Tuesdays at 4:10 in
We study weak-* closed masa-bimodules generated by A(G)-invariant subspaces of vN(G). An annihilator formula is established, which is used to characterise the weak-* closed subspaces of B(L(G)) which are invariant under both Schur multipliers and the action of M(G) via completely bounded maps. We study the special cases of extremal ideals having a given null set and, for a large class of groups, establish a link between relative spectral synthesis and relative operator
Nate Dean (Texas State) Some edge length problems and applications
A graph is a combinatorial structure consisting of a set of objects called vertices and a set of edges defined as unordered pairs of vertices. First, we consider certain unsolved problems where length is associated with the edges of an infinite graph (i.e., chromatic number of the plane and Ulam's problem). Then we investigate the number theoretic and geometric structure of related finite graphs (i.e., unit and rational distance graphs). Finally, we discuss how attempts to solve these problems, to generalize them, and to adapt them for certain applications leads to ongoing research in mathematical programming, nanotechnology, algebraic geometry, data visualization, and other areas.
Dean Beate Schmittmann on First attempts at characterizing interacting networks
Many physical networks, such as the internet, the power grid, or the interstate system, have been characterized in considerable detail, but in isolation from each other. Yet, each of these networks supports the functions of the others, and so far, little is known about how their interactions affect their structure and functionality.
We will present a first study of two coupled model networks. Each individual network is generated by a simple set of dynamic rules: Given a set of nodes, links are created and destroyed according to a prescribed rate function which favors a specific degree. The resulting stationary degree distribution can be computed analytically, in excellent agreement with simulation results. We consider two such networks, with different preferred degrees, reminiscent of two social groups, e.g., extroverts and introverts. The two networks are allowed to interact by establishing a controllable fraction of cross links. The resulting distribution of links, both within and across the two model networks, is investigated and discussed, along with some potential consequences for real networks. Large fluctuations and slow dynamics are observed under certain conditions.
In this presentation we discuss the guiding principles of active learning, particularly in mathematics, and discuss three different pedagogical methods. Further, we provide some empirical and anecdotal evidence of the benefits of these methods.
Chelsea Walton (MIT) An introduction to noncommutative invariant theory
In the first part of this talk, I aim to convince you that invariant theory is a beautiful field. This area dates back over 100 years to the work of Hilbert, Klein, Gauss, and many others. It is a very active area of research today, particularly from the viewpoint of algebraic geometry and combinatorics. It also has far reaching applications in representation theory, coding theory, mathematical modelling, and even air target recognition. (I just happened to run across this last application on google; it will *not* be explained.)
In the remainder of this talk, I hope to illustrate the wonders of a new and fast-paced field: noncommutative invariant theory. All basic notions will defined. To say, I will explain the noncommutative analogues of each of the following terms: "groups", "acting on", and "polynomial rings". I will then contrast classical results of (commutative) invariant theory with the current results of several individuals in noncommutative invariant theory. Many examples will be provided.
Zhimin Zhang (Wayne State) Superconvergence of polynomial spectral interpolation
Superconvergence properties for some high-order orthogonal polynomial interpolations are studied. The results are twofold: When interpolating function values, we identify those points where the first and second derivatives of the interpolant converge faster; When interpolating the first derivative, we locate those points where the function value of the interpolant superconverges.
Daniel Toundykov (Nebraska) Geometry and stability of the 2D von Karman plate model
I will begin by revisiting the classical result known as Alexandrov's maximum principle which helps estimate the values of a function via the curvature of its graph. This inequality arises in the study of the von Karman model describing mechanical vibrations of thin plates. I will give an overview of the von Karman equations and will discuss their geometric meaning in relation to the Alexandrov's principle. These results provide insights into the asymptotic stability of solutions to the von Karman system. I will present some of the recent developments in this direction.
