Mathematics Department Colloquium (2006-2007)

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

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.

Speaker: Shane Henderson, Cornell University 

Title: Planning External Radiation Therapy for Cancer Treatment 

Abstract

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

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.