Speaker: Thomas H. Banks
Title: HIV Models: Cellular to Systemic.
Abstract: PS, PDF.
We consider several classes of mathematical models used to aid in
understanding of disease progression and possible treatment
regimes. These range from cellular level models investigated with in
vitro mouse data to systemic models supported with longitudinal
clinical data for HIV patients undergoing treatment interruptions. In
the first class we employ delay dynamical systems to investigate
intercellular pathways. Inverse problem techniques along with
sensitivity analyses are discussed. In the second class we use
nonlinear dynamical mathematical models in attempts to fit individual
patient data as well as infer population level characteristics. A
statistically-based censored data method is combined with inverse
problem techniques to estimate dynamic parameters. The predictive
capabilities of this approach are demonstrated by comparing
simulations based on estimation of parameters using only partial
longitudinal observations to the full longitudinal data sets. The
viral blips observed in patients undergoing therapy are investigated
using an expanded immune response component of the systemic models. As
time permits we will also discuss use of control theory to design
Structured Treatment Interruption (STI) therapies.
Speaker: Messoud Efendiev
Title: On some class of degenerate parabolic systems arising in the modelling of biofilms.
Abstract: PS, PDF.
In my talk a nonlinear density dependent system of reaction-diffusion
equations describing spatial spreading of biomass during the
development of microbial films is analyzed. It comprises two
non-standard diffusion effects, degeneracy as in the porous medium
equations and fast diffusion. In particular, it is shown that the global
existence (in time)of the model solutions depends on the boundary
conditions. The existence of a global attractor of the corresponding
semigroup is proved and its fractal dimension is estimated. Recent
developments related to multi-component systems as well as
hydrodynamical and chemotaxis effects will be discussed.
Speaker: Rupert Frank
Title: Lieb-Thirring-Hardy inequalities and the stability of relativistic matter.
Abstract: PS, PDF.
We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schr\"odinger-like operators remain true, with possible different constants, when the critical Hardy-weight is subtracted from the Laplace operator. Similar results are true for fractional powers of the Laplacian and, in particular, for relativistic Schr\"odinger operators. We also allow for the inclusion of magnetic vector potentials. As an application, we extend the proof of stability of relativistic matter with magnetic fields to the critical value of the nuclear charge. The talk is based on joint works with T. Ekholm and with E. H. Lieb and R. Seiringer.
Speaker: Avner Friedman
Title: Mathematical Challenges Arising in Modeling of Tumor Growth.
Abstract: PS, PDF.
I will present several mathematical models of tumor growth by a system
of PDEs in a domain with free boundary.
For some of the simplest models, I will consider the existence of
stationary solutions with many "fingers", representing a state of
metastasis.
Such solutions are obtained as bifurcation branches of spherical
solutions.
Asymptotic stability results will be stated for the spherical solution
and for the first bifurcation branch.
For more general models I shall state local existence and uniqueness
theorems, but more detailed analysis regarding the existence of
stationary spherical solutions and bifurcation branches remains to be
developed.
Speaker: Philip J. Holmes
Title: Models of legged locomotion, or How cockroaches run stably without thinking about it.
Abstract: PS, PDF.
I will discuss joint work with John Schmitt, Raffaele Ghigliazza,
Justin Seipel, Raghavendra Kukillaya and Manoj Srivivasan,
in which nonlinear mechanics and dynamical systems theory meet
biology.
Motivated by Robert Full's experimental studies of running
insects at UC Berkeley, we propose a heirarchy of models for the
dynamics of legged locomotion in the horizontal plane. We start with
energetically-conservative bipedal models (each leg corresponding to
the front/rear/opposite-middle tripod used by many insect species),
and move on to describe a central pattern generator of bursting
neurons linked via simplified muscles to more realistic hexapedal leg
geometries. We show that piecewise-holonomic mechanics due to
intermittent foot contacts can confer strong asymptotic stability. We
stress the relevance of simple models, and show how phase reduction
and averaging allow significant simplification of complex
neuromechanical models.
Speaker: Kim Knudsen
Title: Inverse boundary value problems with partial data.
Abstract: PS, PDF.
In this talk I will discuss recent progress on the inverse
conductivity problem, which is the mathematical problem behind
electrical impedance tomography. I will show that for the
three-dimensional problem measurements of the
Dirichlet-to-Neumann map on particular subsets of the boundary
uniquely determine a non-smooth conductivity. Moreover, I will
discuss the unique determination of a non-smooth magnetic field
from partial measurements of the associated Dirichlet-to-Neumann map.
Speaker: Mathieu Lewin
Title: The Thermodynamic Limit of Quantum Coulomb Systems.
Abstract: PS, PDF.
