Location: Minard 304A Time: Wednesday 4:00 - 4:50 PM Organizer: Dr. Maria Alfonseca-Cubero

Spring 2008 Schedule January 9 Organizational Meeting January 30 An introduction to Lebesgue spaces Riley Casper February 6 Introduction to $\Gamma$-convergence and applications Cristina Popovici Abstract: In these two talks I will define De Giorgi's $\Gamma$-convergence, discuss some of its main properties, and describe some recent applications to problems arising in Continuum Physics. February 13 Introduction to $\Gamma$-convergence and applications (part II) Cristina Popovici Abstract: We continue discussing De Giorgi's $\Gamma$-convergence and its applications to problems arising in Continuum Physics. February 20 No Seminar February 27 Weakly Convergent Sequences in $L^{p}$ spaces: Oscillation and Concentration Effects Marian Bocea Abstract: An understanding of the possible oscillation and concentration effects developed in weakly converging sequences is crucial for the study of certain variational problems arising in applications, most notably to Materials Science. In order to describe the limiting behavior of nonlinear functionals acting on such sequences we introduce the notions of Young measure and reduced defect measure, discuss their basic properties, and describe some applications to non-convex variational problems. March 12 We will meet in Minard 340 Weakly Convergent Sequences in $L^{p}$ spaces: Oscillation and Concentration Effects (Part II) Marian Bocea Abstract: An understanding of the possible oscillation and concentration effects developed in weakly converging sequences is crucial for the study of certain variational problems arising in applications, most notably to Materials Science. In order to describe the limiting behavior of nonlinear functionals acting on such sequences we introduce the notions of Young measure and reduced defect measure, discuss their basic properties, and describe some applications to non-convex variational problems. March 19 Extremal Signatures Friedrich Littmann Abstract: One of the basic theorems in the theory of L^1 approximation states that the signature p(x) of the difference between a given function and its best L^1 approximation from a vector space has to be orthogonal (in the norm sense) to the approximation space.       Consider approximations that are functions whose Fourier transforms are supported in a symmetric interval about the origin. In this case the above theorem is essentially the statement that the signature p(x) needs to be a high-pass function. Except in special cases it is difficult to check directly whether a given signature p(x) is such a function.       Let p(x) be a function that assumes only values +1 and -1 on the real line. Using the harmonic extension of p(x) into the upper half-plane, B. Logan formulated and proved in the 1960's a necessary and sufficient condition that allows for the construction of entire functions B(x) such that sign(B(x)) is a high-pass function. I will sketch Logan's construction and some of its applications. March 26 Extremal Signatures Friedrich Littmann Abstract: One of the basic theorems in the theory of L^1 approximation states that the signature p(x) of the difference between a given function and its best L^1 approximation from a vector space has to be orthogonal (in the norm sense) to the approximation space.       Consider approximations that are functions whose Fourier transforms are supported in a symmetric interval about the origin. In this case the above theorem is essentially the statement that the signature p(x) needs to be a high-pass function. Except in special cases it is difficult to check directly whether a given signature p(x) is such a function.       Let p(x) be a function that assumes only values +1 and -1 on the real line. Using the harmonic extension of p(x) into the upper half-plane, B. Logan formulated and proved in the 1960's a necessary and sufficient condition that allows for the construction of entire functions B(x) such that sign(B(x)) is a high-pass function. I will sketch Logan's construction and some of its applications. April 2 Extremal Signatures (Part II) Friedrich Littmann April 9 Extremal Signatures (Part III) Friedrich Littmann April 16 Master thesis results Umar Islambekov Abstract: We will prove the existence of the ergodic Hilbert transform for strongly bounded symmetric admissible processes defined on a sigma-finite measure space. April 23 Master thesis results Umar Islambekov Abstract: We will continue with the proof of the existence of the ergodic Hilbert transform for strongly bounded symmetric admissible processes defined on a sigma-finite measure space. April 30 Master thesis results Umar Islambekov Abstract: We will finish the proof of the existence of the ergodic Hilbert transform for strongly bounded symmetric admissible processes defined on a sigma-finite measure space.