Songting Luo (ISU) on Numerical methods for Helmholtz Equations in the high frequency regime beyond geometric optics
in this talk. We will discuss a new numerical method for computing the highly oscillating waves of the Helmholtz equations in the high frequency regime. The method is based on the-state-of-art geometrical optics, along with new treatments to overcome the limitations of geometrical optics. Numerical examples will be presented to demonstrate the new method. (joint with Profs. R. Burridge (UNM), J. Qian(MSU), and H. Zhao (UCIrvine).
Ilya Krishtal (NIU) on Dynamic sampling
We will discuss a few ideas for sampling signals that evolve in time. The standard sampling theory is not designed to take into account the additional information provided by sampling a signal at different time levels. Using this dependency it is sometimes possible to recover the signal at a finer resolution from multiple coarse snapshots obtained at appropriate time intervals. It may also be possible to identify an unknown underlying evolution operator or its key parameters. We shall present several simple examples illustrating these ideas. The talk is based on joint work with A. Aldroubi and J. Davis. The talk is accessible to a broad mathematical audience as it relies mostly on linear algebra and basic Fourier analysis.
Chi-Kwong Li (College of William and Mary) Factorization of permutation matrices
We discuss some results and problems concerning factorization of permutation matrices arising in various areas including algebra, combinatorics, computer science, numerical linear algebra, optimization, genomics, quantum computing, etc.
We report on two Illinois State University programs that are designed to engage mathematics undergraduates and high school teachers in research. The first program is a course that has been run at Illinois State every spring since 2004. The second program is a Research Experiences for Undergraduates (REU) Site for pre-service and in-service teachers. The REU Site is funded by the National Science Foundation and has been run every year since 2007. This presentation will describe the successes and challenges of these programs, sample research topics, components designed to help teachers translate their research experience to the classroom, and suggestions for implementation.
Amanda Ruiz (Binghamtom U) Realization of phased matroids
A phased matroid, recently defined by Anderson and Delucchi, is a combinatorial abstraction of a finite set of vectors in complex space. The phased matroid is a tool for keeping track of some of the geometric information of the set of vectors. Just as complex numbers are an extension of real numbers, phased matroids are an extension of oriented matroids, which are a well studied field.
The realization space of an oriented (resp. phased) matroid is the sets of vectors in $\R^n$ (resp $\C^n$) that correspond to oriented (resp. phased) matroid. A big question in oriented matroid theory is what kind of topology can the realization space of an oriented matroid have. In this talk I address the analogous question for phased matroids.
According to Mnëv's Universality Theorem, the realization space of an oriented matroid can be arbitrarily complicated. Mnëv's result also applies to phased matroids which are complexified oriented matroids. In contrast, for most other phased matroids, the realization space is remarkably simple. I will focus on the uniform case to demonstrate some properties and proofs of phased matroids
Algorithmic randomness defines what it means for a single mathematical object to be random. This active area of computability theory has been particularly fruitful in the past several decades, both in terms of expanding theory and increasing interaction with other areas of math and computer science. Randomness can be equivalently understood in terms of measure theory, Kolmogorov complexity (incompressibility), and martingales.
In this context, we present a novel definition of betting strategies that uses probabilistic algorithms also studied in complexity theory. This definition leads to new characterizations of several central notions in algorithmic randomness and addresses Schnorr's critique, a longstanding philosophical question in algorithmic randomness. Moreover, these techniques suggest new approaches for tackling one of the biggest open questions in the field (KL = ML?). This is joint work with Sam Buss.
Consider two random subspaces V and W of C^d. Their intersection has dimension at least dim(V)+dim(W)-d; if there is equality (or 0 if dim(V)+dim(W)<d) then V and W are said to be in general position. If V and W are random subspaces, they are almost surely in general position. Depending on the definition of ``almost sure'', this easy theorem dates back 150 years or more. A more modern form could be stated thus: let U_t be a Brownian motion on the unitary group (of rotations of C^d), independent from V and W; then, for any t>0, U_t(V) and W are in general position with probability 1.
What happens when the dimension d is infinite? While there is no unitarily invariant measure in infinite dimensions, it is still possible to make sense of the Brownian motion acting on some subspaces. However, the easy techniques for proving the general position theorem are unavailable.