I will review the methods for proving the existence of a thermodynamic
limit for quantum systems composed of electrons and nuclei, like
ordinary matter. I will also present a new approach (joint work
with Jan Philip Solovej and Christian Hainzl) which provides a
general setting for the study of many different quantum systems.
Speaker: Benjamin Schlein
Title: Dynamics of Bose-Einstein Condensates.
Abstract: PS, PDF.
In this talk I am going to present a rigorous
derivation, starting from microscopic quantum dynamics,
of a cubic nonlinear Schroedinger equation, known as the
Gross-Piteavskii equation, for the macroscopic time evolution of
Bose-Einstein condensates. This is joint work with
L. Erdos and H.-T. Yau.
Speaker: Daniel Ueltschi
Title: A Feynman cycle approach to the Bose gas.
Abstract: PS, PDF.
Large systems of bosonic quantum particles undergo a special phase
transition called "Bose-Einstein condensation". I will recall
the basics and discuss Feynman's approach. It involves the
length of cycles of permutations that appear in the Feynman-Kac
representation. I will describe a model of random permutations
on random points, and its relation with the original Bose gas.
Speaker: Jakob Yngvason
Title: Rotating Bose Gases.
Abstract: PS, PDF.
The talk will give a brief overview of the mathematical physics of rotating Bose gases and present some recent results on gases in rapid rotation.
Speaker: Viggo Andreasen
Title: Modeling Spread and Control of Methicillin Resistant Staphylococcus Aureus (MRSA).
Abstract: PS, PDF.
Staphylococcus aureus is a wide-spread bacteria colonizing the nostrils of 25-50% of the human population. In addition the bacteria can infect soft tissue through wounds and other damages to the skin barrier; if Staphylococcus enters the blood stream, it may cause a life-threatening bacteremia ("blood poisoning").
Resistance to the preferred types of antimicrobials (methicillin) arose
shortly after the first introduction of these penicillins. In these early types of methicillin resistant staphylococcus aureus (MRSA) resistance was carried by a large gene-segment which conferred reduced fitness in an antibiotic-free environment and which effectively prohibited vertical ("plasmid") transfer of the resistance gene.
Thus MRSA was spread by solely infection processes and in countries with a low consumption of antibiotics, MRSA spread primarily in hospitals.
Restricted use of antibiotics combined with quarantine measures have
allowed the Scandinavian countries and the Netherlands to keep MRSA-prevalence at around 1% of the total staphylococcus cases. In the rest of the industrialized world MRSA accounts for 30-80% of all staphylococcus infections.
In the last decade a new and smaller resistance gene has spread rapidly
over the world. It is believed to cause little or no cost of fitness and to spread outside the hospital environment. The new MRSA-type could threaten the current control efforts in the low prevalence countries.
Together with Robert Skov at Statens Serum Institut (the Danish center for disease control) a group of mathematicians at Roskilde University have started at project that will investigate a) if the new MRSA does indeed spread outside the hospitals and b) which new control measures could be implemented.
I will report on this work and in particular discuss the two modeling
approaches which are currently used to describe the situation: detailed simulation type models describing the behavior of health care workers (which are believed to be the primary vector moving MRSA around between patients) and population level models describing the movement of individuals between hospitals and the general population. Our preliminary results suggest that MRSA is still primarily a hospital acquired infection.
Speaker: Bernhelm Booss-Bavnbek
Title: The geometry and dynamics of the bilayer membrane-vesicle fusion event in animal cells.
Abstract: PS, PDF.
I shall address a free boundary value problem related to the processes of exocytosis and endocytosis in animal cells. This is joint ongoing work with Darya Apushkinskaya (Saarbrücken) and Nina Uraltseva (St. Petersburg).
Speaker: Anna Dall'Acqua
Title: The clamped plate equation and positivity.
Abstract: PS, PDF.
A partial differential equation has the positivity preserving property if a positive
source term leads to a positive solution. The clamped plate equation
2u = f in
,
u = @
@ u = 0 on @
,
does not have this property in general planar domains. We will present some examples
of convex and non-convex domains on which it holds. Moreover, we show that the
solution operator for the clamped plate equation in two-dimensional regular bounded
domains can be split in a positive part and a possible negative part which both satisfy
the Dirichlet boundary condition. The positive part contains the singularity and the
other inherits the full regularity of the boundary.
Speaker: Michael Vogelius
Title: Cloaking in the context of Electrical Impedance Tomography.
Abstract: PS, PDF.
I shall discuss some recent results that employ a "cloak" of a highly
singular anisotropic material in order to hide, or nearly hide,
any object at a given location from detection by electrostatic
boundary measurement. This is joint work with R.Kohn, H. Shen and M. Weinstein.