Tuesday April 29th. 2.30pm. Prof Niko Marola. Seminar room. Room 273 WCharlton.
On the growth of infinity harmonic functions.
Abstract: I will discuss the growth rate of a solution to the nonlinear highly degenerate elliptic equation, the infinity harmonic equation, $-\sum_{i,j=1}^n u_{x_i}u_{x_j}u_{x_ix_j} = 0$ in unbounded convex domains of $\mathbb{R}^n$.
Thursday April 16th. 2.30pm. Marcos Lopez. Seminar room. 4206 French Hall.
Discretizing metric measure spaces of controlled geometry
Abstract: This talk will be about providing an alternative method to verify a Poincare Inequality on a complete doubling metric measure space (X,d,mu). First, I will provide a method of discretizing X that preserves a (1,p)-Poincar e inequality. Secondly, we show that such discretezations can be extended to a sequence of connected graphs that will converge in the pointed measured Gromov Hausdorff sense to a metric measure space that is bi-Lipschitz equivalent to the (X,d,mu). Furthermore, if these discretizations have (1, p)-Poincare inequalities with uniform data, then the original space has such an inequality as well with data derived only from the discretizations.
Thursday April 3rd. 2.30pm. Dave Smith. Seminar room. 4206 French Hall.
Fokas's method for linear evolution equations: a spectral interpretation.
Abstract: We study linear, constant-coefficient evolution PDE, of arbitrary spatial order, in 1 space and 1 time dimension, equipped with an initial condition and arbitrary linear boundary conditions. Fokas' method allows us to derive a transform-inverse transform pair, tailored to any such well-posed problem and which may be used to solve that problem. By analogy with the classical trigonometric Fourier transform methods for the heat equation, we provide a spectral interpretation of the Fokas transforms. In the process, we define a new species of spectral functional and show how they diagonalize the non-self-adjoint spatial differential operator.
Thursday 26th March. 2.30pm. Robbie Buckingham. Seminar room. 4206 French Hall.
Eigenvalue Distributions for Gaussian Randon Normal Matrices.
Thursday 6th March. 2.30pm. Micheal Goldberg. Seminar room. 4206 French Hall.
The Helmholtz equation with L^p data.
Thursday 27th Jan. 2.30pm. Andrew Lorent. Seminar room. 4206 French Hall.
Differential inclusions with applications to PDE and Calculus
of Variations.
Abstract: This will be a survey talk. Simple seeming the problem of solving the differential inclusion Du\in K where u is an vector function of n variables and K is a subset of M^{m\times n} is discussed. It turns out that functions solving or approximately solving this different inclusion have unexpected rigidity and stability properties for (many) sets K that lack rank-1 connections. Indeed Tara conjectured this was a necessary and sufficient for "stability" and later produced a counter example to his own conjecture. For exact inclusions it might be conjecture that sets K with no rank-1 connections have no non-trivial solutions - this too is false and can be counter exampled by methods (know as convex integration) originating in work of Gromov and Kuipier on embeddings. Many PDE can be reformulated as differential inclusions into sets without rank-1 connections and the ability to solve such inclusions has lead to some of the most exciting counter examples in regularity theory in the last few years.
Thursday 20th Feb. 2.30pm. David Herron. Seminar room. 4206 French Hall.
QuasiHyperbolic Type Metrics and Ball Convexity II.
Thursday 13th Feb. 2.30pm. David Herron.
QuasiHyperbolic Type Metrics and Ball Convexity.
Abstract: The problem of finding a best possible route, seen in applied and pure mathematics, is both old and hard. In metric geometry, this is the "geodesic problem", to find a shortest path between two points. Due to its difficulty, it is worthwhile to know information about the location of geodesics.Suppose we conformally deform a "nice" space. Do the geodesics in the deformed space stay inside balls in the original space? We answer this question for the case of certain conformal metrics defined on domains in Euclidean n-space (or in the n-sphere).
Thursday 6th Feb. 2.30pm. Professor Xiaoyu Fu. School of Mathematics, Sichuan University. Room 130 WCharlton.
Carleman estimates and their applications. Part II
Thursday 30th Jan. 2.30pm. Professor Xiaoyu Fu. School of Mathematics, Sichuan University. Room 135 WCharlton.
Carleman estimates and their applications
Abstract: In 1939, T. Carleman introduced some energy estimates with exponential weights to prove a strong unique continuation property for some elliptic partial differential equations (PDEs for short) in dimension two. This type of weighted energy estimates, now referred to as Carleman estimates, have become one of the major tools in the study of unique continuation property, inverse problems and control theory for PDEs . In this talk, I will introduce the ideas of Carleman estimates and show how Carleman estimates play important roles in studying control problems of wave equation and heat equation.
