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Colloquium: 1999-2000

FALL 1999

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September 2

On the Fundamental Role of Interior-Point Methodology in Constrained Optimization

Richard Tapia Rice University

Abstract: Recently primal-dual interior-point methodology has proven to be an effective tool in linear programming applications and is now being extended, with great enthusiasm to general nonlinear programming applications. The primary purpose of this current study is to develop and promote the belief that since Newton's method is a tool for square nonlinear systems of equations, the fundamental role of interior-point methodology in inequality constrained optimization is to produce, in a meaningful and effective manner, a square system of nonlinear equations that represents the inequality constrained optimization problem sufficiently well that the application of Newton's method methodology to this square system is effective and successful. The two most popular interior- point formulations, the logarithmic barrier function formulation and the perturbed KKT conditions formulation, will be compared from both a theoretical and numerical point of view.

September 10

On the History of Topological and Geometric Semigroups

Karl HofmannTulane University and Technical University of Darmstadt

Abstract: We trace some aspects of the history of semigroup theory in the domain of topology and analysis and highlight some of the developments beginning with the origins of the theory at Tulane University and LSU (Wallace, Koch, Hunter Mostert), in the fifties, passing by order theoretic aspect (developed by J.D.Lawson, Mislove et al), and leading to the more recent Lie theory of semigroups which has a more geometric than topological flavor (Hilgert, J.D.Lawson, Neeb).

September 17

Integrals on the Moduli Space of Curves

Rahul PandharipandeCalifornia Institute of Technology

Abstract: The moduli space of curves (or equivalently, the moduli space of complex structures on a fixed Riemann surface) has been a central topic in algebraic geometry for last 30 years. In the last decade, this moduli space has played a role in string theoretic physics. Mathematical techniques motivated by string theory have led to a much better understanding of natural integrals over the moduli space of curves. I will start with an introduction to these integrals describing the basic role they play in the geometry of the moduli space. Then, I'll describe the techniques found in Gromov-Witten theory for calculation.

September 23

Syzygies and Graph Colorings

Dave BayerBarnard College Columbia University

Abstract: When combinatorial problems are encoded as polynomial ideals, one counts by understanding the structure of the chain of syzygies. We describe recent progress in viewing chains of syzygies as topological cell complexes, for monomial and toric ideals. With these methods, graphs yield periodic hyperplane arrangements which form the "universal cover" of a corresponding chain of syzygies. Individual syzygies correspond to cells of this arrangement, which define toric varieties giving the syzygies distinctive identities. Graph coloring problems translate to questions about these toric varieties.

October 7

Surgery Formula for Spectral Invariants of 3-Manifolds

Ronnie LeeYale University

Abstract: Not available

October 21

Ellipsoids, Complete Integrability and Hyperbolic Geometry

Serge TabachnikovUniversity of Arkansas

Abstract: We will discuss new proofs of the two related classical facts that go back to Jacobi: the geodesic flow on the ellipsoid and the billiard ball map inside it are completely integrable. It is well known that the geodesic flow and the billiard ball map preserve natural symplectic structures associated with Euclidean geometry of the ambient space. The proofs are based on the observation that they also preserve symplectic structures associated with the projective model of the hyperbolic geometry inside the ellipsoid.

October 28

Modular Forms, Toric Varieties, and Nonvanishing of L-Functions

Paul GunnellsColumbia University

Abstract: A modular form is a holomorphic function that satisfies a certain symmetry with respect to a discrete group action. Modular forms are important in number theory because they package arithmetic information into an analytic object. For example, the Taniyama-Shimura-Weil conjecture (now a theorem) asserts that if a modular form f satisfies certain conditions, then its Fourier coefficients equal the number of points of an elliptic curve E over finite fields. Another example concerns the L-function L (f,s) associated with a modular form. According to the Birch and Swinnerton-Dyer conjecture, the order of vanishing of L (f,s) at s=1 is related to the complexity of the set of rational points of E.

A toric variety is an algebraic variety built out of the combinatorial data of a collection of cones in a lattice. Examples include affine space Cⁿ and projective space CPⁿ. In contrast to elliptic curves, toric varieties are arithmetically very simple objects, and so one might expect them not to have much to do with number theory.

