Georgia Institute of Technology College of Sciences

We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin subgroup quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi-isometric embedding for both of the standard metrics. This is joint work with Chris Leininger and Johanna Mangahas.

A region of space is cloaked for a class of measurements if observers are not only unaware of its contents, but also unaware of the presence of the cloak using such measurements. One approach to cloaking is the change of variables scheme introduced by Greenleaf, Lassas, and Uhlmann for electrical impedance tomography and by Pendry, Schurig, and Smith for the Maxwell equations. They used a singular change of variables which blows up a point into the cloaked region. To avoid this singularity, various regularized schemes have been proposed. In this talk I present results related to cloaking via change of variables for the Helmholtz equation using the natural regularized scheme introduced by Kohn, Shen, Vogelius, and Weintein, where the authors used a transformation which blows up a small ball instead of a point into the cloaked region. I will discuss the degree of invisibility for a finite range or the full range of frequencies, and the possibility of achieving perfect cloaking. At the end of my talk, I will also discuss some results related to the wave equation in 3d.

Recent experiments indicate that one- and two-dimensionalinstabilities, bunching and meandering, respectively, coexist duringepitaxial growth of a thin film in the step-flow regime. This is in contrastto the predictions of existing Burton–Cabrera–Frank (BCF) models. Indeed, inthe BCF framework, meandering is predicated on an Ehrlich–Schwoebel (ES)barrier whereas bunching requires an inverse ES effect. Hence, the twoinstabilities appear to be a priori mutually exclusive. In this talk, analternative theory is presented that resolves this apparent paradox. Itsmain ingredient is a generalized Gibbs–Thomson relation for the stepchemical potential resulting in jump conditions along the steps that coupleadatom diffusions on adjacent terraces. Specialization to periodic steptrains reveals a competition between the stabilizing ES kinetics and adestabilizing energetic correction that can lead to step collisions. Theaforementioned instabilities can therefore be understood in terms of thetendency of the crystal to lower, away from equilibrium and in the presenceof dissipation, its total free energy. The presentation will be self-contained and no a priori knowledge of theunderlying physics is needed.

A well-known conjecture of Lovasz and Plummer asserts that the number of perfect matchings in 2-edge-connected cubic graph is exponential in the number of vertices. Voorhoeve has shown in 1979 that the conjecture holds for bipartite graphs, and Chudnovsky and Seymour have recently shown that it holds for planar graphs. In general case, however, the best known lower bound has been until now barely super-linear. In this talk we sketch a proof of the conjecture. The main non-elementary ingredient of the proof is Edmonds' perfect matching polytope theorem. This is joint work with Louis Esperet, Frantisek Kardos, Andrew King and Daniel Kral.