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Geometry & Analysis Seminar Spring 2018

Thursdays - 4:00pm
McHenry Library Room 4130
For further information please contact Professor Francois Monard or call 831-459-2400
Thursday, April 5, 2018

Alejandruo Bravo-Doddoli, University of California, Santa Cruz
The dynamics of an articulated $n$-trailer vehicle
The dynamics of an articulated $n$-trailer vehicle that moves under its own inertia. Such system consists of a leading car, or truck, that is pulling $n$ trailers, like a luggage carrier in the airport. The leading car and the trailers form a convoy that is subjected to $(n+1)$-nonholonomic constraints, one for each body. This system is a canonical example in nonholonomic motion planning, which is fundamental in robotics, and has been extensively considered from the control perspective.
 
A cualitative study of dynamic will be discussed, the related $SE(2)$ symmetry of the problem and the reduced space. Finally, the persistent of invariant manifolds under the perturbation of one parameter will be presented as an open problem. 
Thursday, April 12, 2018

Laura Starkston, Stanford University
Symplectic isotopy problems
The symplectic isotopy problem asks whether a symplectic surface in the complex projective plane can be isotoped through symplectic surfaces to a complex curve. We will discuss both smooth and singular versions of this problem, and give a new idea to approach the problem. We will also discuss comparisons of the space of complex curves versus symplectic surfaces when we impose certain singularities on the curves.
Thursday, April 19, 2018

TBA
Thursday, April 26, 2018

TBA
Thursday, May 3, 2018

Sean Curry, University California San Diego
Strictly pseudoconvex domains in C^2 with obstruction flat boundary
On a bounded strictly pseudoconvex domain in C^n, n>1, the smoothness of the Cheng-Yau solution to Fefferman's Monge-Ampere equation up to the boundary is obstructed by a local curvature invariant of the boundary. For bounded domains in C^2 diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball.
Thursday, May 10, 2018

Katya Krupchyk, University California Irvine
Inverse boundary problems for elliptic PDE in low regularity setting
In this talk, we shall discuss recent progress in the global uniqueness issues for inverse boundary problems for second order elliptic equations, such as the conductivity and magnetic Schr\"odinger equations, with low regularity coefficients. Generally speaking, in an inverse boundary problem, one wishes to determine the coefficients of a PDE inside a domain from the knowledge of its solutions along the boundary of the domain. While ubiquitous in practice, the mathematical analysis of such problems is quite challenging, and the consideration of the low regularity setting, motivated by applications, brings additional substantial difficulties. In this talk, we shall discuss the case of full, as well as partial, measurements, both for domains in the Euclidean space, as well as in the more general setting of transversally anisotropic compact Riemannian manifolds with boundary. Some of the important ingredients in our approach are semiclassical Carleman estimates with limiting Carleman weights with an optimal gain of derivatives, precise smoothing estimates, as well as a construction of Gaussian beam quasimodes in a low regularity setting. This is joint work with Gunther Uhlmann. 
Thursday, May 17, 2018

Steven Flynn, University California Santa Cruz
The Heisenberg X-ray Transform
The classical X-ray transform in two dimensions is the same as the planar Radon transform. Its injectivity allows one to recover a sufficiently regular function through integrals over lines. The n-dimensional Euclidean X-ray transform is defined by integrating functions over Euclidean lines and is also injective. I will formulate a subRiemannian version of the X-ray transform on the simplest subRiemannian manifold, the Heisenberg group. I will describe progress towards proving the injectivity of this transform.
Thursday, May 24, 2018

TBA
Thursday, May 31, 2018

TBA
Thursday, June 7, 2018

TBA