UIUC Dept. of Mathematics Seminar Calendar

     October 2009          November 2009          December 2009    
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
              1  2  3    1  2  3  4  5  6  7          1  2  3  4  5
  4  5  6  7  8  9 10    8  9 10 11 12 13 14    6  7  8  9 10 11 12
 11 12 13 14 15 16 17   15 16 17 18 19 20 21   13 14 15 16 17 18 19
 18 19 20 21 22 23 24   22 23 24 25 26 27 28   20 21 22 23 24 25 26
 25 26 27 28 29 30 31   29 30                  27 28 29 30 31

Tuesday, February 3, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, February 3, 2009

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Submitted by berdogan.

Betsy Stovall (UC Berkeley)
Quasi-extremals for convolution with surface measure on the sphere
Abstract: Let $T$ be the operator which maps functions on $\mathbb{R}^d$ to their convolution with surface measure on the unit sphere. Then a classical result from harmonic analysis states that there exists a constant $C$ so that for any functions $f,g \in L^{(d+1)/d}(\mathbb{R}^d)$, \[ |\langle Tf, g \rangle | \leq C \|f\|_{L^{(d+1)/d}} \|g\|_{L^{(d+1)/d}}. \] But what happens if we try to reverse this inequality? Can we describe the pairs of functions $f,g \in L^{(d+1)/d}(\mathbb{R}^d)$ satisfying \[ | \langle Tf,g \rangle | \geq \varepsilon \|f\|_{L^{(d+1)/d}} \|g\|_{L^{(d+1)/d}}? \] In this talk, we will give a partial answer to this question, sketch its proof, and highlight the differences between convolution with surface measure on the sphere and convolution with surface measure on the paraboloid.

Tuesday, March 10, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, March 10, 2009

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Submitted by dhundert.

Volker Betz (Warwick, UK)
Superadiabatic transition histories in quantum molecular dynamics
Abstract: We are interested in the dynamics of a molecule's nuclear wave function near an avoided crossing of two electronic energy levels. More precisely, we study the time development of the wave function's component in an initially unoccupied energy subspace, when the wave packet crosses the transition region. In the optimal superadiabatic representation, which we define, this component builds up monotonously, and has the approximate shape of an error function; thus, its norm displays the same behaviour as observed by Michael Berry in a simplified, time-adiabtic model in 1990. Finally, we give a simple, explicit formula for the transmitted wave packet in the scattering region, which is in excellent agreement with high precision ab initio numerical computations.

Tuesday, March 17, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 245 Altgeld Hall, Tuesday, March 17, 2009

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Submitted by ekirr.

Arghir Zarnescu [email] (University of Oxford Mathematical Institute )
Landau-de Gennes theory on nematic liquid crystals: the Oseen-Frank limit
Abstract: The complexity of nematic liquid crystals is described, in Landau-de Gennes theory, by functions defined in a three-dimensional open set, with values in the set of traceless symmetric three-by-three matrices. A global energy minimizer solves a system of five coupled nonlinear elliptic partial differential equations in dimension three. Mathematically this problem can be regarded, in a certain sense, as a high dimensional version of the 2D Ginzburg-Landau problem. Our problem shares certain features with the 2D problem and also exhibits new phenomena, specific to the high dimensionality of the problem. We show that in the (physically relevant) limit of low elastic constant the solutions converge in H^1 to the underlying harmonic map problem, and uniformly away from the defects of the limiting harmonic map problem. We also study the interplay between biaxiality and uniaxiality in global energy minimizers and obtain estimates for various quantities such as the biaxiality parameter, the size of admissible strongly biaxial regions and the size of uniaxial regions.

Tuesday, March 31, 2009

Tuesday, April 7, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, April 7, 2009

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Submitted by tyson.

Erin Pearse (University of Iowa)
Resistance analysis on infinite networks
Abstract: A network is a graph with weighted edges, and is a basic object of study in discrete harmonic analysis and potential theory, probability and percolation, group theory, numerical analysis, and many other areas. I will discuss the effective resistance distance, a natural metric which is sensitive to the local and global topology of the network, and its close relationship with the "energy space" (the space of functions of finite energy). The discrete Gauss-Green formula tells how to do "integration by parts" on the energy space, and in cases where the network supports nonconstant harmonic functions of finite energy, a somewhat mysterious and intrinsic boundary term arises. In particular, harmonic functions have a boundary representation akin to that of Poisson or Martin. I will describe a way to construct a boundary representation (and a kernel supported on this boundary) using the operator theory of the Laplacian. The result is closely analogous to Martin boundary, but the methods are entirely analytic rather than probabilistic. It is known that the extension of resistance metric to infinite networks is not unique. If time permits, I will explain how the discrepancy is a result of nontrivial boundary, and how the "free" and "wired" effective resistances may be understood in terms of the Hilbert space structure of the energy space.

