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Speaker: John Dever (Georgia Institute of Technology)
Time: Monday, January 9, 2017 at 1:30 pm
Place: MONT 414
Title: Probability and Geometry on Quantum Graphs and Resistance Spaces
Speaker: Dan Kelleher (Purdue University)
Time: Friday, January 27, 2017 at 1:30 pm
Place: MONT 226Abstract: Resistance forms are crucial in the study of diffusions and analysis on post-critically finite fractals, from the Sierpinski triangle to continuous random trees. These forms can be understood from the limit of a discrete electrical networks. We shall discuss Sub-Gaussian heat kernel estimates on these resistance spaces. We are interested the geometry associated with quantum graphs -- continuous spaces made from a network of wires. Of particular focus is gradient estimates for these spaces and the functional inequalities which are implied. Finally, we shall talk about fractals which are best understood as the limits of quantum graphs.
Title: Weyl's eigenvalue asymptotics for the Laplacian on the Apollonian gasket and on circle packing limit sets of certain Kleinian groups
Speaker: Naotaka Kajino (Kobe University)
Time: Friday, February 17, 2017 at 1:30 pm
Place: MONT 226Abstract: The purpose of this talk is to present the speaker's recent results on the construction of a ``canonical'' Laplacian on circle packing fractals invariant under the action of certain Kleinian groups (discrete subgroups of the group of Moebius transformations on the Riemann sphere) and on the asymptotic behavior of its eigenvalues. The simplest example is the Apollonian gasket, which is constructed from a given ideal triangle (the closed subset of the plane enclosed by mutually tangent three circles) by repeating indefinitely the process of removing the interior of the inner tangent circles of the ideal triangles. On the Apollonian gasket, a ``canonical'' Laplacian (to be more precise, a ``canonical'' Dirichlet form) was constructed by Teplyaev (2004) as the unique one with respect to which the coordinate functions on the Apollonian gasket are harmonic. The speaker has recently discovered an explicit expression of this Dirichlet form in terms of the circle packing structure of the gasket, which immediately extends to general circle packing fractals and defines (a candidate of) a ``canonical'' Laplacian on such fractals. Then the speaker has further studied this Laplacian on more general circle packing fractals. When the circle packing fractal is the limit set of a certainclass of Kleinian group (the smallest non-empty closed subset of the Riemann sphere invariant under the action of the group), some explicit combinatorial structure of the fractal is known, which makes it possible to prove Weyl's asymptotic formula for the eigenvalues of this Laplacian. The asymptotic formula involves the Hausdorff dimension and measure of the fractals and is of the same form as the circle-counting asymptotic formula by Oh and Shah (Invent. Math., 2012).
Title: Coupled supersymmetric quantum mechanics: unifying the harmonic oscillator and supersymmetric quantum mechanics
Speaker: Cameron Williams (University of Houston)
Time: Friday, February 24, 2017 at 1:30 pm
Place: MONT 226Abstract: In this talk, we will briefly cover the standard quantum mechanical harmonic oscillator and the usual ladder operator technique for solving it. The ladder operator technique will be used to motivate traditional supersymmetric quantum mechanics. Supersymmetric quantum mechanics is useful for finding excited state energies of quantum systems, but has inherent flaws. Exact solutions via supersymmetric techniques are hard to come by and the (super)algebra of supersymmtric quantum mechanics is not very rich. Supersymmetric quantum mechanics is meant to mimic the quantum mechanical harmonic oscillator, however in general no true ladder structure exists. Returning to a treatment of the quantum mechanical harmonic oscillator under the guise of supersymmetry leads to a natural new structure which has flavors of both supersymmetry and the usual ladder structure from the harmonic oscillator. We call this new structure coupled supersymmetric quantum mechanics (coupled SUSY for short). Some results regarding the (Lie) algebraic structure underlying coupled SUSY, uncertainty principles, and coherent states for coupled SUSY will be discussed.
Title: Concentration Compactness for Critical Radial Wave Maps
Speaker: Jonas Lührmann (Johns Hopkins University)
Time: Friday, March 24, 2017 at 1:30 pm
Place: MONT 226Abstract: The wave maps equation is the natural generalization of the linear wave equation for scalar-valued fields to fields that take values in a Riemannian manifold. In this talk we consider radially symmetric, energy critical wave maps from (1+2)-dimensional Minkowski space into the unit sphere and prove global existence and scattering for essentially arbitrary smooth data of finite energy. In addition, we establish a priori bounds on a suitable scattering norm of the radial wave maps and exhibit concentration compactness properties of sequences of radial wave maps with uniformly bounded energies. The proof proceeds along the so-called concentration compactness/rigidity method of Kenig-Merle which in recent years has turned out to be a powerful strategy to study the long-time dynamics of solutions to critical dispersive or hyperbolic equations. I will explain the basic ideas of this method and the severe difficulties of implementing this method for energy critical wave maps due to the strong interactions in the wave maps nonlinearity. Then I will outline how these difficulties can be overcome by introducing a "twisted" profile decomposition. This is joint work with Elisabetta Chiodaroli and Joachim Krieger.
Title: Transition probabilities for degenerate diffusions arising in population genetics
Speaker: Camelia Pop (University of Minnesota)
Time: Friday, April 7, 2017 at 1:30 pm
Place: MONT 226Abstract: We provide a detailed description of the structure of the transition probabilities and of the hitting distributions of boundary components of a manifold with corners for a degenerate strong Markov process arising in population genetics. The Markov processes that we study are a generalization of the classical Wright-Fisher process. The main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward Kolmogorov equation defined on a compact manifold with corners, which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities.
Title: Two-Weight Inequalities for Commutators with Calderon-Zygmund Operators.
Speaker: Irina Holmes (Washington University at St. Louis)
Time: Friday, April 14, 2017 at 1:30 pm
Place: MONT 226Abstract: In 1976, a foundational paper by Coifman, Rochberg and Weiss characterized the norm of a commutator [b, T] with a Calderon-Zygmund operator T, in terms of the BMO norm of b. We discuss a recent extension of this result to the two-weight setting, and ongoing research to extend this result to the multiparameter setting using modern dyadic methods.
Title: The prescribed Ricci curvature problem on homogeneous spaces
Speaker: Artem Pulemotov (University of Queensland)
Time: Friday, April 21, 2017 at 1:30 pm
Place: MONT 226Abstract: We will discuss the problem of recovering the ``shape" of a Riemannian manifold $M$ from its Ricci curvature. After reviewing the relevant background and the history of the subject, we will focus on the case where $M$ is a homogeneous space for a compact Lie group.
Title: Zero-sum stochastic differential games without the Isaacs condition: random rules of priority and intermediate Hamiltonians
Speaker: Mihai Sirbu (University of Texas at Austin)
Time: Friday, April 28, 2017 at 1:30 pm
Place: MONT 226Abstract: For a zero-sum stochastic game which does not satisfy the Isaacs condition, we provide a value function representation for an Isaacs-type equation whose Hamiltonian lies in between the lower and upper Hamiltonians, as a convex combination of the two. For the general case (i.e. the convex combination is time and state dependent) our representation amounts to a random change of the rules of the game, to allow each player at any moment to see the other player's action or not, according to a coin toss with probabilities of heads and tails given by the convex combination appearing in the PDE. If the combination is state independent, then the rules can be set all in advance, in a deterministic way. This means that tossing the coin along the game, or tossing it repeatedly right at the beginning leads to the same value. The representations are asymptotic, over time discretizations. Space discretization is possible as well, leading to similar results. The talk is based on joint work with Daniel Hernandez-Hernandez.