Georgia Institute of Technology College of Sciences

I will discuss a new general framework for cutting and gluing manifolds in topological quantum field theory (TQFT). Applying this method to Chern-Simons theory with gauge group SL(2,C) on a knot complement M leads to a systematic quantization of the SL(2,C) character variety of M. In particular, the classical A-polynomial of M becomes an operator "A-hat", the same operator that appears in the recursion relations of Garoufalidis et al. for colored Jones polynomials.

Gauge theory is a beautiful subject that studies the space of connections on a vector bundle. It is also the natural language in which theories of particle physics are formulated. In fact, the word "gauge" has its origins in electromagnetism, and in this talk, we explore the basic geometric objects of gauge theory and show how one explicitly recovers the classical Maxwell's equations as a special case of the equations of gauge theory . Next, generalizing Maxwell's equations to a ``nonabelian" setting, we obtain the Yang-Mills equations, which describe the electroweak force in nature. Surprisingly, these equations were used by Simon Donaldson in the 1980s to prove spectacular results for the topology of smooth four-manifolds. We conclude this talk by describing some of the beautiful geometry and analysis behind gauge theory that goes into the work of Donaldson (for which we awarded a Fields Medal), and time permitting, we hope to say a bit about other gauge-theoretic applications to low-dimensional topology, for instance, instanton Floer homology.

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.

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.