If you want to receive e-mail announcements of talks please contact
Dionisios Margetis (dio@math.umd.edu).
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January 31
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NO SEMINAR
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Abstract:
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February 7
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Alina Chertock
Department of Mathematics -- North Carolina State University
A Particle Method for the EPDiff Equation
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We consider a model of active fluid transport described by an evolutionary equation, known as the the Euler-Poincar´e
(EPDiff) equation. The EPDiff equation arises in many scientific applications. In particular, it appears in the nonlinear
dynamic of shallow water waves, and coincides, for example, with the Camassa-Holm equation of shallow water in 1-D
and 2-D, and with the averaged template matching equation for computer vision in higher dimensions. The EPDiff singular
solutions are contact discontinuities, called peakons. The key feature of the peakons is that they carry momentum; so the wave
front interactions they represent are collisions, in which momentum is exchanged. This is very reminiscent to the KdV solitons
behavior in 1-D.
We numerically investigate the EPDiff dynamics of contact interactions using particle methods. We show that he discretization
by means of the particle method preserves the basic Hamiltonian, the weak and variational structure of the original problem and
respects the conservation laws associated to the symmetry under the Euclidean group. We use this symmetry to reduce the
dimensionality of the system, and thus to simplify the theoretical and, especially, numerical analysis of the dynamics of peakons
and their interactions.
This is a joint work with Jerrold E. Marsden, California Institute of Technology.
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February 14
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Steve Shkoller -- TALK POSTPONED -- RESCHEDULED FOR MAY 1
Free-boundary problems for the incompressible and
compressible Euler equations
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February 21
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RESERVED FOR THE N. WIENER CTR FOURIER TALKS
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Abstract:
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SPECIAL APPLIED MATH SEMINAR
(Note special day and time)
MONDAY February 25
Room MTH 3206, 3:00PM
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Mac Hyman
Los Alamos National Laboratory
Paradigm Shifts in Science-Based Simulations
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Today's scientific world is experiencing a paradigm shift
where the sophistication of mathematical models, the accuracy and
efficiency of numerical algorithms, the robustness of computer software,
and the power of computation have become so great that numerical
simulations are now considered a third pillar, along with theory and
experiment, in the triad of tools used for scientific discovery. The
rate of advances in these fields, and our ability to simulate complex
physical systems, will increasingly be the limiting factors in our ability
to solve many of our most pressing scientific challenges. I will describe
recent advances in mathematical models, numerical algorithms, software,
and hardware that have allowed computer simulations of complex
multidisciplinary problems to have unprecedented impact in guiding
scientific discoveries.
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SPECIAL PDE/APPLIED MATH SEMINAR
(Note special day and time)
WEDNESDAY February 27
Room MTH 3206, 11:00AM
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Diego Cordoba
Department of Mathematics -- Institute of Mathematics and Fundamental Physics
Consejo Superior de Investigaciones Científicas, Spain
Interfase evolution: the Hele-Shaw and Muskat problem
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We study the dynamics of the interface between two
incompressible 2-D flows where the evolution equation is obtained from
Darcy~Rs law. The free
boundary is given by the differences between the densities and
viscosities of the
fluids. This physical scenario is known as the two dimensional Muskat
problem or the two-phase
Hele-Shaw flow. We prove local well-posedness in Sobolev spaces when,
initially, the
difference of the gradients of the pressure in the normal direction has
the proper sign, an
assumption which is also known as the Rayleigh-Taylor condition.
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February 28
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Xiaoyi Zhang
Institute for Advanced Study,
Princeton
The characterization of minimal mass blow up solution of focusing
mass critical NLS.
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Let u be a global solution to the focusing mass critical nonlinear schrodinger equation for radially symmetrical H^1 initial data with ground
state mass in dimension bigger or equal to 4. We prove that if u does not scatter, then up to symmetries, u is the solitrary wave. This together with
the results from F. Merle shows that the pseudo-conformal blow up and the solitary wave are the only two minimal mass blow up solutions.
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March 6
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Derek Moulton
Department of Mathematical Sciences -- University of Delaware
Mathematical modeling of electrostatic interactions with mean
curvature surfaces
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We investigate the interaction of static electric fields with surfaces of zero and constant mean curvature. Such systems have as their driving components surface
tension and electrostatic forces. The interaction of these forces is of key importance in many applications, including micro- and nanoelectromechanical systems,
self-assembly, nanolithography, and microfluidic processes. The partcicular system we explore involves a minimal surface catenoid membrane deflected by an axially
symmetric electric field. A model is formulated via variational techniques to describe equilibrium shapes of the membrane. A detailed analysis of the general
solution set is performed, with emphasis on stability and the effect of dimensionless parameters. Experimental analysis is performed using soap-film bridges and a
high voltage power source. The general effect of the electric field is determined and the utility of an electric field in such systems examined.