Tuesday, April 14, 2009

Tuesday, April 21, 2009

Tuesday, May 5, 2009

Tuesday, September 1, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, September 1, 2009

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Submitted by dhundert.

Johannes Zimmer (Department of Mathematics, University of Bath)
Moving interfaces in solids: from conservative lattice models forphase transitions to macroscopic dissipation
Abstract: Moving interfaces in solids can dissipate energy, but are commonly described by conservative (Hamiltonian) equations on the lattice scale. How can conservative lattice models generate dissipation on the continuum scale? To understand this, we consider a model problem, namely that of a moving phase boundary in a solid (the talk will start with a short survey on phase transitions in solids). A well-accepted microscopic model is then that of a one-dimensional chain of atoms with nearest neighbor interaction. To describe phase transitions, the elastic potential is chosen to be nonconvex; we will consider a piecewise quadratic energy with two wells. A simple solution class describing the motion of an interface is then the class of travelling waves. A solution which explores both wells of the energy will have an interface, moving with the speed of the wave. We show that for suitable fixed subsonic velocities, there is a family of ``heteroclinic" travelling waves (heteroclinic means here that they connect both wells of the energy). Though the microscopic picture is Hamiltonian, we derive non-trivial so-called kinetic relations on the continuum scale; they can be related to the dissipation generated by a moving phase boundary. We then investigate the question of when the kinetic relation does not vanish (dissipation is generated). It turns out that a microscopic asymmetry determines here the macroscopic dissipation. This is joint work with Hartmut Schwetlick (Bath). If time permits, the talk will finish with with a glimpse on a similar model of dislocation dynamics, proposed by Frenkel and Kontorova in 1939. We will sketch a rigorous existence result for travelling waves, obtained in collaboration with Carl-Friedrich Kreiner (Munich).

Tuesday, September 8, 2009

Tuesday, September 22, 2009

Tuesday, September 29, 2009

Tuesday, October 13, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, October 13, 2009

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Submitted by verahur.

Francisco Gancebo (University of Chicago)
Contour dynamics for 2D active scalars
Abstract: In this talk we discuss two free boundary problems given by fluid domains which are weak solutions of incompressible equations. We consider the contour dynamics Muskat problem and the evolution of a sharp front by the 2D surface Quasi-geostrophic equation. Both systems are described by means of a transport equation for the active scalar \rho(x,t) which takes constant values on complementary domains. The velocity field is determined by \rho(x,t) by singular integral operators. However the solutions of these two physical scenarios have completely different outcomes regarding well-posedness and regularity issues.

Tuesday, October 20, 2009

Tuesday, October 27, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, October 27, 2009

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Submitted by ekirr.

Boaz Ilan [email] (U. of California, Merced)
Solitons at the interface between bands and gaps
Abstract: Solitons or localized bound states arise in nonlinear wave systems including nonlinear optics, ultra-cold atomic systems, and water waves. In some cases solitons can be very stable while in others highly unstable and can undergo collapse (singularity formation). We study solitons in focusing Nonlinear Schr�dinger (NLS) equations with periodic potentials. Rigorous asymptotic analysis reveals that when a soliton bifurcates from a band edge into a gap, the soliton profile is constructed from a linear Bloch wave that is slowly modulated by a bound state solution of a homogenized NLS equation. A consequence of the analysis is that in the L2-critical case, the soliton power (L2 norm) is below the threshold for collapse. Direct computations of soliton dynamics in L2-critical NLS equations elucidate these results.

Tuesday, November 3, 2009

Harmonic Analysis and Mathematical Physics
1:00 pm in 347 Altgeld Hall, Tuesday, November 3, 2009

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Submitted by laugesen.

Joe Lakey [email] (New Mexico State University)
Time- and band-limiting
Abstract: This talk will survey some of the classical and recent results concerning operators composed of a projection onto a compact set in time, followed by a projection onto a compact set in frequency. Such "time- and band-limiting" operators were studied by Landau, Slepian, and Pollak in a series of papers published in the Bell Systems Tech. Journal in the early 1960s. Among other important results, Landau and Pollak gave an initial precise statement of the "folklore" observation that the dimension of the space of signals that are essentially timelimited to a given duration and bandlimited to a given frequency bandwidth is the time-bandwidth product. Other useful versions were proved by Slepian in the early 1970s and by Landau and Widom in 1980.
Further progress on time- and bandlimiting has been intermittent, but genuine recent progress has been made in terms of numerical analysis, sampling theory, and extensions to multiband signals, all driven to some extent by potential applications in wireless communications.
After providing a brief outline of the historical developments in the mathematical theory of time- and bandlimiting, some details of the sampling theory and multiband setting will be given.