In this talk we present recent joint work with Lev Borisov connecting toric varieties with modular forms. We construct a sub-ring T (l) of the modular forms, the toric modular forms. Our main result is that T (l) is a natural sub-ring, in the sense that it is stable under various operations from the classical theory of modular forms. We also characterize the space of toric modular forms associated to toric surfaces: it coincides (modulo Eisenstein series) with the space of cusp forms with non-vanishing L-functions.

The talk will be directed to a general mathematical audience. In particular, we will not assume that anyone has ever seen a modular form, an L-function, or a toric variety before.

November 4

Co-algebras in Mathematics and Computer Science

Larry MossIndiana University

Abstract: Co-algebras are simple but fundamental mathematical structures capturing the essentials of dynamical systems in a broad sense, including (possibly infinite) behavior, invariance, and indistinguishability. Their importance is beginning to be recognized by people in both mathematics and theoretical computer science. The area of co-algebra has connections to the semantics of programming languages, and to studies of concurrency and interacting systems. It also may be considered as an offshoot of the study of non-well-founded sets.

The co-algebras considered are generally of functors on the category of sets. Examples include automata, streams, Markov chains, systems of equations for sets, and Kripke models. The conceptual point that these have in common is that the notion of `observation' makes more sense than the notion of 'construction'. The general theory provides a uniform notion of bisimulation. It also leads to 'co-recursion', a dual to recursion, which does not need a base case. The theory features final co-algebras, a dual notion to the initial algebras that one finds wherever the notion of construction is primary. Despite the fact that it uses duals of well-known notions, results in the subject are generally not obtained by dualizing.

This is a survey talk on co-algebra, stressing the examples, parts of the general theory, and areas of current research. It also goes into co-algebraic logic and into the foundational aspects of co-recursion.

November 18

Collective Dynamics of Swimming Bacteria: from Bioconvection to 2d(+)-Turbulence

John KesslerDepartment of Physics, University of Arizona

Abstract: Thin layers of approximately close-packed populations of the swimming bacteria Bacillus subtilis exhibit collective dynamics that have the appearance of turbulence. ~Bacteria-sized latex spheres, used as passive markers embedded in the seething mass of cells exhibit transport properties reminiscent of 2d turbulence, e.g. "superdiffusion", Levy Flights,.. Since these phenomena occur at Reynolds number <<1, modeling them calls for some innovative approaches. Self-propelled fluidized beds with some liquid-crystal-like properties and non-rigid porous media are being considered, and will be discussed -- after 1) showing pertinent videos, and 2) reminders, plus some new discoveries, on the properties and behavior of single bacterial cells, of many cells in constraining environments, and of bioconvection, the latter two at low volume fraction.

December 3

Degenerate Conformal Structures and Singular Quotients of Manifolds

Ricardo Perez-MarcoUCLA

Abstract: All mathematicians learn in their first topology course that an arbitrary quotient of a topological space has a canonical topological space structure. A Hausdorff equivalent relation is necessary in order to stay in the category of Hausdorff topological spaces. Then in our first differential geometry course we learn that a manifold is a Hausdorff topological space endowed with a local smooth structure provided by an atlas. And next we learn with dismay (for true believers in locality!) that in order to quotient a manifold we need a non-local structure: A fundamental domain. For example, when one collapses in [0,1] the components of the complement of the triadic Cantor set, one gets a topological segment but, from the classical point of view, no canonical analytic structure.

We present in this talk simple examples from holomorphic dynamics that lead naturally to quotients of the Riemann sphere with no fundamental domain. We have developed recently new analytic tools that make sense for the first time of these quotients in Riemann surfaces. This provides a canonical theory of renormalization for polynomials.

December 9

Topics in Random Graphs

Alan FriezeCarnegie Mellon University

Abstract: Random graphs were introduced as a subject of study by Erdos and Renyi on 1960. Since then there have been many papers describing the properties and applications of random graphs. We will survey some of the results that have been obtained so far.

