicial complexes. As in the graphical case, the critical group of a simplicial complex (if its codimension 1 skeleton has a suitably nice spanning tree) can be computed directly from the reduced Laplacian, and its order is given by a weighted count of the spanning trees.

This is joint work with Carly Klivans and Jeremy Martin.

I will present work about self-similar solutions of the binormal flow.
This is a flow in three dimensions of curves that move in the direction of their
binormal with a speed proportional to their curvature. The flow was
obtained in 1906 by Da Rios as an approximation of the evolution of a
vortex filament. I will first justify the validity and limitations of
this approximation to the Euler equations. Then, I will characterize
the self-similar solutions (joint work with S. Gutierrez) from a
geometric point of view. Finally I will give some results obtained
with V. Banica about the stability of these solutions. These results
are based on the connection of this equation with the one-dimenstional cubic non-linear Schrödinger equation. As a byproduct of our analysis we obtain some new scenarios of dispersive break down.
Refreshments preceding the lecture, 3:30-4:00pm, POT745.

In this talk, we consider the design of practical adaptive multilevel finite element methods for semilinear elliptic partial differential equations. At each refinement level, the nonlinear system of equations is solved inexactly by Newton/multilevel methods. Under certain assumptions on the inexact solver, we are able to show that the adaptive algorithm still satisfies the a contraction property between two successive refinements. We will also show some numerical evidence of the convergence and accuracy of the overall AFEM algorithm.

NIH is an enormous institution, or rather collection of institutions, which funds nearly $30 Billion dollars of biomedical research annually on behalf of Congress and the US taxpayers. Many of its long-standing customs and traditions can seem impenetrable to those who are not already entrenched, particularly when it comes to winning funding for research that features mathematical and computational approaches to biological problems. This talk will present a broad picture of research at NIH in these areas, along with current funding opportunities and strategies for finding the right niche for your research proposals within NIH.
Flyer with more information is posted on the 7th floor bulletin board