Georgia Institute of Technology College of Sciences

I will discuss inverse problems involving elliptic partial differential equations with highly oscillating coefficients. The multiscale nature of such problems poses a challenge in both the mathematical formulation and the numerical modeling, which is hard even for forward computations. I will discuss uniqueness of the inverse in certain problem classes and give numerical methods for inversion that can be applied to problems in medical imaging and exploration seismology.

All students interested in graduate studies in the School of Math are invited to attend the "prospective student day." This event will offer the opportunity to hear about our graduate degree options, requirements for admission, as well as meet our Faculty and current graduate students. Prospective students from underrepresented groups in the Mathematical Sciences and students from the Atlanta area are particularly encouraged to attend. If you plan to attend, please send your name, the year you plan to graduate, and the college you are attending to dgs@math.gatech.edu. See the schedule for more details.

Many statistical physics models are defined on an infinite lattice by taking appropriate limits of the model on finite lattice regions. A key consideration is which boundary to use when taking these limits, since the boundary can have significant influence on properties of the limit. Fixed boundary conditions assume that the boundary cells are given a fixed assignment, and free boundary conditions allow these cells to vary, taking the union of all possible fixed boundaries. It is known that these two boundary conditions can cause significant differences in physical properties, such as whether there is a phase transition, as well as computational properties, including whether local Markov chain algorithms used to sample and approximately count are efficient. We consider configurations with free or partially free boundary conditions and show that by randomly extending the boundary by a few layers, choosing among only a constant number of allowable extensions, we can generalize the arguments used in the fixed boundary setting to infer bounds on the mixing time for free boundaries. We demonstrate this principled approach using randomized extensions for 3-colorings of regions of Z2 and lozenge tilings of regions of the triangle lattice, building on arguments for the fixed boundary cases due to Luby et.al. Our approach yields an efficient algorithm for sampling free boundary 3-colorings of regions with one reflex corner, the first result to efficiently sample free boundary 3-colorings of any nonconvex region. We also consider self-reducibility of free boundary 3-colorings of rectangles, and show our algorithm can be used to approximately count the number of free-boundary 3-colorings of a rectangle.

The Abelian sandpile was invented as a "self-organized critical" model whose stationary behavior is similar to that of a classical statistical mechanical system at a critical point. On the d-dimensional lattice, many variables measuring correlations in the sandpile are expected to exhibit power-law decay. Among these are various measures of the size of an avalanche when a grain is added at stationarity: the probability that a particular site topples in an avalanche, the diameter of an avalanche, and the number of sites toppled in an avalanche. Various predictions about these exist in the physics literature, but relatively little is known rigorously. We provide some power-law upper and lower bounds for these avalanche size variables and a new approach to the question of stabilizability in two dimensions.