variational integrators preserve geometric and topological
structure when applied to hamiltonian systems. most of the research into
variational integrators has focused upon their derivation by discretizing
hamilton's principle as a type i generating function of the symplectic
map. in this talk we examine the derivation of variational integrators
from a type ii generating function. even when the maps resulting from
different generating functions are analytically equivalent there can be
important numerical differences.
we introduce a new class of variational integrators based on the
taylor method and an augmented shooting method. the role of automatic
differentiation for an efficient implementation is discussed. finally,
a new framework for adaptive variational integrators is presented,
which is dependent upon hamiltonian variational integrators.

tensor categories have myriad uses in mathematics and physics, for
instance they appear as algebraic data associated to topological quantum
field theories and provide the framework for topological quantum
computation. what are all the tensor categories with given fusion rules?
this question can't be answered in full generality at the moment (by me)
but in this talk we discuss the classification of braided tensor
categories whose fusion rings are those of the representation rings of
classical lie groups.

a few years ago in a landmark paper, guionnet and shlyakhtenko proved the existence of free monotone transport maps from the free group factors to von neumann algebras generated by elements which have a joint law ``close'' to that of the free semicircular law. in this talk, i will discuss how to modify their idea to obtain similar results for interpolated free group factors using an operator-valued framework. this is joint work with brent nelson.

for a bounded convex domain on a riemannian manifold, the fundamental gap is the difference of the first two non-trivial dirichlet eigenvalues. in their celebrated work, b. andrews and j. clutterbuck proved the fundamental gap conjecture for convex domains in the euclidean space, showing that the gap is at least as large as the one for a one-dimensional model. they also conjectured that similar results hold for spaces with constant sectional curvature. very recently, on the unit sphere, seto-wang-wei proved that the fundamental gap is greater than the gap of the one dimensional sphere model, in particular, $\geq 3\frac{\pi^2}{d^2}$($n \geq 3$), provided the diameter of the domain $d \leq \pi/2$. in a joint work with guofang wei at ucsb, we extend seto-wang-wei's lower bound estimate to all convex domains in the hemisphere.

recent advances in single cell technology enable researchers to study heterogeneity of cell populations and dynamics of gene expression in individual cell level. one of the main challenges is to extract the salient features in a manner that reveals the underlying dynamics process.

an optimization method, single-cell low rank similarity-based method (sclrsm), is proposed for identifying cell types associated with cell differentiation and detecting cell lineage from single-cell gene expression data. sclrsm constructs structured cell-to-cell similarity matrix based on a low rank optimization model and cell types can be derived directly through the similarity matrix using non-negative matrix factorization.
the number of cell types is determined automatically via computing the eigenvalue gaps of the constructed consensus matrix while the vast majority of algorithms require the prior knowledge of such a number. in particular, the temporal order of cells is estimated by the non-negative rank one approximation of the cell-to-cell similarity matrix, which captures the global structure of the whole data. cell lineage is inferred by constructing the minimum spanning tree of the weighted cluster-to-cluster graph.

we applied our method to three different single cell data sets with known lineage and developmental time information from both mouse early embryo and human early embryo. sclrsm successfully identifies the cell subpopulations within different developmental time stages and reconstructs cell differentiation trajectories which is agreed with the previously experiments. the current results demonstrate the potential and high accuracy of the proposed method in determining cellular differentiation states and reconstructing cell lineages from single cell gene expression data.

the aim of this work (joint with amy ward [usc, marshall school of business]) is to determine fluid asymptotically optimal policies for many server queues with general reneging distributions. for exponential reneging distributions, it has been shown that static priority policies are optimal in a variety of settings, that include generally distributed interarrival and service times. moreover, in these cases, the priority ranking is determined by a simple rule known as the c-mu-theta rule. for non-exponential reneging distributions, the story is more complex. we study reneging distributions with monotone hazard rates. for reneging distributions with bounded, nonincreasing hazard rates, we prove that static priority is not necessarily asymptotically optimal. we identify a new class of policies, which we are calling random buffer selection and prove that these are asymptotically optimal in the fluid limit. we further identify a fluid approximation for the limiting cost as the optimal value of a certain optimization problem. for reneging distributions with nondecreasing hazard rates, our work suggests that static priority policies are in fact optimal, but the rule for determining the priority ranking seems more complex in general. it is work in progress to prove this.

free entropy dimension and the first $l^2$ betti number are both numeric invariants of discrete groups; one comes from voiculescu’s free probability theory and is defined by using finite matrices to ``approximate’’ the group, while the other comes from geometric group theory and is of cohomological nature. somewhat surprisingly, the two numbers are related. i will describe this connection and talk about some applications to von neumann algebras.

let $*$ be a binary operation on a set $x$ and let $x_0,x_1,\ldots,x_n$ be $x$-valued indeterminate.
define two parenthesizations of $x_0*x_1*\cdots*x_n$ to be equivalent if they give the same function from $x^{n+1}$ to $x$. under this equivalence relation, we study the number $c_{*,n}$ of equivalence classes and the largest size $\widetilde c_{*,n}$ of an equivalence class. we have $1\le c_{*,n}\le c_n$ and $1\le \widetilde c_{*,n}\le c_n$, where $c_n := \frac{1}{n+1}{2n\choose n}$ is the ubiquitous catalan number. moreover, $c_{*,n}=1 \leftrightarrow$ $*$ is associative $\leftrightarrow \widetilde c_{*,n}=c_n$. thus $c_{*,n}$ and $\widetilde c_{*,n}$ measure how far the operation $*$ is away from being associative. in this talk we will present various results on the nonassociativity measurements $c_{*,n}$ and $\widetilde c_{*,n}$, and show their connections to many known combinatorial results, assuming $*$ satisfies some multiparameter generalizations of associativity.

the graber-harris-starr theorem says that any family of smooth
rationally connected varieties over a complex curve has a section. a
natural analogue of this statement in mixed characteristic would be that
every rationally connected variety over the maximal unramified extension
of a p-adic field has a rational point. i will discuss a geometric
approach to this problem, as well as a proof of this statement for del
pezzo surfaces (for ${p>3}$). this is joint work with david zureick-brown.