Location: Room 383N.
Time: Wednesdays, 3-4pm, or 4-5pm if the geometry seminar does not meet.
Abstract: I will describe several results in connection with a conjecture by de Giorgi dealing with some symmetry properties of solutions of some elliptic equations. In particular, I will focus on equations involving nonlocal operators and on equations with nonlinear boundary conditions. A link between these two types of equations will be described. We prove that in low dimension the level sets of bounded stable solutions are hyperplanes. We will present several methods to obtain this type of result, focusing particularly on a proof of geometric flavor. I will also mention other contexts in which this geometric proof can be carried over, like elliptic equations on riemannian manifolds.
Abstract: I will discuss recent work with Blair and Sogge establishing Lp estimates for solutions to the wave and Schrodinger equations in the setting of manifolds with boundary, along with some applications to corresponding nonlinear equations. At a point of convexity on the boundary an example of Ivanovici shows that, at least for the wave equation, the full range of Strichartz estimates cannot hold. Nevertheless, we can use microlocal parametrix constructions to obtain a range of important estimates, including ones used to establish well-posedness for energy critical nonlinear equations.and
Abstract: Locally, coordinates can be selected such that a canonical relation is prescribed by the gradient of a phase. I will discuss which choices of coordinates give the most interesting realizations of certain Fourier integral operators as oscillatory integrals. Combined with old almost-orthogonality ideas and new matrix factorization tools, we will see that these considerations go a long way towards solving the important practical problems of optimal-complexity computation of linear hyperbolic propagators and seismic imaging operators.
Abstract: In this talk, I will explain some similar results and interaction between locally symmetric spaces and moduli spaces of curves.
For example, let A_g be the moduli space of principally polarized abelian varieties of dimension g, the quotient of the Siegel upper space by Sp(2g, Z), and M_g be the moduli space of projective curves of genus g. Then there is a Jacobian map J: M_g \to A_g, by associating to each curve its Jacobian.
The celebrated Schottky problem is to characterize the image J(M_g). Buser and Sarnak viewed A_g as a complete metric space and showed that J(M_g) lies in a very small neighborhood of the boundary of A_g as g goes to infinity. Motivated by this, Farb formulated the coarse Schottky problem: determine the image of J(M_g) in the asymptotic cone (or tangent space at infinity) C_\infty(A_g) of A_g, as defined by Gromov in large scale geometry. In a joint work with Enrico Leuzinger, we showed that J(M_g) is d-dense in A_g for some constant d and hence its image in the asymptotic cone C_\infty(A_g) is equal to the whole cone.
Another example is that the symmetric space SL(n, R)/SO(n) admits several important equivariant cell decompositions with respect to the arithmetic group SL(n, Z) and hence a cell decomposition of the locally symmetric space SL(n, Z)\SL(n, R)/SO(n). One such decomposition comes from the Minkowski reduction of quadratic forms (or marked lattices). We generalize the Minkowski reduction to marked hyperbolic Riemann surfaces and obtain an equivariant cell decomposition of the Teichmuller space T_g with respect to the mapping class groups Mod_g.
If time permits, I will also discuss other results on similarities between the two classes of spaces.