We consider only two fractals: Sierpinski and Apollonian gaskets. The
idea is to show on these two examples how geometry, analysis, algebra and
number theory are tied together in the simplest problems, related to
We start with definitions, speculate on the general matrix numerical
systems, consider the analytic properties and the p-adic behavior of
harmonic functions, analyse the spectrum of the Laplace operator on the
Sierpinski gasket. Then we describe the geometry, group-theoretic
structure and arithmetic properties of the Apollonian gasket.
The final idea is to draw a parallel between the two fractals - an
I will try to give a broad review of the amasing
developments in the theory of infinite groups during the last 25 years. These include the
emergence of Monsters and flourishing of the Asymptotic Theory of Finite Groups. We will focus on important examples and formulate some open problems.
Gross and Pitaevskii proposed to model the dynamics of the Bose-Einstein
condensate by a nonlinear Schrdinger equation, the Gross-Pitaevskii
equation. This equation plays a key role in the theory and experiments of
the Bose-Einstein condensation. The fundamental mathematical question is
to derive this equation from the first principle physics law, the
many-body Schrdinger equation. In the time-independent setting, this
problem was solved by Lieb-Seiringer-Yngvason. In this lecture, we shall
review the recent progress concerning the dynamical aspects of this
problem and the analytic methods developed for quantum dynamics of
We shall review recent progress obtained in the understanding of localization
properties of random Schrodinger operators. The hard issue of the Anderson
transition is stated in terms of the spreading and of the non spreading of a
wave-packet initially located at the origin. It particular it is shown that
slow transport cannot happen for ergodic random operators. As an application,
we study quantum Hall systems, that is the Hamiltonian of an electron confined
to a two dimensional plane and subjected to a constant transverse magnetic
field. We prove delocalization around each Landau level, and localization
outside a small neighborhood of these levels.
The total variation based image denoising model of Rudin, Osher,
has been generalized and modified in many ways in the literature; one of
these modifications is to use the L1 norm as the fidelity term. We study the
interesting consequences of this modification, especially from the point of
view of geometric properties of its solutions. It turns out to have
new implications for data driven scale selection and multiscale image
This talk will describe the simulation, design and optimization of a qubit
for use in quantum communication or quantum computation. The qubit is
realized as the spin of a single trapped electron in a semi-conductor
quantum dot. The quantum dot and a quantum wire are formed by the
combination of quantum wells and gates. The design goal for this system is a
"double pinchoff", in which there is a single trapped electron in the dot
and a single (or small number of) conduction states in the wire. Because of
considerable experimental uncertainty in the system parameters, the optimal
design should be "robust", in the sense that it is far away from
unsuccessful designs. We use a Poisson-Schrodinger model for the
electrostatic potential and electron wave function and a semi-analytic
solution of this model. Through a Monte Carlo search, aided by an analysis
of singular points on the design boundary, we find successful designs and
optimize them to achieve maximal robustness.
In this talk, I will give a survey on
some of recent advances in orbifold theory and focus
on the application. It includes the computation for
cohomology of Hilbert scheme of points of algebraic surface,
symplectic resolution, twisted K-theory and many other stuff.
Tumor growth has been modeled at the macroscopic level by using established physical laws coup
led with biological processes which are described in a phenomenological fashion. Such model c
onsists of a system of PDEs in the tumor region; this region is changing in time, and thus its
boundary is a "free boundary." In this talk, I shall introduce basic material on free bounda
ry problems, and then proceed to describe models of tumor growth. I shall state results on exi
stence theorems, the shape of the free boundary, and on its asymptotic behavior as time goes t
I will discuss the proofs of some conjectural formulas
about Hodge integrals on moduli spaces of curves.
The generating series for all genera and all marked
points of such integrals are expressed in terms of
finite closed formulas from Chern-Simons knot invariants.
Such conjectures were made by string theorists based
on large N duality in string theory. I will explain
our proofs from localization techniques. Their relation
to toric Calabi-Yau manifolds and equivariant index
theory in gauge theory will also be discussed.
These are joint works with C.-C. Liu, J. Zhou and J. Li.