Symplectic harmonic theory was initiated by Ehresmann and Libermann in 1940's, and was rediscoverd by Brylinski in late 1980's. More recently, Bahramgiri showed in his MIT thesis that symplectic harmonic representatives of Thom classes exhibited some interesting global feature of symplectic geometry. In this talk, we discuss a new approach to symplectic Harmonic theory via geometric measure theory. The new method allows us to establish a fundamental property on symplectic harmonic forms, which is a non-trivial generalization of Bahramgiri's result, and enables us to provide a complete solution to an open question asked by V. Guillemin concerning symplectic harmonic representatives of Thom classes.  This talk is based on a very recent work of the speaker.

We will discuss recent results obtained for the one-dimensional discrete Schrodinger operator with potential given by a primitive invertible substitution sequence. This talk focuses on the methods used to obtain these results, similarities and differences from previous methods, and obstructions to further generalization.

We study the spectrum of discrete Schrodinger operators with potential given by a primitive invertible substitution sequence (and in fact our results hold for a larger class of potentials). We show this family of operators has a spectrum which is a dynamically defined Cantor set of zero Lebesgue measure. We also show that the Hausdorff dimension of this set depends analytically on the coping constant lambda and tends to 1 as lambda tends to 0. Finally, we show that at small coupling constant, all gaps allowed by the gap labeling theorem are open and furthermore open linearly.

Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long ranged elastic fields with a much larger region that cannot be computed atomistically. Many methods have recently been proposed to compute solutions to these multiscale problems by coupling atomistic models near a localized defect with continuum models where the deformation is nearly uniform. During the past several years, we have given a theoretical structure to the description and formulation of atomistic-to-continuum coupling that has clarified the relation between the various methods and the sources of error. Our theoretical analysis and benchmark simulations have guided the development of optimally accurate and efficient coupling methods.

To a Hilbert modular form one may attach a p-adic analytic
L-function interpolating certain special values of the usual L-function.
Conjectures in the style of Mazur, Tate and Teitelbaum prescribe the order
of vanishing and first Taylor coefficient of such p-adic L-functions, the
first coefficient being controlled by an L-invariant which has conjectural
(arithmetic) value defined by Greenberg and Benois. I will explain how to
compute arithmetic L-invariants for (critical, exceptional) symmetric
powers of non-CM Iwahori level Hilbert modular forms via triangulations on
eigenvarieties. This is based on joint work with Robert Harron.