The harmonic oscillator problem in quantum mechanics is to find
operators $a$ and $b$ acting on a Hilbert space satisfying the relation
$ab-ba=1$. This is one of the physical motivations behind studying the
Weylalgebra and the enveloping algebra of the Heisenberg Lie algebra.
In this talk, I will present a two-parameter version of this problem and discuss some of the subtleties in looking for simple, primitive factor rings in quantum enveloping algebras.

We discuss how randomization and rank structures can be used in the direct solution of some large dense or sparse linear systems with nearly $O(n)$ complexity and storage. Randomized sampling and related adaptive strategies help significantly improve both the efficiency and the flexibility of structured solution. We also demonstrate how these can be extended to the development of matrix-free direct solvers based on matrix-vector products only. This is especially interesting for problems with few varying parameters (e.g., frequency or shift). We show that such structured solvers also have significantly better stability than classical LU factorizations. They are then very suitable for ill-conditioned problems and can provide solutions with controllable accuracies. Applications to some difficult situations will be demonstrated and the effectiveness will be justified: 1. preconditioning certain indefinite problems where only matrix-vector products are available; 2. solving ill-conditioned Toeplitz least squares problems; 3. finding good initial estimates for iterative (e.g., Newton) eigenvalue solutions.

I will talk about the problem of separating multiple signals from each other when we only have access to a few linear (or non-linear) combinations of them. An example of this type of problem is at a cocktail party when you are trying to have a conversation with a friend but there are several conversations happening around you. Your ears provide you with a superposition of all the voices, and your brain does remarkably well at focusing on your friend's voice and drowning out all the others. We will talk about one computer algorithm (or time permitting, more) that does such a task (reasonably) successfully. Along the way, we will talk about important tools in mathematical signal processing, including the Fourier transform and sparsity.

A shortest remaining processing time (SRPT) queue is a single server queue in
which the server dedicates its effort to the job that has the least remaining
processing time among all jobs in the system. This is done with preemption
so that when the total processing time of an arriving job is less than that
remaining for the job in service, service of the job in service is paused and
the newly arriving job enters service. Shortest remaining processing time is of interest because it is performance optimal in the sense that over all non-idling service disciplines it minimizes queue length. However, this may come at the expense of long delays for jobs with large total processing times. From a mathematical point of view, SRPT is challenging to analyze, in part because the preemptive nature of SRPT leads to an infinite dimensional state space. By formulating and analyzing a probabilistic model for SRPT that employs measure-valued processes, we investigate this performance trade off. The talk will begin with a discussion of a fluid limit (first order approximation) for the stochastic model. This will yield some interesting insights about the anticipated performance trade-off. Next work in progress on a diffusion limit (second order approximation) will be discussed. Various parts of this work are joint with D. Down (McMaster University), H. C. Gromoll (U Virginia), and L. Kruk (Maria Curie-Sklodowska University)

General Relativity (GR) suffers, in several ways -- from source modeling to
data analysis--, from the "curse of dimensionality", by which it is here
roughly meant that the complexity of the system grows beyond practical
control as more physical parameters of interest are taken into account.

This is a very concerning, practical bottleneck for the upcoming generation
of advanced gravitational wave detectors, worth billions of dollars, which
is expected to detect within a few years gravitational waves in a direct
way for the first time in history and reach unexplored portions of the
universe. This is the most anticipated era of general relativity since the
study of Hulse and Taylor which lead to their Nobel prize in 1993.

Due to the low signal-to-noise ratio of any expected detection, matched
filtering and catalogs of templates are needed, both for detection and
parameter estimation of the source of any trigger.

Numerical relativity simulations of the Einstein equations typically take
hundreds of thousands of hours, making a survey of the full parameter space
intractable with standard search methods. Parameter estimation algorithms
are just impractical, would take years of computing time.

I will first summarize previous work I have involved in and the
state-of-the-art of Einstein's equations as an initial-boundary value
problem, both at the continuum and discrete level. Next I will discuss my
current research program, in collaboration with many colleagues, to tackle
the curse of dimensionality in GR, existing results, and plans for the
future. The effort is essentially about dealing with parametrized systems
using reduced order models (ROM), and the techniques are generic and
applicable to many areas.

In our driving field, general relativity, we typically obtain several (from
3 to 11) orders of magnitude of computational speedup compared to direct
approaches, both on the modeling and data analysis sides. As an example, we
can now substitute a typical months-long supercomputer simulation of
colliding black holes with a surrogate model that can run on a smartphone
within a tenth of a millisecond without loss of accuracy. Real time
calculations on mobile devices not only provide a huge opportunity of
outreach about complicated systems such as colliding black holes, but also
application-specific ones in many areas, such as on-site or remote design
and control.

The effort is a combination of theoretical physics, analytical and
numerical methods for partial differential equations, scientific computing,
large scale computations, reduced order modeling, approximation theory,
sparse representations and signal processing.

It is well know that water-mediated interactions play critical roles in biomolecular recognition processes. Decades of theoretical and experimental studies showed us a very complicated picture. No single reliable model can provide simple and consistent descriptions to its role in kinetics, thermodynamic, and structural characterizations yet.

However, a joint explicit solvent molecular dynamics (MD) simulations and the variational implicit-solvent model (VISM) may still provide semi-quantitative insight into these complicated heterogeneous hydration, the solute-solvent interface, and individual interesting water molecules around proteins.

In this study, we used this combination approach to study the hydration properties of the biologically important p53/MDM2 complex. Unlike simple model solutes, in such a realistic and heterogeneous solute-solvent system with both geometrical and chemical complexity, it occurs that the local water distribution sensitively depends on nearby amino acid properties and the geometric shape of the protein. We show that the VISM can accurately describe the locations of high and low density solvation shells identified by the MD simulations, and can explain them by a local coupling balance of solvent-solute interaction potentials and curvature. In particular, capillary transitions between local dry and wet hydration states in the binding pocket are captured for inter-domain distance between 4 to 6 � right at the onset of binding. The underlying physical connection between geometry and polarity are illustrated and quantified. Our study offers a microscopic and physical insight into the heterogeneous hydration behavior of the biologically highly relevant p53/MDM2 system and demonstrates the fundamental importance of hydrophobic effects for biological binding processes. We hope this study may help to establish new design rules for drugs and medical substances.

A character on a group is a class function of positive type. For finite groups, the classification of characters is directly connected to the representation theory of the group and plays a key role in the classification of finite simple groups. Based on the rigidity results of Mostow, Margulis, and Zimmer, it was conjectured by Connes that for lattices in higher rank Lie groups the space of characters should be completely determined by the finite dimensional representations of the lattice. In this talk, I will give an introduction to this conjecture (which has now been solved in a number of cases), and I will discuss its relationship to ergodic theory, abstract harmonic analysis, invariant random subgroups, and von Neumann algebras.