In this talk, I will address my recent joint work with Benoit Collins in this infinite dimensional setting. Using the tools of random matrix theory and free probability, the question can be reduced to one of regularity for the solution of a certain family of complex PDEs. This allows us to prove not only the general position theorem, but also (a special case of) a long-standing conjecture about free entropy and information.
In this talk I will talk about the study of stochastic differential equations that have components that evolve and interact at different time scales (multi-scale SDE's). I will review and give recent developments on the theory and practice of systems of this kind with especial emphasis on the theory of large deviations, its applications, its connections with dynamical systems, its limitations and possible extensions. Also, we will review the connections (and formulation) of multi-scale stochastic differential equations with the universality conjecture from Statistical Physics and the stochastic PDE known as KPZ (Kardar-Parisi-Zhang) equation.
We consider the unique infi nite connected component of
supercritical bond percolation on the square lattice and
study the geometric properties of isoperimetric sets, i.e., sets with minimal boundary for a given volume. For almost
every realization of the infi nite connected component we prove that, as the volume of the isoperimetric set tends
to infi nity, its asymptotic shape can be characterized by an isoperimetric problem in the plane with respect to a
particular norm. As an application we then show that the anchored isoperimetric profi le with respect to a given point
as well as the Cheeger constant of the giant component in fi nite boxes scale to deterministic quantities. This settles a
conjecture of Itai Benjamini for the plane.
Stochastic networks arise as models in various areas including
computer systems, telecommunications, manufacturing,
fi nance, and service industry. The networks are often too complex to be analyzed directly and thus one seeks suitable
approximate models. One class of such approximations are diffusion models that can be rigorously justifi ed when networks
are operating in heavy traffi c, i.e., when the network capacity is roughly balanced with network load. In this talk, some recent
study on Markov modulated stochastic networks in heavy traffi c will be presented. We fi rst consider generalized Jackson
networks with Markov modulated arrival and service rates and routing structure, and develop suitable reduced models using
techniques from diffusion approximations and heavy traffi c theory. We then develop a comprehensive stability theory
for such Markov modulated stochastic networks and their diffusion limit. At last, we study optimal control problems for
Markov modulated multiclass single-server queueing systems in heavy traffi c, and establish an asymptotically optimal policy
which is an “average” cμ rule.
Linear algebra is very useful in many areas of combinatorics such as the theory of expander graphs, the theory of strongly and distance-regular graphs or graph and hypergraph decomposition.
I will describe some of the applications of linear algebra in these areas. The talk will be accessible to undergraduate students.
Edmond Jonckheere (USC) Decoherence splitting manifold--a geometric approach to decoherence control
The geometry of decoherence control relies on the concept of Decoherence Splitting Manifold (DSM). It is a real analytic submanifold of the space of density operators characterized by a tangent space that splits into an eigenspace of unitarily preserved eigenvalues, not subject to decoherence, and an eigenspace of time-varying eigenvalues, subject to decoherence. There are many such DSM’s in the space of density operators and it is noteworthy that their algebraic geometric properties are decoupled from the master equations. The crucial condition to achieve decoherence control is that the reachability operator of the Lindbladian is included in the DSM. Naturally, one would like to choose the DSM in such a way that its eigenspace of unitarily preserved eigenvalues is maximum dimensional—with the help of control. Our major focus has been to characterize what controls are able to enlarge this eigenspace. Local magnetic fields cannot achieve this objective, but the back-action of measurements can do it. Our future work will focus on the algebraic- topological properties of the DSM, e.g., the characteristic classes of the vector bundle of unitarily preserved eigenvalues (in cooperation with D’Alessandro), and the utilization of an auxiliary system as a mean to achieve decoherence control (in cooperation with K. Jacobs.)
The question of whether one can recover a function from its integral has a long history. For example, Lebesgue's Differentiation theorem tells us that we can recover a function f from averages of of its integral taken over successively smaller intervals. In higher dimensions, this becomes a statement about balls-- but a succession of balls of decreasing volumes is not the only possible scenario. We will look at how the geometry of the sets over which averages are taken influences their convergence or divergence, paying special attention to averages taken over sequences of rectangles.