SPRING 2000

January 13

Counting Something That Counts the Rationals

Herbert WilfUniversity of Pennsylvania

Abstract: There is a function f(n), which counts something interesting, combinatorially speaking, and which also has the following property: the sequence { f(n)/f(n+1) } consists of all positive rational numbers, each appearing just once. The function f(n) counts certain restricted integer partitions. We'll discuss many properties of this remarkable function.

January 21

Dealing with Discreteness: Improved Confidence Intervals for Proportions, Differences of Proportions, and Odds Ratios

Alan AgrestiDepartment of Statistics, University of Florida, Gainesville

Abstract: The standard large-sample confidence intervals for proportions and their differences used in introductory statistics courses have poor performance, the actual confidence level possibly being much lower than the nominal level. `Exact' intervals have limited use because the discreteness implies very conservative performance. However, simple adjustments of the large-sample intervals based on adding two successes and two failures have surprisingly good performance even for small samples. To illustrate, for n1 = n2 = 10, a nominal 95% confidence interval for p1 - p2 has actual coverage probability below .93 for 88% of (p1, p2) pairs in the unit square with the standard interval but in only 1% with the adjusted interval; the mean distance between the nominal and actual coverage probabilities is .06 for the standard interval but .01 for the adjusted one. In teaching with these adjusted confidence intervals, one can bypass awkward sample size guidelines and use the same formulas for small and large samples. Similar adjustments (and related Bayesian methods) work well in other discrete problems, such as confidence intervals for Poisson means and for odds ratios.

January 27

A New Class of Laplace Inverses and Their Application to Dipolar Fluids

Pedro M. JordanNaval Research Lab, Stennis Space Center, MS

Abstract: A new class of inverse Laplace transforms of exponential functions involving doubly nested square roots is determined. These inverses are then used to determine exact solutions to Stokes' first problem and the Couette flow problem for incompressible dipolar fluids (i.e., a class non-Newtonian fluids). It is shown that in the case of both problems, the flow achieves its steady-state configuration instantaneously for critical values of the physical parameters. Moreover, in considering special/limiting cases of the dipolar constants, the exact solutions are also determined for Rivlin--Ericksen fluids, fluids with coupled stresses, and viscous Newtonian fluids. Results obtained for these fluid types are then compared to those of dipolar fluids. In addition, a number of new three-parameter definite integrals, which evaluate to simple closed-form expressions, are generated from these inverses. Lastly, asymptotic results for these inverses and integrals are presented and extensions of this work are noted. E-mail: pjordan@nrlssc.navy.mil

February 3

Cell Cycle Control: Molecular Mechanisms and Mathematical Models

John TysonVirginia Tech Department of Biology

Abstract: The cell cycle is the sequence of events by which a growing cell duplicates all its components and partitions them more-or-less equally between two daughter cells. In the last 12 years, molecular biologists have made great progress in identifying the genes, proteins and molecular interactions that control the basic events of the cell cycle (DNA synthesis and mitosis). The control system is so complex that its behavior cannot be understood by casual, hand-waving arguments. We use biochemical kinetics and dynamical systems theory to convert hypothetical molecular mechanisms of cell cycle control into quantitative computational models. By testing our models against experimental observations, we gain new insights into how the control system works. The approach is generally applicable to any complex gene-protein network that regulates some physiological characteristics of a living cell.

Ref: Tyson et al. (1996) Trends in Biochemical Sciences 21:89-96.

February 10

Phase Space and Path Integral Methods in Classical Elliptic Wave Propagation Modeling

Louis FishmanNaval Research Lab, Stennis Space Center, MS

Abstract: The n-dimensional, elliptic, two-way Helmholtz wave propagation problem can be exactly reformulated in a well-posed manner as a one-way wave propagation problem in terms of appropriate square-root Helmholtz and Dirichlet-to-Neumann (DtN) operators. These operators are formally defined and constructed in an appropriate pseudo-differential operator calculus in terms of their corresponding operator symbols. The analysis and computation of both direct and inverse wave propagation problems can then be largely reduced to the understanding and exploitation of the operator symbol (singularity and oscillatory) structure, and the subsequent construction of uniform (over phase space) asymptotic operator symbol approximations. These operators and their corresponding operator symbol asymptotics lie outside of the elliptic pseudo-differential operator calculus. Examples from both direct and inverse scattering will be given.

