Talks

In this talk, first we will show that for every spacelike CMC surface of revolution (except spacelike cylinders and standard hyperboloids) about a timelike axis or spacelike axis, which is either an unduloid or a nodoid, in the Lorentz-Minkowski space L^3 there is an associated Weierstrass-P function. Next, using this association, we will show that unduloid and nodoid cannot be algebraic. A similar result is obtained for CMC surfaces of revolution in the Euclidean space E^3.

We describe the asymptotics of high energy harmonic maps from Riemann surfaces to locally symmetric spaces in special classes in two settings: surface group actions on PSL(2,\R) and on SL(3,\R).  The goal is to highlight some aspects of technique, though inevitably we will state some results that follow from the methods. Joint work with Dumas, Loftin, Tamburelli, and Pan, if not others.

I shall consider minimal hypersurfaces inside the unit ball whose boundary on the sphere is a small perturbation of the link of a minimizing quadratic cone. In this talk, I will discuss a recent result where I show that such minimal surfaces are uniquely determined by their boundary condition. In particular the solutions of the Plateau problem are unique for boundary conditions given by small perturbations of the link of a quadratic cone. Building on previous results we can characterize these unique surfaces. This is a joint work with Gábor Székelyhidi.

We will discuss $\rho$-equivariant conformal minimal immersions of finite energy into $\mathbb{CH}^2$. We prove that the induced metric is complete at a puncture $p$ if and only if the holonomy of $\rho$ around $p$ is parabolic. In this case $f$ is proper and $\partial f$ has pole at each puncture, so on the compactification $\Sigma$ we obtain a parabolic $\mathrm{PU}(2,1)$–Higgs bundle $(E,\Phi)$ with nilpotent residues and zero parabolic weights. This yields an Osserman-type extension principle for minimal surfaces in $\mathbb{CH}^2$ with complete ends. We further provide an algebro–geometric characterization of these immersions in terms of their associated parabolic Higgs data. This is a joint work with Indranil Biswas and John Loftin.

The unit sphere minimizes the first positive Laplacian eigenvalue among all compact n-dimensional manifolds with Ricci curvature bounded below by n−1, as a consequence of Lichnerowicz's Theorem. This naturally raises a question about the higher Laplacian eigenvalues: Does the unit sphere minimize all the Laplacian eigenvalues? In this talk, we will explore a class of manifolds whose Laplacian eigenvalues are strictly greater than the corresponding eigenvalues of the unit sphere. These examples provide evidence for such a general minimization in dimensions two and three.

Using a special Euler–Ramanujan identity and Wick rotation, we reveal how Scherk-type zero mean curvature surfaces in Lorentz–Minkowski space can be decomposed into superpositions of dilated helicoids and hyperbolic helicoids. These decompositions also extend to maximal codimension-2 surfaces, linking them to weakly untrapped and ∗-surfaces.

We consider the problem of finding the optimal value of n in the n-step temporal difference (TD) learning algorithm. Our objective function for the optimization problem is the average root mean squared error (RMSE). We find the optimal n by resorting to a model-free optimization technique involving a one-simulation simultaneous perturbation stochastic approximation (SPSA) based procedure. Whereas SPSA is a zeroth-order continuous optimization procedure, we adapt it to the discrete optimization setting by using a random projection operator. We prove the asymptotic convergence of the recursion by showing that the sequence of n-updates obtained using zeroth-order stochastic gradient search converges almost surely to an internally chain transitive invariant set of an associated differential inclusion. This results in convergence of the discrete parameter sequence to the optimal n in n-step TD. Through experiments, we show that the optimal value of n is achieved with our algorithm for arbitrary initial values. We further show using numerical evaluations that the proposed algorithm outperforms a well known discrete parameter stochastic optimization algorithm ‘Optimal Computing Budget Allocation (OCBA)’ on benchmark RL tasks.