I'll first review some of the milestones in the development of Control Theory in
application to partial differential equations. Then I'll present an overview of some results
and mathematical issues that arise in the analysis and control theory of some layered beam and plate models that are of interest in the modeling of composite structures.
We study a reduced Poisson--Nernst--Planck (PNP) system for a charged spherical solute immersed in a solvent with multiple ionic or molecular species that are electrostatically neutralized in the far field. Some of these species are assumed to be in equilibrium. The concentrations of such species are described by the Boltzmann distributions that are further linearized. Others are assumed to be reactive, meaning that their concentrations vanish when in contact with the charged solute. We present both semi-analytical solutions and numerical iterative solutions to the underlying reduced PNP system, and calculate the reaction rate for the reactive species. We give a rigorous analysis on the convergence of our simple iteration algorithm. Our numerical results show the strong dependence of the reaction rates of the reactive species on the magnitude of its far field concentration as well as on the ionic strength of all the chemical species.
We also find non-monotonicity of electrostatic potential in certain parameter regimes.The results for the reactive system and those for the non-reactive system are compared to show the significant differences between the two cases. Our approach provides a means of solving a PNP system which in general does not have a closed-form solution even with a special geometrical symmetry. Our findings can also be used to test other numerical methods in large-scale computational modeling of electro-diffusion in biological systems.
The discontinuous Galerkin (DG) method is a class of finite element method that use completely discontinuous piecewise function space for the numerical solution and the test function. Without the continuity restriction, DG methods have the flexibility which is not shared by classical finite element methods, such as the allowance of arbitrary triangulation with hanging nodes, complete freedom in changing the polynomial degrees in each element, and extremely local data structure and the resulting high parallel efficiency.
In this talk, I will discuss our recent studies on the Direct Discontinuous Galerkin(DDG) methods for convection diffusion equations. We proposed a general numerical flux formula for the solution derivative and a new DG method based on the direct weak formulation of diffusion equation was developed. Then I will present some variations of the DDG method. The symmetric DDG gives us the optimal L2(L2) error estimate and the nonsymmetric DDG leads to the recovery of optimal convergence when comparing to Baumann-Oden method and NIPG method.
We then prove the DDG method and its variations all satisfy the strict maximum principle with quadratic polynomial
approximations. Sufficient conditions are given to guarantee the polynomial solutions bounded above and below by the
given constants. Extension to two-dimensional rectangular meshes and triangular meshes will be
discussed. Numerical examples will be presented to verify the optimal 3rd order of accuracy is maintained with the maximum principle limiter. The positivity of the polynomial solutions are maintained sharply for nonlinear porous medium equations.
Research in biological motors and recent advances in DNA nanofabrication technology have spurred a lot of interest in biomimetic nanomotor designs and DNA-based devices, such as nanomechanical switches and DNA templates for the growth of semiconductor nanocrystals, to name a few. Research activity in this area has been focused on designing and controlling dynamic DNA nanomachines that can be activated by and respond to specific chemical signals in their environment.
In this talk, we formulate and analyze a Markov process modeling the motion of DNA nanomechanical walking devices. We consider a molecular biped restricted to a well-defined one-dimensional track and study its asymptotic behavior. Our main result is a functional central limit theorem for the biped with an explicit formula for the effective diffusion coefficient in terms of the parameters of the model. A law of large numbers and a recurrence/transience characterization are also obtained. Our approach is applicable to a variety of other biological motors such as myosin and motor proteins on polymer filaments.
This is a talk about mathematics, probability, and some neat biological applications, but also about some improbable collaborations, and the friendships that emerged.
This is joint work with Iddo Ben-Ari and Alexander Roitershtein.
Quantum probability and non-locality have opened the question about the completeness of quantum mechanics, and have triggered the development of theoretical schemes which should complete it (the so-called hidden variables program). While, following the results of Bell, non-locality has been widely accepted as an intrinsic property of the microscopic world, the hidden variable program still represents an important field of research. In this talk an introduction to the topic is presented, with special emphasis on some recent results on crypto-nonlocal theories.
 J. S. Bell, "On the Einstein-Podolsky-Rosen paradox", Physics 1,
 G.C. Ghirardi and R. Romano, "Local parts of hidden variable models for maximally entangled bipartite qudits", Physical Review A 86, (2012) 022107
In this talk, I will discuss our recent work on Recursive Sparse Recovery (RecSparsRec) and show how it provides novel solutions to two very different problems in dynamic imaging. RecSparsRec refers to recursive approaches to causally recover a time sequence of signals/images from a greatly reduced number of measurements (compared to existing approaches), by utilizing their sparsity.
The motivating application for RecSparsRec is fast recursive dynamic magnetic resonance imaging (MRI) for real-time applications like MRI-guided surgery. MRI is a technique for cross-sectional imaging that acquires Fourier projections of the cross-section to be reconstructed, one-at-a-time. Thus, the ability to accurately reconstruct using fewer measurements directly translates into reduced scan times. This, along with online (causal) and fast (recursive) reconstruction algorithms, can enable real-time imaging of fast changing physiological phenomena, and thus make real-time MRI feasible. Cross-sectional images of the brain, heart, or other organs are known to be wavelet sparse. Our recent work was the first to observe that, in a time sequence, their sparsity pattern changes quite slowly. Using this fact, we were able to reformulate the RecSparsRec problem as one of sparse reconstruction with partially known support. We introduced a simple, but very powerful, approach called Modified-CS that achieves provably exact reconstruction (in the noise-free case) and whose error is provably stable over time (in the noisy case), with using much fewer measurements than existing work. Our preliminary experiments indicate that Modified-CS needs roughly 5-times fewer measurements than existing MR scanner technology and 1.5-times fewer than existing research literature.
I will briefly also discuss our ongoing work on the difficult video analysis problem of separating foreground moving objects from a background scene that is itself is changing and dong this in real-time. This can be posed as a recursive robust principal components analysis (PCA) problem in the presence of correlated sparse outliers or equivalently, as a problem of recursive sparse recovery in the presence of very large, but ``low rank" noise (noise with a low rank covariance matrix).
We investigate certain random processes on graphs which are related to the so-called Tsetlin library random walks as well as to some variants of classical voter model. A specific example can be described as a hypergraph coloring game. Each node represents a voter and is colored according to its preferred candidate, or undecided.
Each hyperedge is a subset of nodes and can be viewed as a chat group.
In each round of the game, one chat group is chosen randomly, and voters in the group can change colors based on interactions. We analyze the game as a random walk on the associated weighted directed state graph. Under certain "memoryless" conditions, the spectrum of the state graph can be explicitly determined by using semigroup spectral theory. It can be shown that random walk on the state graph converges to its stationary distribution in O(m log n) time, where n denotes the number of voters and m denotes the number of chat groups.
This can then be used to determine the appropriate cut-off time for voting. We also consider a partial memoryless version which can be used to approximate general voter games
September 14 (Friday) - Tong Li (UI) on Global entropy solutions to a quasilinear hyperbolic system modeling blood flow
This talk is concerned with an initial-boundary value problem on bounded
domains for a one dimensional quasilinear hyperbolic model of blood flow
with viscous damping. It is shown that L entropy weak solutions exist globally in time when the initial data are large, rough and contains vacuum
states. Furthermore, based on entropy principle and the theory of divergence measure field, it is shown that any Lentropy weak solution converges to
a constant equilibrium state exponentially fast as time goes to infinity. The physiological relevance of the theoretical results obtained in this paper is
demonstrated. This is a joint work with Kun Zhao.
September 14 ( Lihe Wang (UI) on Connection between continuity, integrality and critical regularity of elliptic equations
This talk will give a survey of the classical regularity theory as continuity and integral estimates.
A proof of Hopf lemma is discussed in the exact conditions.