Thursday 16th Jan. David Freeman. UC Blue Ash. 2.30pm Rm 119 WCharlton
Invertible Carnot Groups.
Abstract: We will discuss a characterization of the ideal boundaries of non-compact rank one symmetric spaces in terms of metric inversions. Time permitting, we will also discuss a related characterization of Euclidean space (up to bi-Lipschitz distortion).
Tuesday 10th Dec. Adam Osekowski. University of Warsaw. 4206 French Hall. 1pm.
Inequalities for the dyadic maximal operator.
Abstract: The dyadic version of Hardy-Littlewood maximal operator plays an important role in analysis and it is of interest to study its action between various spaces. The objective of the talk is to identify the norm of this object as an operator from $L^{p,\infty}$ to $L^q$. The proof will rest on the exploitation of the appropriate Bellman function corresponding to the problem.
Thursday 5th Dec. Micheal Goldberg. 4206 French Hall, 12.30pm.
The discrete Schrodinger equation in one and two dimensions.
Abstract: The fundamental solution to the discrete Schrodinger equation on a regular lattice is easy to express in terms of Fourier transforms. The function itself is then given as the value of an oscillatory integral. For one-dimensional domains the result is a familiar function with well-known asymptotics. For rectangular lattices in the plane, the problem can be reduced to one dimension by separation of variables, thus the answer is also well known. The Schrodinger equation on a triangular lattice does not separate variables, and its asymptotic decay depends on the coupling between adjacent vertices in a highly nontrivial way. This is joint work with Vita Borovyk.
Thursday 14th Nov. Roman Shterenberg of the University of Alabama-Birmingham. Room 4206 French Hall.
On the inverse resonance problem for continuous and discrete models.
Abstract: We consider uniqueness and stability in the inverse resonance problem for the Schrodinger operator on the half-line, its discrete analogue and other generalizations.
Thursday 7th Nov. Andrew Lorent. Room 4206 French Hall.
Rigidity of pairs of Quasiregular mappings whose symmetric part of gradient are close.
Abstract: I will survey a line of generalization of Friesecke-Muller-James rigidity estimate suggested by Muller and Cialet-Mardare, then state a result in this direction and sketch the methods of proof. This involves studying the Beltrami equation, Stoilow decompositions and making explicit estimates.
Thursday 17th Oct. Andrew Silverio. Room 4206 French Hall.
Introduction to Higher Dimensional Real Hyperbolic Geometry.
Abstract: I will discuss few properties of higher dimensional real hyperbolic space that are common to and different from dimensions 2 and 3.
Thursday 3rd Oct: Nadya Askaripour. Room 4206 French Hall.
On k-differentials and the Poincare series map.
Abstract: k-differentials can be seen in different subjects of mathematics, in this talk I will discuss two different approaches to define them. Poincare series map is an important technique to construct automorphic k- differentials on Riemann surfaces. I will talk about some properties and applications of Poincare series map.
Thursday Sept 26: David Smith. Room 273 West Charlton.
Well-posedness and spectral representation of linear initial-boundary value problems.
Abstract: We study initial-boundary value problems for linear constant-coefficient evolution equations on a finite 1-space, 1-time domain. The best known examples are the Dirichlet, Neumann, Robin & Space-Periodic problems for the heat equation in a finite rod—any student can use separation of variables or Fourier transform techniques to find a solution. However, the third order equation, Linearized KdV, sees these methods fail, for all but very special boundary conditions. The method of Fokas & Pelloni for solving all well-posed problems of this form is described. Some of the well-posedness criteria obtained by Pelloni & Smith are given. Finally, a functional-analytic view of the failure of classical methods & success of Fokas’ method is described.
Thursday Sept 19: Alexandra Zapadinskaya. Room 273 West Charlton.
Generalized dimension estimates for images of small sets under Sobolev mappings II.
Thursday Sept 12: Alexandra Zapadinskaya. Room 140 West Charlton.
Generalized dimension estimates for images of small sets under Sobolev mappings I:
Abstract: This is a joint work with Pekka Koskela. Firstly, we give an essentially sharp dimension estimate for images of porous sets under monotone Sobolev mappings, satisfying suitable Orlicz-Sobolev conditions. Secondly, we give a similar essentially sharp estimate for images of boundaries of Hölder domains under continuous Orlicz-Sobolev mappings. This estimate marks a dimension gap, which was first observed by Hencl, Koskela and Nieminen in 2011 for conformal mappings.