February 18

On Topologies for General Function Spaces

Martin EscardoUniversity of St. Andrews, Scotland

Abstract: It is well-known that the exponentiable Hausdorff spaces are precisely the locally compact spaces, and that the exponential topology is the compact-open topology. Since non-Hausdorff spaces are often regarded as uninteresting and not very well-behaved, it is less well-known that among arbitrary topological spaces, the exponentiable spaces are precisely the core-compact spaces. While the spaces considered in analysis happen to be Hausdorff, interesting and quite well-behaved non-Hausdorff spaces arise frequently in applications of topology to algebra via Stone duality and to the theory of computation. As function spaces play a fundamental role in the theory of computation, it is important to have exponentiability criteria for general spaces. The available approaches to the general characterization are based on either category theory or continuous-lattice theory, or even both. It is the purpose of this expository talk to provide a self-contained, elementary and brief development of general function spaces. The only prerequisite is a basic knowledge of topology (continuous functions, product topology and compactness).

February 24

Control Categories and the Lambda-Mu Calculus

Peter SelingerUniversity of Michigan

Abstract: In this talk, I will describe a class of categorical models for functional programming languages with control operators, and specifically for Parigot's lambda mu calculus. The beauty of these models is that they generalize the well-known correspondence between the simply-typed lambda calculus and cartesian-closed categories. I will introduce the class of "control categories", which is based on Power and Robinson's premonoidal categories. I will show that the call-by-name lambda mu calculus forms an internal language for these categories. Moreover, the call-by-value lambda mu calculus forms an internal language for the dual co-control categories. As a consequence, one obtains a syntactic, isomorphic translation between call-by-name and call-by-value which preserves the operational semantics, answering a question of Streicher and Reus. This result makes precise the intuitive duality between data-driven and demand-driven computation.

March 2

The Viscous Nonlinear Dynamics of Twist and Writhe

Chris WigginsCourant Institute

Abstract: In this talk, I will describe a class of categorical models for functional programming languages with control operators, and specifically for Parigot's lambda mu calculus. The beauty of these models is that they generalize the well-known correspondence between the simply-typed lambda calculus and cartesian-closed categories. I will introduce the class of "control categories", which is based on Power and Robinson's premonoidal categories. I will show that the call-by-name lambda mu calculus forms an internal language for these categories. Moreover, the call-by-value lambda mu calculus forms an internal language for the dual co-control categories. As a consequence, one obtains a syntactic, isomorphic translation between call-by-name and call-by-value which preserves the operational semantics, answering a question of Streicher and Reus. This result makes precise the intuitive duality between data-driven and demand-driven computation.

The Viscous Nonlinear Dynamics of Twist and Writhe

Chris Wiggins, Courant Institute

Abstract: Exploiting the "natural" frame of space curves, we formulate an intrinsic dynamics of a twisted elastic filament in a viscous fluid. Coupled nonlinear equations describing the temporal evolution of the filament's complex curvature and twist density capture the dynamic interplay of twist and writhe. These equations an used to illustrate a remarkable nonlinear phenomenon: geometric untwisting of open filaments, whereby twisting strains relax through a transient writhing instability without axial rotation. Experimentally observed writhing motions of fibers of the bacterium B. subtilis [N. Mendelson et al., J. Bacteriol. 177, 7060 (1995)] may be examples of this untwisting process.

March 16

An Approach to Computing Singular Integrals

Tom BealeDuke University

Abstract: Mathematical models of many problems in science can be formulated in terms of singular integrals. Thus there is a need for accurate and efficient numerical methods for calculating such integrals. We will describe one approach, in which we replace a singularity, or near singularity, with a regularized version, compute a sum in a standard way, and then add correction terms, which found by asymptotic analysis near the singularity. We will first review the role of singular integrals for Laplace's equation and then describe a general principle which predicts the accuracy of a standard descretization applied to a singular integral. We will explain our approach for a singular integral over a plane, or for a double layer potential on a curve, evaluated at a point near the curve. The latter case could be applied when a computed boundary moves through a region in which unknowns are evaluated at grid points, some of which will be near the boundary.

