we apply spectral theory to study random processes involving directed graphs. in the first half of this talk, we apply spectral tools to study orientations of graphs. we focus on counting orientations yielding strongly connected directed graphs, called strong orientations. namely, we show that under a mild spectral and minimum degree condition, a possibly irregular, sparse graph has ``many'' strong orientations. furthermore, we provide constructions that show our conditions are essentially best possible. in the second half, we examine random walks on directed graphs, which is rooted in the study of non-reversible markov chains. we prove bounds on key spectral invariants which play a role in bounding the rate of convergence of the walk and capture isoperimetric properties of the directed graph. these invariants include the principal ratio of the stationary distribution and the first nontrivial laplacian eigenvalue. finally, we conclude by briefly exploring future related work.

the problem of classifying modular categories is motivated by applications to
topological quantum computation as algebraic models for topological phases of matter.
these categories have also applications in different areas of mathematics like topological
quantum field theory, von neumann algebras, representation theory, and others.

in this talk, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular categories, and we will also give some concrete examples to have a better understanding of their structures. we will empathize some of the interesting properties that modular categories carry with them. we will give a brief overview on the situation of the classification program for this kind of categories.

in our talk we will discuss the old and celebrated question regarding the pointwise behavior of fourier series near $l^1$. this presentation will include

\begin{itemize}

\item the resolution of konyagin's conjecture (icm, madrid 2006) on the pointwise convergence of fourier series along lacunary subsequences;

\item the $l^1$-strong convergence of fourier series along lacunary subsequences.

\item recent progress on the $l^1$-strong convergence of (the full) fourier series.

\end{itemize}

we end with several considerations on the relevance/impact of the above items on the subject of the pointwise convergence of fourier series.

in this talk, a new sequential quadratically constrained quadratic programming (sqcqp) algorithm is presented for nonlinear programming. at each iteration of an sqcqp method, a quadratically constrained quadratic program (qcqp) subproblem is solved followed by a line search. if an l-infinity penalty function is used as a merit function, this method is shown to have global convergent property under the mfcq and other mild conditions. no convexity assumptions are made concerning the objective and constraints. finally numerical results from the cutest test collection will be given to justify our theoretical prediction.

we investigate the collinearity of vector time series in the frequency domain,
by examining the
rank of the spectral density matrix at a given frequency of interest. rank reduction
corresponds to collinearity at the given frequency. when the time series data
is nonstationary and has been differenced to stationarity, collinearity corresponds to
co-integration
at a particular frequency. we pursue a full understanding of rank through the
schur complements of
the spectral density matrix, and test for rank reduction via assessing the positivity of
these schur complements,
which are obtained from a nonparametric estimator of the spectral density. we provide new
asymptotic results
for the schur complements, under the fixed bandwidth ratio paradigm. the test statistics
are $o_p (1)$ under
the alternative, but under the null hypothesis of collinearity the test statistics are
$o_p (t^{-1})$, and the
limiting distribution is non-standard. subsampling is used to obtain the limiting null
quantiles. simulation study and an empirical illustration for six-variate time series
data are provided.

alongside derivative-based methods, which scale better to higher-dimensional problems, derivative-free methods play an essential role in the optimization of many practical engineering systems, especially those in which function evaluations are determined by statistical averaging, and those for which the function of interest is nonconvex in the adjustable parameters. this talk focuses on the development of a new family of surrogate-based derivative-free optimization schemes, namely delta-dogs schemes. the idea unifying this efficient and (under the appropriate assumptions) provably-globally-convergent family of schemes is the minimization of a search function which linearly combines a computationally inexpensive ''surrogate`` (that is, an interpolation or in some cases a regression, of recent function evaluations - we generally favor some variant of polyharmonic splines for this purpose), to summarize the trends evident in the data available thus far, with a synthetic piecewise-quadratic ''uncertainty function`` (built on the framework of a delaunay triangulation of existing datapoints), to characterize the reliability of the surrogate by quantifying the distance of any given point in parameter space to the nearest function evaluations.

this talk introduces a handful of new schemes in the delta-dogs family:

(a) delta-dogs(omega) designs for nonconvex (even, disconnected) feasible domains defined by computable constraint functions within a bound search domain.

(b) delta-dogs(lambda) accelerates the convergence of delta-dogs family by restricting function evaluations at each iteration to lie on a dense lattice (derived from an n-dimensional sphere packing) in a linear constraint search domain. the lattice size is successively refined as convergence is approached.

