Past Seminars: Spring 2014


  • February 27, 2014
    Dongbin Xiu, University of Utah
    Multi-dimensional polynomial interpolation on arbitrary nodes
    Abstract: Polynomial interpolation is well understood on the real line. In multi-dimensional spaces, one often adopts a well-established one-dimensional method and fills up the space using certain tensor product rule. Examples like this include full tensor construction and sparse grids construction. This approach typically results in fast growth of the total number of interpolation nodes and certain fixed geometrical structure of the nodal sets. This imposes difficulties for practical applications, where obtaining function values at a large number of nodes is infeasible. Also, one often has function data from nodal locations that are not by "mathematical design" and are ``unstructured''. In this talk, we present a mathematical framework for conducting polynomial interpolation in multiple dimensions using arbitrary set of unstructured nodes. The resulting method, least orthogonal interpolation, is rigorous and has a straightforward numerical implementation. It can faithfully interpolate any function data on any nodal sets, even on those that are considered degenerate by the traditional methods. We also present a strategy to choose ``optimal'' nodes that result in robust interpolation. The strategy is based on optimization of Lebesgue function and has certain highly desirable mathematical properties.

  • TUESDAY March 11, 2014, 5:00pm, ROOM 6417 (Note special time, and room!)
    Friedemann Schuricht, TU-Dresden
    Non-smooth variational problems
    Abstract: Several problems arising in physics lead to natural lack of differentiability in the models used to describe them. Appropriate tools are needed to handle the questions of existence of solutions in such framework. We present some fundamental ideas that allow to treat some convex functional that are not necessarily differentiable.

  • March 13, 2014
    Edger Sterjo, Graduate Center at CUNY
    An introduction to Morse Theory and its applications
    Abstract: Given a function f defined on a manifold M, both sufficiently differentiable, what is the topological significance of the critical points of f? In one dimension, for example, the level sets of the parabola f(x) = x^2 all have 2 components, except at the critical value y=0, at which the level set is a point. Similarly on R^2, the function f(x,y) = x^2 - y^2 has level sets which are hyperbolas, except at the critical value z=0, where the level set is the union of two diagonals. The idea of Morse theory is to relate the critical point structure of f to the topological characteristics of it's (sub)level sets. In all of this compactness properties play a big role. However, in the infinite dimensional setting, where our manifold is no longer locally compact, we must make up for this lack of compactness by imposing a further condition on our function.

  • March 20, 2014
    Alessandro Ottazzi, CIRM, Bruno Kessler Foundation, Trento, Italy
    Maps on stratified Lie groups
    Abstract: In this talk I discuss some new results concerning the study of special maps on Carnot-Carathéodory spaces, obtained in different collaborations. I shall begin with the definition of Carnot-Carathéodory space and that of Carnot group. Then I describe the classes of maps in which I am interested: isometries, conformal maps, quasiconformal maps. Finally, I will briefly comment on the techniques that are used in the proofs, with particular attention on Tanaka prolongation theory.

  • March 27, 2014
    Cyril Joel Batkam, University of Sherbrooke (Canada)
    Multiple solutions to some differential systems with strongly indefinite variational structure
    Abstract

  • April 10, 2014
    Michael McCourt, University of Colorado, Denver
    Positive Definite Kernels, Opportunities and Challenges
    Abstract: Scattered data approximation, the process of fitting a function to given data, is an important tool in applications including spatial statistics, econometrics, machine learning, computer graphics and others. In one dimension, polynomials and splines are often used; however, when higher dimensions are considered, possibly with oddly-shaped domains, polynomials and splines can prove problematic. Positive definite kernels (sometimes called radial basis functions) provide a mechanism for conducting mesh free approximation in higher dimensions and complicated domains without the fear of non-uniqueness that accompanies polynomials. In this talk we will introduce positive definite kernels and focus on the properties of the Gaussian, long the preferred kernel of many applications because of its spectrally accurate convergence properties. We will explain how interpolation with a Gaussian basis has the potential to outperform polynomial interpolation, even in one dimension, and also show how numerical instabilities can sabotage this superior performance. We will then describe a new technique to stably evaluate Gaussian interpolants, allowing for their theoretically optimal behavior to emerge.

  • April 24, 2014
    Dhanya Rajendran, Tata Institute, Bangalore (India)
    Critical growth elliptic problem with singular discontinuous nonlinearity in R2
    Abstract: Elliptic problems with discontinuous nonlinearity has its own difficulties due to the non-differentiability of the associated functional. Hence, a generalized gradient approach developed by Chang has been used to solve such problems if the associated functional is known to be Lipchitz continuous. In this talk, we will consider critical elliptic problem in a bounded domain in R^2 with the simultaneous presence of a Heaviside type discontinuity and a power-law type singularity and investigate the existence of multiple positive solutions. Here discontinuity coupled with singularity does not fit into any of the known framework and we will discuss our approach employed to obtain multiple positive solutions.

  • May 1, 2014
    One day symposium in Room 4102 from 9:30am-4:00pm
    Hyperbolic Conservation Laws: Recent Progress
    Speakers: Alberto Bressan, Geng Chen, Sebastian Noelle, Ronghua Pan
    An event sponsored by the Initiative for the Theoretical Sciences at CUNY (ITS)

  • May 8, 2014
    No seminar: Undergraduate Lecture at CSI

  • May 15, 2014 (!! Meeting at 3:00pm !!)
    Abbas Bahri, Rutgers University
    Periodic orbits of Reeb vector fields in dimension three
    Abstract:

  • May 22, 2014
    Mythily Ramaswamy, Tata Institute, Bangalore
    Control aspects of Navier-Stokes equations
    Abstract: We recall the concepts of contrability and stabilization for PDEs' and apply it for the compressible Navier-Stokes system in one dimension. i