Mathematical theory of solids: From atomic to macroscopic scales

Speaker: 

Professor Weinan E

Institution: 

Princeton

Time: 

Thursday, November 29, 2007 - 4:00pm

Location: 

MSTB 254

There are no analogs of Navier-Stokes equations for solid mechanics. One reason is that information at the atomic scale seems to play a much more important role for solids than for fluids. A satisfactory mathematical theory for solids has to taken into account the behavior of solids at different scales, from electronic to atomic, to macroscopic scales.

I will discuss some of the fundamental problems that we have to resolve in order to build such a theory. I will start by reviewing the geometry of crystal lattices, the quantum as well as classical atomistic models of solids. I will then focus on a few selected problems:

(1) The crystallization problem -- why the ground states of solids are crystals and which crystal structure do they select?

(2) the microscopic foundation of elasticity theory;

(3) stability and instability of crystals;

(4) the electronic structure and density functional theory.

Subelliptic Cordes estimates

Speaker: 

Professor Juan Manfredi

Institution: 

University of Pittsburgh

Time: 

Thursday, March 13, 2008 - 4:00pm

Location: 

MSTB 254

The classical Friedrichs identity states that for $u\in C_0^{\infty}(\mathbb{R}^n)$ we have
$$\int_{\mathbb{R}^n} |D^2 u|^2\, dx = \int_{\mathbb{R}^n} |\Delta u|^2\, dx.$$
From this inequality we immediately get $W^{2,2}$-estimates for
solutions of $\Delta u =f$ and also for solutions of measurable perturbations
of the form $\sum_{ij}a_{ij}(x) u_{ij}(x)=f(x)$, when the matrix
$A=(a_{ij})$ is closed to the identity in sense made precise
by Cordes.
In this talk we first explore extensions of the Friedrichs identity in
the form of sharp inequalities
$$\int_{X} |\mathfrak{X}^2 u|^2\, dx \le C_1 \int_{X} |\Delta_{\mathfrak{X}} u|^2\, dx +C_2\int_{X} |\mathfrak{X}u|^2 \, dx $$
where $X$ is a Riemannian manifold, the Heisenberg group, and certain types of CR manifolds.
\par
We then show how to use these estimates to study quasilinear subelliptic equations.\par

This is joint work with Andr\`as Domokos (PAMS 133, 2005) and Sagun
Chanillo (2007 preprint.)

Solving the Cauchy-Riemann equations: old paradigms and new phenomena

Speaker: 

Professor Alexander Nagel

Institution: 

University of Wisconsin, Madison

Time: 

Thursday, May 17, 2007 - 4:00pm

Location: 

MSTB 254

Holomorphic functions on a domain are the solutions to a set of homogeneous partial differential
equations called the Cauchy-Riemann equation, and CR functions are the solutions to the analogous
equations on the boundary. Many problems in complex analysis can be reduced to finding appropriate
solutions to the inhomogeneous versions of these equations. These solutions have been successfully
constructed when the geometry of the domain is sufficiently simple. I hope to show how these constructions
follow a pattern based on the singular integral operators of Calder\'on and Zygmund. I then plan to
discuss some more recent examples where this well-understood paradigm breaks down.

The 16th Hilbert problem, a story of mystery, mistakes and solution

Speaker: 

Professor Oleg Viro

Institution: 

Uppsala University

Time: 

Thursday, March 8, 2007 - 4:00pm

Location: 

MSTB 254

Hilbert's problem of the topology of algebraic curves and surfaces (the
16th problem from the famous list presented at the second International
Congress of Mathematicians in 1900) was difficult to formulate. The way it
was formulated made it difficult to anticipate that it has been solved. I
believe it has, and this happened more than thirty years ago, although the
World Mathematical Community missed to acknowledge this.

Reaction-diffusion equations and biological invasions

Speaker: 

Professor Henri Berestycki

Institution: 

Ecole des hautes etudes en sciences sociales, Paris.

Time: 

Tuesday, February 6, 2007 - 4:00pm

Location: 

MSTB 254

Diffusion, along with transport and reaction effects, is the main factor explaining changes
or transitions in a wide array of situations such as flames, some phase transitions, tumours
or other biological invasions. In these systems, two or several possible states coexist, and
one observes certain states expanding or receding or patterns being formed.

This lecture, meant for a general audience, will describe some mathematical properties of
reaction-diffusion equations as an approach to spatial propagation and diffusion. After
describing the mechanism of reaction and diffusion and giving several illustrations, I will
review some classical results. In the context of ecology of populations, I will then mention
some recent works dealing with non homogeneous media. In this framework, I will describe a
model addressing the question of how a species keeps pace with a shifting climate.

Modelling dynamics of T cells in Type 1 Diabetes

Speaker: 

Professor Leah Keshet

Institution: 

UBC

Time: 

Friday, February 2, 2007 - 3:00pm

Location: 

MSTB 254

Type 1 diabetes (T1D) is an autoimmune disease in which immune cells
target and kill the insulin-secreting pancreatic beta cells.
Recent investigation of diabetes-prone (NOD) mice reveals large cyclic
fluctuations in the levels of T cells (cells of the adaptive
immune system) weeks before the onset of the disease. We extend
a previous mathematical model for T-cell dynamics to account for the
gradual killing of beta cells, and show how such cycles can arise
as a natural consquence of feedback between self-antigen and T-cell
populations. The model has interesting nonlinear dynamics
including Hopf and homoclinic bifurcations in biologically reasonable
regimes of parameters. The model fits into a larger program of
investigation of type 1 diabetes, and suggests experimental tests.

Bodies interacting with and through fluids

Speaker: 

Professor Mike Shelley

Institution: 

Courant Institute

Time: 

Thursday, February 15, 2007 - 4:00pm

Location: 

MSTB 254

The interaction of flowing fluids with free bodies -- sometimes
compliant, sometimes active, sometimes multiple -- constitutes a class
of beautiful dynamic boundary problems that are central to biology and
engineering. Examples range from how organisms locomote in fluids
(which depends strongly on scale) to how non-Newtonian stresses
develop in complex liquids (strongly dependent on the nature of
fluidic microstructure). I will discuss several interesting examples,
emphasizing how they are formulated mathematically so as to yield
models tractable for analysis or simulation, and show how this work
has interacted with experimental studies.

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