We study how to solve sum of squares (SOS) and Lasserre's
relaxations for large scale
polynomial optimization. When interior-point type methods are used,
typically only small
or moderately large problems could be solved. This paper proposes the
regularization
type methods which would solve significantly larger problems. We first
describe these
methods for general conic semidefinite optimization, and then apply
them to solve large
scale polynomial optimization. Their efficiency is demonstrated by
extensive numerical
computations. In particular, a general dense quartic polynomial
optimization with 100
variables would be solved on a regular computer, which is almost
impossible by applying
prior existing SOS solvers.

Venn diagrams are tools used to represent relationships among
sets. They are easy to understand but can be difficult to draw if they
involve more than three sets. The quest for a method to construct symmetric
Venn diagrams has led to some interesting theorems about partially ordered
sets. We describe several of these theorems, their relationship to Venn
diagrams, and a conjecture that unifies this research.

There has been much recent progress in the study of Ricci solitons on nilpotent and solvable Lie groups. In this talk, I will define the Ricci Yang-Mills flow which is related to the Ricci flow. I will also define Ricci Yang-Mills solitons, which are generalized fixed points of the Ricci Yang-Mills flow. These metrics are related to Ricci solitons; however, they are defined on principal G-bundles and are designed to detect more of the bundle structure. On nilpotent Lie groups, one can say precisely in what sense Ricci Yang-Mills solitons are related to Ricci solitons. I will provide examples of 2-step nilpotent Lie groups that admit Ricci Yang-Mills solitons but that do not admit Ricci solitons. This is joint work with Mike Jablonski.

The Wiener Chaos is a natural orthogonal decomposition of the $L^2$ space of a Brownian motion, naturally associated to stochastic integration theory; the orders of chaos are given by the range of multiple Wiener-Ito integrals.

In 2006, Nualart and collaborators proved a remarkable central limit theorem in the context of the chaos. If $X_k$ is a sequence of $n$th Wiener-Ito integrals (in the $n$th chaos), then necessary and sufficient conditions that $X_k$ converge weakly to a normal law are that its (second and) fourth moments converge -- all other moments are controlled by these.

In this lecture, I will discuss recent joint work with Roland Speicher in which we prove an analogous theorem for the empirical eigenvalue laws of high-dimensional random matrices.

The concept of average is highly useful (and much maligned) concept in all of mathematics and in life. However, few people stop to think about what an average really \emph{is}. As it turns out, it is a very important theoretical concept in mathematics, and it isn't just something that helps one lie with statistics. It is really the heart of measure and integration theory. In this talk we'll learn how measure theory and integration unifies various different kinds of averages, and one big result: Jensen's inequality, and its applications to relating more exotic means to one another.

\footnotesize Single-molecule force spectroscopy experiments involve imposition of controlled forces at the single molecule level and observing the corresponding mechanical behavior of the molecule. The molecular resistance to deformation can be utilized for studying transition pathways of molecules in terms of energy, time scales and even number of transition states. These have found applications in a wide variety of problems, for instance, to understand folding-unfolding dynamics of biomolecules, ligand-receptor binding, transcription of DNA by RNA polymerase, motion of molecular motors to name a few.
Existing analyses of force measurements rely heavily on theoretical models for reliable extraction of kinetic and energetic properties. Despite significant advances, there remain gaps in fully exploiting the experiments and their analyses. Specifically, the effect of pulling device stiffness or compliance has not been comprehensively captured. Hence, the best models for extracting molecular parameters can only be applied to measurements obtained from soft pulling devices (e.g., optical tweezers) and result in well-documented discrepancies when applied to stiff devices (e.g., AFM). This restriction makes pulling speed the only control parameter in the experiments, making reliable extraction of molecular properties problematic and prone to error. \\

Here, we present a one-dimensional analytical model derived from physical principles for extracting the intrinsic rates and activation free energies from rupture force measurements that is applicable to the entire range of pulling speeds and device stiffnesses. The model therefore is not restricted to the analyses of force measurements performed with soft pulling devices only. Further, the model allows better design of experiments that specifically exploits device stiffness as a control parameter in addition to pulling speed for a more reliable estimation of energetic and kinetic parameters. The model also helps explain previous discrepancies noted in rupture forces measured with devices of different effective stiffnesses and provides a framework for modeling other stiffness-related issues in single-molecule force spectroscopy.

\footnotesize Single-molecule force spectroscopy experiments involve imposition of controlled forces at the single molecule level and observing the corresponding mechanical behavior of the molecule. The molecular resistance to deformation can be utilized for studying transition pathways of molecules in terms of energy, time scales and even number of transition states. These have found applications in a wide variety of problems, for instance, to understand folding-unfolding dynamics of biomolecules, ligand-receptor binding, transcription of DNA by RNA polymerase, motion of molecular motors to name a few.
Existing analyses of force measurements rely heavily on theoretical models for reliable extraction of kinetic and energetic properties. Despite significant advances, there remain gaps in fully exploiting the experiments and their analyses. Specifically, the effect of pulling device stiffness or compliance has not been comprehensively captured. Hence, the best models for extracting molecular parameters can only be applied to measurements obtained from soft pulling devices (e.g., optical tweezers) and result in well-documented discrepancies when applied to stiff devices (e.g., AFM). This restriction makes pulling speed the only control parameter in the experiments, making reliable extraction of molecular properties problematic and prone to error. \\

Here, we present a one-dimensional analytical model derived from physical principles for extracting the intrinsic rates and activation free energies from rupture force measurements that is applicable to the entire range of pulling speeds and device stiffnesses. The model therefore is not restricted to the analyses of force measurements performed with soft pulling devices only. Further, the model allows better design of experiments that specifically exploits device stiffness as a control parameter in addition to pulling speed for a more reliable estimation of energetic and kinetic parameters. The model also helps explain previous discrepancies noted in rupture forces measured with devices of different effective stiffnesses and provides a framework for modeling other stiffness-related issues in single-molecule force spectroscopy.