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March 13
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OPEN
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Abstract:
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March 20
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SPRING BREAK -- NO SEMINAR
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March 27
Aziz Lecture
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Richard James
Dept. of Aerospace Engineering & Mechanics -- University of Minnesota
Objective molecular dynamics
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Perhaps the most important deformations in solid mechanics are those that
represent the bending, twisting and extension of beams. The most important
flows in fluid mechanics are viscometric flows. In both cases these are the
motions that, when compared with the corresponding experiments, are used to
measure the material constants. We give a universal molecular level
interpretation of these motions. It is argued that all these motions are
associated at molecular level with a time-dependent invariant manifold of
the equations of molecular dynamics. The presence of this manifold can be
used to simplify molecular-level computations. Its presence also suggests a
modification of the principle of material frame-indifference, a cornerstone
of nonlinear continuum mechanics. Interesting links to theories of
turbulence, to the kinetic theory of gases (i.e., the Boltzmann equation),
to the dynamics of nanostructures, and to the Langevin equation will be
briefly discussed. Joint work with Traian Dumitrica and Kaushik Dayal.
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April 3
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Nikolaos Tzirakis
Department of Mathematics -- University of Illinois at Urbana-Champaign
Correlation Estimates and applications to Nonlinear Schroedinger
equations
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In this talk I will show how one can obtain new interaction Morawetz type
(correlation) estimates in one and two dimensions. These estimates
correspond to the nonlinear diagonal analogue of Bourgain's bilinear
refinement of Strichartz. The framework is quite general and we can
simultaneously obtain the global a priori estimates that have been
obtained before in dimensions higher than three. In higher dimensions the
proof
uses commutator vector operators acting on the conservation laws of the
equation. In one dimension we use the Gauss-Weierstrass summability method
acting on the conservation laws. I will then show several applications of
the new estimates to the semilinear Schroedinger equation.
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JOINT STATISTICS/PDE/NWC EVENT
April 10, Rm MATH 3206
(Note different, prolonged duration)
3:30PM-5:00PM
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DOCUMENTARY FILM SCREENING:
"Wolfgang Doeblin - a mathematician rediscovered''
a film by Agnes Handwerk and Harrie Willems
See also this
Statistics Announcement
Duration: 82 min
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April 17
APPLIED MATH COLLOQUIUM
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Robert V. Kohn
Courant Institute -- New York University
Cloaking by Change of Variables
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We say a region of space is "cloaked" with respect
to electromagnetic measurements if its contents -- and
even the existence of the cloak -- are inaccessible
to such measurements. One recent proposal for achieving
cloaking takes advantage of the coordinate-invariance
of Maxwell's equations. I shall explain this scheme,
including its mathematical basis and its apparent
limitations.
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April 24
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OPEN
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Abstract:
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JOINT NUMERICAL ANALYSIS/PDE APPL. MATH SEMINAR
(Note
day and time)
TUESDAY April 29
Room MTH 3206, 3:30PM--4:30PM
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Houman Owhadi
Applied & Computational Mathematics
California Institute of Technology
From Stochastic Variational Integrators to Ballistic Diffusion at
Uniform Temperature
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We present a continuous and discrete Lagrangian theory for
stochastic Hamiltonian systems on manifolds. Analogously to discrete
mechanics and variational integrators, this theory leads to structure
preserving numerical integrators for noisy mechanical systems by extremizing a discrete stochastic action. In this talk, we will focus on a particular one, designed
to approximate the solutions of Langevin equations with uniform friction and noise. This new integrator is characterized by the following two properties:
* It exactly preserves the exponential rate of decay of the symplectic
form.
* It inherits a near Boltzmann-Gibbs measure preservation property from
the near energy preservation property of its associated noise/friction-
free symplectic Euler integrator.
By applying our integrator to a fluctuation-driven magnetic motor we exhibit a paradigm for isothermal, mechanical rectification of stochastic
fluctuations.
The central idea is to transform energy injected by (random/thermal)
fluctuations into rigid-body rotational kinetic energy. We show that
although directed motion is not possible at uniform temperature, thermal
noise can be used to obtain ballistic diffusion. This is a joint work with
Nawaf Bou-Rabee.
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May 1
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Steve Shkoller
Department of Mathematics -- University of California, Davis
Free-boundary problems for the incompressible and
compressible Euler equations
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We describe a new method for treating free boundary problems in perfect
incompressible and compressible fluids, and prove local-in-time
well-posedness in Sobolev spaces for the free-surface 3D Euler equations
for arbitrary initial data, and without any irrotationality assumption on
the fluid. This is a free boundary problem for the motion of a
perfect liquid in vacuum, wherein the motion of the fluid interacts
with the motion of the free-surface at highest-order. We will describe the geometry
behind the a priori nonlinear estimates and the approximation schemes that
must be developed in order to prove existence of solutions. This is joint work
with D. Coutand and H. Lindblad.
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May 8
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OPEN
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Abstract:
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May 15
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Henrique Versieux
Instituto de Matematica Pura e Aplicada (IMPA), Brazil
Numerical analysis of a steepest-descent PDE model for surface relaxation
below the roughening temperature
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We study the numerical solution of a PDE describing the relaxation of a
crystal surface to a flat facet. The PDE is a singular, nonlinear, fourth
order evolution equation. It can be seen as the gradient flow of a convex
but non-smooth energy with respect to the $H^{-1}$ inner product.
Our numerical scheme uses implicit discretization in time and a mixed
finite-element approximation in space. The singular character of the energy
is handled using regularization, combined with a primal-dual method that
remains robust as the regularization parameter tends to zero.
We study the convergence of this scheme, both theoretically and
numerically.
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Special accomodations for individuals with disabilities can be made
by calling in advance (301) 405-5048. It would be appreciated if we are
notified at least one week in advance.