March 24

Model Checking Systems with Many Processes

Allen EmersonUT Austin

Abstract: Model checking is an algorithmic method for verifying correctness of finite state systems that originated as part of the speaker's dissertation work. Now, twenty years later, it is widely used in the computer hardware industry to formally verify and debug microprocessor designs, and is showing promise for software verification. The chief limitation to the more widespread application of model checking is the state explosion problem, where the size of the system's global state graph grows exponentially with the number of processes running concurrently in the system. This talk will examine various strategies for ameliorating state explosion. Some of these depend on exploiting the symmetry inherent in systems composed of many homogeneous processes, and provide an interesting application of elementary aspects of group theory.

March 31

Dealing with Drift in Sequences of Auto-correlated Observations

Todd OgdenDepartment of Statistics, University of South Carolina

Abstract: A common assumption of time series analysis is that of stationarity (i.e., constancy of the mean, variance, and auto-correlation function over time). However, data in several fields of study (financial, social sciences, etc.) exhibit some form of non-stationarity. This talk will focus on methods for handling non-stationary drift (changing mean) in such sequences. Specifically, the effect of drift on estimators of model parameters when a specific (no drift) model is assumed will be discussed, along with routines for testing and estimating drift and incorporating this into the estimation of model parameters. One technique worthy of particular attention is local linear regression, and the application of this technique to these situations will be described. The motivating example used throughout is a study of rhythmic ability which has come to play an increasingly central role in psychological studies of human motor skills as well as in temporal auditory processing in audiology.

April 6

Calibrated Embeddings in the Special Lagrangian and Co-associative Cases

Robert BryantDuke University

Abstract: Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a co-associative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of compact calibrated submanifolds with nontrivial deformation spaces.

April 14

Quadratic Functions in Geometry, Topology, and String Theory

Isadore SingerNational Medal of Science Recipient, M.I.T.

Abstract: Not available

April 18

Mirror Symmetry

Eric ZaslowNorthwestern University

Abstract: Mirror symmetry, a phenomenon from string theory, predicts unusual relationships between mathematical fields—especially complex and symplectic geometry. In this talk, I will review the "classical" results of mirror symmetry, then discuss more recent approaches to the problem, including Kontsevich's conjecture relating sheaves on one manifold to minimal submanifolds on its "mirror". This talk will be accessible to a general audience.

April 27

Computing Fluid Flows in Complicated Geometry

Marsha BergerCourant Institute

Abstract: We give an overview of the difficulties of simulating fluid flow in complicated geometry. The principal approaches to this problem use either overlapping or patched body-fitted grids, unstructured grids, or Cartesian (non-body-fitted) grids, with our work focusing on the latter approach. Cartesian meshes transform the mesh generation problem into a simpler problem in computational geometry, where many recently developed tools can be used. They greatly automate the mesh generation process, and reduce the human effort. In addition, difference schemes based on regular Cartesian grids can be used in the interior of the flow region, so that high order accuracy, robustness and vectorizability are easily achieved. However, it is a challenge to find stable and accurate difference formulas for the irregular Cartesian cells cut by the boundary. We present several approaches to this problem, and give numerical convergence results for test problems in 2D steady and unsteady flow. Computational results for full 3D aircraft will also be presented.

May 5

Calculus on Metric Measure Spaces

Jeff CheegerCourant Institute

Abstract: We will discuss first order calculus on a space equipped with a metric and a measure. If the measure is doubling and a Poincare inequality holds in a suitable sense, then it turns out Rademacher's theorem asserting the almost everywhere differentiability of real-valued Lipschitz functions has a meaningful generalization. Examples of spaces satisfying our assumptions include Gromov-Hausdorff limits of manifolds with Ricci curvature bounded below, Carnot-Caratheodory spaces, and boundaries at infinity of certain hyperbolic buildings. The former examples are rectifiable with respect to the natural measure, while the latter two are fractal.

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu