2006 - 2007 Colloquium Archive
Talks in Spring 2007
Munchies are served 15 minutes beforehand in Jones 131April 6, Friday, 3:00 pm in Jones 302
Speaker: Barry Nelson, Northwestern University
Title: Reliable COMPASS for Optimizing Simulated Systemsn
Abstract: Computer simulation is a standard engineering tool for improving and optimizing system design and performance. Stochastic computer simulations incorporate probability models to represent the uncertainty in a system, and then generate samples from these models to drive the simulation and estimate the system's performance. "Optimization," in the stochastic setting, usually means maximizing or minimizing some measure of long-run average performance; due to sampling, the performance that can only be estimated subject to statistical error. As a result, convergence of optimization algorithms for stochastic simulation problems with discrete decision variables (e.g., number of machines in a work center, staffing levels per time period in a call center, order-up-to levels in a supply chain) typically requires that all feasible solutions are simulated, and simulated infinitely often; this is unrealistic for practical problems with large numbers of feasible solutions.
In this talk we describe Convergent Optimization via Most-Promising Area Stochastic Search (COMPASS), an approach that is simple and easy to understand, is provably convergent, but requires simulating only a small fraction of the available solutions (even when there are infinitely many feasible solutions). The talk will emphasize the difficulties that arise in stochastic problems, and the intuition behind COMPASS.
March 26, Monday, 3:00 pm in Jones 131
Speaker: Anthony Mendes, California Polytechnic University
Title: How to take old bijections and make them new again
Abstract: Say A and B are finite sets with the same number of elements. Finding an explicit bijection f : A -> B can help in fully understanding the relationship between A and B. However, it is often difficult to construct such a function. To create these new, awesome bijections, I will take old, boring bijections and compose them. These ideas can be understood by both undergraduates and non-combinatorialists. Composing functions may seem trivial, but I will discuss nontrivial open problems which have been recently solved using these methods.
March 2, Friday, 3:00 pm in Jones 131
Speaker: Man-Duen Choi, University of Toronto
Title: My Adventures in Wonderland
Abstract: In the early 70's, I started off my mathematical journey in the wonderland of completely positive linear maps between matrix algebras. Now, in an unexpected era of quantum computers, time runs backwards in an alternate world. As I have to come back to the same scene, I shall report what I found there, through the Looking-Glass.
References:
Lewis Carroll, Alice's Adventures in Wonderland, 1865.
Lewis Carroll, Through the Looking-Glass, and What Alice Found There, 1871.
February 26, Monday , 3:00 pm in Jones 301
Speaker: Helena Smigoc, University College Dublin, Ireland
Title: Existence of a common solution to the Lyapunov equation
Abstract: A matrix A 2 Cn£n is called (Hurwitz) stable if all its eigenvalues lie in the open left half of the complex plane. A classical result of Lyapunov states that a matrix A is stable if and only if for arbitrary Hermitian positive de¯nite Q; the Lyapunov equation
AP + PA¤ = ¡Q
admits a positive defnite solution P: The problem of determining when for given stable matrices A1;A2; : : : ;Ak in Cn£n there exists a common solution P > 0 to the Lyapunov equations:
AjP + PA¤j < 0
for j = 1; 2; : : : ; k; is important in applied and theoretical research.
We will present some conditions for two stable complex matrices to have a common solution to the Lyapunov equation. We will apply these conditions to the case when a common weak solution exists and show that necessary and su±cient conditions for the existence of a common solution for 2 £ 2 complex matrices A and B is that matrices (A + i®I)(B + i¯I) and (A + i®I)¡1(B + i¯I) have no negative real eigenvalues for all real numbers ® and ¯:
February 23, Friday, 12:50 pm in Jones 301
Speaker: Jani Virtanen University of Helsinki, Finland
Title: Toeplitz and Hankel operators on Hardy and Bergman spaces with p=1
Abstract: Toeplitz and Hankel operators acting on Hardy spaces Hp and on Bergman spaces Ap have been studied extensively when p>1, whereas the case p=1 has been avoided due to the unboundedness of the Riesz and Bergman projections. We first consider the boundedness and compactness of Hankel operators on the Hardy space H1 and then establish Fredholm theory for Toeplitz operators. We also deal with similar questions when the operators are acting on the Bergman space A1.
Talks in Fall 2006
December 8, Friday, 2pm in Jones 301
Speaker: Shane Henderson, Cornell University
Title: Planning External Radiation Therapy for Cancer Treatment
The radiation therapy treatment planning problem involves devising a radiation treatment plan that delivers a sufficient dose to a target region containing the tumour while sparing, as much as possible, surrounding organs. Traditional treatment planning models handle uncertainties arising from organ motion etc. by inflating the target region. We instead use a probabilistic model and show that it is equivalent to using robust linear programming. For a sample prostate case, our results show that the method is computationally feasible, and finds plans that are adept at sparing healthy tissue while maintaining the prescribed dose. I’ll discuss radiation therapy, the idea behind robust linear programming, our formulation and results. No prior knowledge of radiation therapy, cancer or robust linear programming is assumed.
December 1, Friday, 2pm in Jones 131
Speaker: Sanne ter Horst, Vrije Universiteit, Amsterdam
Title: The Nehari extension problem and relaxed commutant liftings
Abstract: The Nehari extension problem is one of the many metric constrained problems that fits in the commutant lifting setting. The commutant lifting theory developed since the 60's provides various explicit state space descriptions of all solutions to the Nehari extension problem. However, for these descriptions the state space will be infinite dimensional. In the special case that the Hankel operator associated with the data has finite rank this can be overcome by constructing a finite dimensional realization for the data. Recently a relaxation of the commutant lifting theorem was introduced, which provides a setting in which a family, indexed by natural numbers, of relaxed versions of the classical interpolation and extension problems can be formulated. In this talk, we consider the classical Nehari extension problem as well as its relaxed versions. We will see how these problems fit in the (relaxed) commutant lifting setting, and obtain an explicit description of all their solutions. For the relaxed versions these descriptions involve state space formulas where the state is finite dimensional with the dimension depending on the index. Moreover, it is observed how the classical Nehari extension problem appears as a limit case of the relaxed versions as the index goes to infinity.
November 17, Friday, 2pm in Jones 131
Speaker: Abbas Salemi, College of William and Mary (visiting from SBU of Kerman, Iran)
Title: Polynomial Numerical Hulls
Abstract: The notion of polynomial numerical hull was introduced by O. Nevanlinna in 1993. In this presentation we determine the polynomial numerical hulls of n-by-n normal matrices. Also, the relationship between polynomial numerical hulls and joint numerical range of matrices is considered.
November 10, Friday, 2pm in Jones 131
Speaker: Timothy Killingback, College of William and Mary
Title: Evolutionary Dynamics of Cooperation
Abstract: Achieving a satisfactory understanding of the evolution of cooperation represents a fundamental problem in biology. Cooperative behaviour can occur in biological systems at all levels of complexity, ranging from simple replicating molecules to highly developed animal and human societies. It is believed that many of the major transitions in evolution depended on cooperation. In this talk I will discuss the evolutionary dynamics of cooperation, and a number of theoretical approached to resolving the problem of cooperation.
November 3, Friday, 2pm in Jones 131
Speaker: Charlie Johnson, College of William and Mary
Title: Bounded Ratios of Products of Principal Minors in a Positive Definite Toeplitz Matrix
Abstract: This will be a VERY informal talk about work done this summer with Alex Porush and Hyo Min Choi. Suppose that S1,..., Sk and T1, ...., Tk are two collections of subsets of N = {1, ..., n}. The titled issue is when does it happen that there is a constant K such that product detA[Si} / product det A[Ti] < K for all positive definite Toeplitz matrices A? Corresponding questions have been raised with respect to other classes of matrices (positive definite, M-matrices and totally positive matrices) with varying degrees of success. There are surprises in this case.
October 27, Friday, 2pm in Jones 131
Speaker: Dan Volok, College of William and Mary
Title: On the Painleve property of the Schlesinger system
Abstract: In 1905 L. Schlesinger has formulated a theorem that a holomorphic deformation of Fuchsian linear differential systems parameterized by the position of singularities is isomonodromic if and only if the coefficients satisfy with respect to the parameter a certain non-linear system, which is known today as the Schlesinger system. In 1981 in a paper by T. Miwa it was stated that the isomonodromic deformations of Fuchsian systems enjoy the Painleve property: they are globally meromorphic with respect to the parameter.
