\indent The talk will review semidefinite programming (SDP) relaxations for polynomial optimization and show how to solve them. We propose regularization type methods to solve such large scale SDP problems. Significantly bigger problems would be solved, which is not possible by using prior existing methods like interior-point algorithms. Numerical examples will also be shown.

We consider the following dynamic Boolean model introduced by
van den Berg, Meester and White (1997). At time $0$, let the nodes of the
graph be a Poisson point process in $R^d$ with constant intensity and let
each node move independently according to Brownian motion. At any time $t$,
we put an edge between every pair of nodes if their distance is at most $r$.
We study two features in this model: detection (the time until a target
point--fixed or moving--is within distance $r$ from some node of the graph),
coverage (the time until all points inside a finite box are detected by
the graph) and percolation (the time until a given node belongs to the
infinite connected component of the graph). We obtain asymptotics for
these features by combining ideas from stochastic geometry, coupling and
multi-scale analysis. This is joint work with Yuval Peres, Alistair
Sinclair and Alexandre Stauffer.

Given a Riemannian manifold $M$, when can it be isometrically
embedded in Euclidean space? When can a local isometry be found, and
when can a global isometry be found? What is the minimum dimension of
the target Euclidean space if $M$ has dimension $n$? These questions have
been extensively studied during the last century, with perhaps the
best known result being the famous Nash embedding theorem from the
1950's. The aim of this talk is to introduce the problem and some of
the well known results. The talk is intended to be more of a history
lesson rather than technical, so there will be minimal discussion of
proofs, and no background required.

\indent This talk will introduce a new class of models which can lead to efficient estimation in multivariate analysis. Some members in the class include the basic envelope model, partial envelope model, inner envelope model, scaled envelope model, and heteroscedastic envelope model. They have the common word ``envelope'' in their names because they are all constructed by enveloping: use reducing subspaces to connect the mean function and the covariance function, so that the number of parameters can be reduced. The application of enveloping is very broad and can be used in many contexts to control parameterization. Theoretical results, simulations and a large number of data examples show that the efficiency gains obtained by enveloping can be substantial.

\indent The eigenvalues of the Laplacian (or Laplacian with potential) on a smoothly bounded domain domain are very natural quantities, arising as the fundamental tones of a drum, the rates of decay of diffusions, and the energy levels of quantum systems. I will discuss some of the history relating to inequalities for low eigenvalues, leading up to the proof of a conjecture of Yau and van den Berg for the `fundamental gap' or excitation energy of a convex domain.

\indent In this talk, we will discuss about Li-Yau-Hamilton type
differential Harnack inequalities, heat kernel estimates and their
applications to study type I ancient solutions of the Ricci flow.