We give a survey of our recent work with collaborators on the construction of uniformly high order accurate discontinuous Galerkin (DG) and weighted essentially non-oscillatory (WENO) finite volume (FV) and finite difference (FD) schemes which satisfy strict maximum principle for nonlinear scalar conservation laws, passive convection in incompressible flows, and nonlinear scalar convection- diffusion equations, and preserve positivity for density and pressure for compressible Euler systems. A general framework (for arbitrary order of accuracy) is established to construct a simple scaling limiter for the DG or FV method involving only evaluations of the polynomial solution at certain quadrature points. The bound preserving property is guaranteed for the first order Euler forward time discretization or strong stability preserving (SSP) high order time discretizations under suitable CFL condition. One remarkable property of this approach is that it is straightforward to extend the method to two and higher dimensions on arbitrary triangulations. We will emphasize recent developments including arbitrary equations of state, source terms, integral terms, shallow water equations, high order accurate finite difference positivity preserving schemes for Euler equations, and positivity-preserving high order finite volume scheme and piecewise linear DG scheme for convection-diffusion equations. Numerical tests demonstrating the good performance of the scheme will be reported.
September 6 (Thursday) - Chi-Wang Shu (Brown) Positivity-preserving high order schemes
for convection dominated equations
305 Carver Host: Jue Yan Poster More info
September 14 (Friday) - Tong Li (UI) on Global entropy solutions to a quasilinear hyperbolic system modeling blood flow at 3:10 p.m. and Lihe Wang (UI) on Connection between continuity, integrality and critical regularity of elliptic equations at 4:10 p.m. in 204 Carver. Host: Hailiang Liu Poster
September 20 (Thursday) - Fan Chung Graham (UCSD) Semigroup spectral theory and graph coloring games
Host: Leslie Hogben 4:10 p.m. in 305 Carver Poster
September 25 - Namrata Vaswani (ISU) on Recursive sparse recovery and applications in dynamic imaging
Host: Leslie Hogben Poster
October 9 in Carver 205 Tasos Matzavinos (ISU) on Stochastic analysis of the motion of DNA nanomechanical bipeds
October 19 (Friday) Zhongming Wang (Florida International University) on Solutions to a model Poisson-Nernst-Planck system and the determination of reaction rates in 268 Carver
November 6 Andrew Parrish (University of Illinois) Pointwise convergence for multidimensional averages
November 9 (Friday) Edmond Jonckheere (USC) Decoherence splitting manifold--a geometric approach to decoherence control
January 23 Xin Liu on Markov modulated stochastic networks in heavy traffic in 202 Carver at 4:10 p.m
Monday, February 4 Sergio Almada Monter (UNC-Chapel Hill) on Multi-scale stochastic differential equations, universality and its applications
4:10 p.m. in 3105 Snedecor Hall Poster
Thursday, February 28 Holl Colloquium featuring Saad El-Zanati (Illinois State) Research experiences for pre-service and in-service secondary teachers: The teacher-scholar concept at 4:10 p.m. in Carver 294 Hosted by Leslie Hogben Poster
Tuesday, April 2 Songting Luo (ISU) on Numerical methods for Helmholtz Equations in the high frequency regime beyond geometric optics
Hosted by Nguyen
Tuesday, April 9 Daniel Toundykov (Nebraska) Geometry and stability of the 2D von Karman plate model
Poster Hosted by Hansen
Monday, April 15 in 282 Carver Zhimin Zhang (Wayne State) Superconvergence of polynomial spectral interpolation
Poster Hosted by Liu
Tuesday, April 16 Chelsea Walton (MIT) An introduction to noncommutative invariant theory
Poster Hosted by Young
Friday, April 19 in Carver 298 Ron Taylor (Berry College) on Implementations of active learning in the mathematics classroom
Hosted by Butler
Tuesday, April 23 Dean Beate Schmittmann on First attempts at characterizing interacting networks
Poster Hosted by Rossmanith
Tuesday, April 30 Aristides Katavalos (University of Athens) Subspaces of vN(G) and bimodules over maximal abelian selfadjoint algebras
Hosted by Peters