In the formulated generality these two well-known results, which played an important role in the study of differential equations in the complex domain, are false: they hold under certain generic assumptions on the coefficients of the deformations, but not in general. Nevertheless, the corollary that the Schlesinger system enjoys the Painleve property holds true without any restrictions.
We shall discuss a proof of this fact and its generalization in the case of arbitrary (possibly non-Fuchsian) linear systems with rational coefficients.
October 20, Friday, 2pm in Jones 131
Speaker: Zhifu Xie, College of William and Mary
Title: Central Configuration, Regularization of Singularity and Periodic Solution with Collisions
Abstract: In this presentation, we discuss central configuration and regularization of singularity due to simultaneous binary collision after we give a brief introduction to N-body problem. The possible region of central configuration in collinear four body problem is given. Newtonian system experiences a singularity of collision because velocities and accelerations of the bodies involving collision approach infinity. We study the motion before, closing, and after collision. Based on the understanding of the regularization, we construct a family of periodic solution with collisions. Lots of numerical animations will be used to show such orbits.
October 6, Friday, 2pm in Jones 131
Speaker: Mihaela Dobrescu, CNU
Title: Wavelet sets and Multiresolution Analysis
Abstract: In this presentation, we consider a special class of wavelets corresponding to wavelet sets. Most of the examples of such wavelets are for dilation sets which are groups. We find a necessary and sufficient condition under which subspace wavelet sets exist for dilation set of the form AB, which is not necessarily a group. We also discuss the multiresolution analysis associated with such dilation sets.
September 29, Friday, 2pm in Jones 131
Speaker: B. Kuzma, University of Ljubljana, Slovenia
Title: Preservers of zero products
Abstract: We discuss maps (possibly nonadditive) on matrices equipped with a product so that matrix pairs having zero products are mapped to matrix pairs having zero products. Extension of results to products of more than two matrices will be discussed. If time permits, we will mention further extensions of the results to other spaces and related problems.
Septrember 21, Thursday, 5pm in Jones 131
Speaker: Hara Charalambous, Department of Mathematics, Aristotle University of Thessaloniki, Greece
Title: Minimal systems of binomial generators and the indispensable complex of a toric ideal
Abstract: Let A = fa1; : : : ; amg ½ Zn be a vector con¯guration and IA ½ K[x1; : : : ; xm] its corresponding toric ideal. We completely determine the number of different minimal systems of binomial generators of IA. We also prove that generic toric
ideals are generated by indispensable binomials. We associate to A a simplicial complex ¢ind(A). We show that the vertices of ¢ind(A) correspond to the indispensable monomials of the toric ideal IA, while one dimensional facets of ¢ind(A) with minimal binomial A-degree correspond to the indispensable binomials of IA.
This talk is based on joint work with A. Katsabekis and A. Thoma.
September 15, Friday, 2pm in Jones 131
Speaker: Th. Schulte-Herbrueggen, Technical University of Munich
Title: UNITARY QUANTUM CONTROL AND BEYOND: The Significance of the C-Numerical Range and Local C-Numerical Range in Quantum Control and Quantum Information
Abstract: C-numerical-range-related new structures arise from practical problems in quantum information and in quantum control. Understanding these structures helps to tackle hot topics in quantum information. We start out from an overview on the role of C-numerical ranges to current research topics in quantum theory: the quantum mechanical task of maximizing the projection of a point on the unitary orbit of an initial state A onto a target state C over the entire unitary group relates to the C-numerical range of A via maximizing the trace function. In quantum control of N qubits one may be interested in (i) U lying in U(2^N) for the entire dynamics or (ii) in restricting the dynamics to local operations on each qubit, i.e. the N-fold tensor product. This relates to a new entity, the local C-numerical range whose intricate geometry is neither star-shaped nor simply connected in contrast to the conventional C-numerical range. We conclude with novel applications of the local C-numerical range in quantum control assisted by gradient flows on the local unitary group: they serve as powerful tools (1) for deciding whether a quantum interaction can be inverted or refocused in a sense generalizing Hahn's famous spin echo; (2) they allow for optimising witnesses of quantum entanglement. (3) Finally, by using the adjoint representation $Ad_U$ to the unitary $U$, the mathematical structures are generalized to an embedding Hilbert space, on which relaxation operators can act. Most recent results cover the control of open systems, whose dynamics are governed by semi-groups.