(c) gradient-based acceleration of delta-dogs combines derivative-free global exploration with derivative-based local refinement.

(d) alpha-dogsx designs to simultaneously increase the sampling time, and refine the numerical approximation, as convergence is approached.

this talk also introduces a method to scale the parameter domain under consideration based on the adaptive variation of the seen data in the optimization process, thereby obtaining a significantly smoother surrogate. this method is called the multivariate adaptive polyharmonic splines (maps) surrogate model. the judicious use of maps to identify variation of the objective function over the parameter space in some of the iterations results in neglecting the less significant parameters, thereby speeding up convergence rate.

these algorithms have been compared with existing state-of-the-art algorithms, particularly the surrogate management framework (smf) using the kriging model and mesh adaptive direct search (mads), on both standard synthetic and computer-aided shape designs such as the design of airfoils and hydrofoils. we showed that in most cases, the new delta-dogs algorithms outperform the existing ones.

in the 1960s, benjamin and feir, and whitham, discovered that a stokes wave would be unstable to long wavelength perturbations, provided that (the carrier wave number) x (the undisturbed water depth) $>$ 1.363.... in the 1990s, bridges and mielke studied the corresponding spectral instability in a rigorous manner. but it leaves some important issues open, such as the spectrum away from the origin. the governing equations of the water wave problem are complicated. one may resort to simpler approximate models to gain insights.

i will begin by whitham's shallow water equation and the modulational instability index for small amplitude and periodic traveling waves, the effects of surface tension and vorticity. i will then discuss higher order corrections, extension to bidirectional propagation and two-dimensional surfaces. this is partly based on joint works with jared bronski (illinois), mat johnson (kansas), and ashish pandey (illinois).

one way to measure the entanglement of a (pure) quantum state ${\psi}$ is the geometric measure of entanglement ${e(\psi)}$, related to the spectral norm of tensor product states. the maximal possible value of ${e(\psi)}$ on an $m$-qubit state ${\psi}$ is $m$; it is $0$ only for product states.

in quantum computation, it is tempting to think ``the more entanglement'', the better. in 2009, gross, flammia, and eisert showed that this intuition is incorrect. they proved that if ${\psi}$ is an $m$-qubit state with near maximal entanglement, ${e(\psi)>m-\delta}$, and if an np problem can be solved by a computer with the power to perform local measurements on ${\psi}$, then there is a purely classical algorithm that can solve the same problem (with positive probability) in a time only about ${2^\delta}$ times longer.

this suggests states with low entanglement are needed to get the exponential speed-up quantum computation is supposed to offer. however, as gross et. al. also show, the situation seems hopeless: with respect to the haar probability measure on all $m$ qubit states, ${e(\psi)}$ is bigger than $m$ minus log factors with very high probability as $m$ grows.

fortunately, this analysis ignores one key fact: the real quantum states that any proposed quantum computers use are boson (symmetric states), since they are built out of photons. hence, the results on entanglement of generic states do not apply.

in this talk, i will discuss my recent work with shmuel friedland, where we prove that the maximal possible entanglement for an $m$-qubit boson state is ${\log_2(m+1)}$. moreover, we show the same concentration phenomenon in this sphere: up to {\em double} log factors, with very high probability boson quantum states are maximally entangled.

transport phenomena in fluid flows are observed ubiquitously in nature
such as smoke rings in the air, pollutants in
the aquifers, plankton blooms in the ocean,
flames in combustion engines, and stirring a few
drops of cream in a cup of coffee.
we begin with examples of two dimensional hamiltonian systems
modeling incompressible planar flows, and illustrate the transition
from ordered to chaotic flows as the hamiltonian becomes more time dependent.
we discuss diffusive, sub-diffusive, and residual diffusive behaviors, and their
analysis via stochastic differential equation and a so called elephant random walk model.
we then turn to level-set hamilton-jacobi models of the flames, and
properties of the effective flame speeds in fluid flows under smoothing (such as
regular diffusion and curvature) as well as stretching.

the talk will indicate the key features in the proof of the
minimal model theorem for foliations by curves, which despite their
possibly chaotic nature more closely parallels semi-stable reduction of
curves (in arbitrary dimension) rather than the mmp for varieties. indeed
since vanishing theorems are false, it is ironically mori theory as mori
intended since everything must be done via the study of invariant rational
curves. highlights include simple local criteria for canonical foliation
singularities, a simple classification of (foliated) fano objects, and an
explicit (foliated) flip theorem by way of the study of formal
neighbourhoods of extremal rays.