# Colloquium: 2001-2002

## FALL 2001

View Spring 2002 »

September 13

## Fast Explicit Methods for Stiff ODEs

Alexei MedovikovInstitute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia

Abstract: Most of known explicit Runge-Kutta methods has small stability domains—their time steps are bounded by a very restrictive CFL condition, and thus standard explicit methods become useless for stiff ODEs. We propose explicit embedded integration schemes with large stability domains. The construction of these methods is achieved in two steps: first, we compute stability polynomials of a given degree with optimal stability domains, i.e., possessing a Chebyshev alternation; second, we realize a corresponding explicit Runge-Kutta method with the help of the theory of Runge-Kutta composition methods.

Stability domain of the method increases by the factor proportional to the square of the degree of the optimal stability polynomial. To construct the stability polynomial of a large degree we use asymptotic formula for polynomial of the least deviation from zero with a weight function.

For example, our third-order explicit Runge-Kutta method uses such polynomials of degrees between 3 and 432. This reduces a stability restriction by a factor of about 200, compared with the forward Euler method. More precisely, the speed up factor of the $p$th-order method can be up to $\Delta_p*s$, where $\Delta_1=2$, $\Delta_2=0.81$, $\Delta_3=0.49$, $\Delta_4=0.35$, and $s$ is the degree of the optimal stability polynomial.

The large stability domains and the explicitness allow the proposed methods to efficiently solve very large systems of stiff ODEs, which arise, for example, after spatial discretization of parabolic PDEs. The high order produces accurate results, and the embedded formulae permit an efficient step size control.

October 4

## Modeling and Verifying Timed Systems Using Discrete Abstractions

Joel OuaknineTulane University

Abstract: Concurrent, or parallel, timed systems are compound entities whose behaviour—communication and evolution over time—is both discrete and continuous in nature. Examples of such systems range from computer networks and secure communication protocols to aircraft controllers and nuclear power plants.

For obvious reasons, verifying properties of such systems - making sure they meet their intended specifications, such as safety or liveness - can be of vital importance. In this talk, I will discuss how such systems are modeled using process algebra, and show how to circumvent the main problem associated with the most realistic models, namely that the state space is invariably infinite (in fact uncountable). I will present some discretisation techniques which have led to efficient verification algorithms, and identify a large class of properties which are stable under the proposed discretisation.

This talk is aimed at a general mathematical audience; in particular, no prior knowledge of timed systems or process algebra will be assumed.

October 11

## Systems of Conservation Laws: Recent Developments in the Well Posedness Theory

Marta LewickaMax-Planck-Institute for Mathematics in the Sciences (MIS), Leipzig, Germany

Abstract: We gather some recent results concerning the well posedness of a class of nonlinear PDEs having the form: $$u_t + f(u)_x=0,$$ where the unknown vector of conserved quantities $u\in R^n$ depends on the one-dimensional space variable $x\in R$ and time $t\geq 0.$ It has been proved that when the initial data $\bar{u}$ in $$u(0,x)=\bar{u}$$ has sufficiently small total variation, then the weak admissible solution to the stated Cauchy problem can be seen as a trajectory of a unique Lipschitz continuous semigroup, constructed by means of the flux function $f$. We outline the concepts and methodology required for this construction and discuss the structure of semigroup solutions., We also indicate the possibilities and major obstacles while extending these results to data with large total variation.

October 25

## Ramanujan's Lost Notebook

Bruce BerndtUniversity of Illinois at Urbana-Champaign

Abstract: In the spring of 1976, among the papers of G. N. Watson in the library at Trinity College, Cambridge, George Andrews found a sheaf of 138 pages of Ramanujan's work, which Andrews naturally called Ramanujan's lost notebook. This work, comprising about 650 results with no proofs, arises from the last year of Ramanujan's life and represents some of his deepest work. About 60 % of the lost notebook is devoted to q-series, including mock theta functions, and Andrews has proved most of these results. Topics in the remaining 40% include theta-function identities, integrals of theta-functions, incomplete elliptic integrals, the Rogers-Ramanujan continued fraction, other continued fractions, other integrals, Eisenstein series, unidentified and unexplained series expansions, and unexplained tables of numbers. A survey of the history and entries from the "remaining 40%" will be given.

November 1

## Open Issues in Formal Methods for Cryptographic Protocol Analysis

Abstract: The history of the application of formal methods to cryptographic protocol analysis spans nearly twenty years, and recently has been showing signs of new maturity and consolidation. A number of specialized tools have been developed, and others have effectively demonstrated that existing general-purpose tools can also be applied to these problems with good results. However, with this better understanding of the field comes new problems that strain against the limits of the existing tools. In this talk we will outline some of these new problem areas, and describe what new research needs to be done to to meet the challenges posed.

November 8

## Homogenized Failure Criteria and the Design of Composite Structures for Maximum Strength and Stiffness

Abstract: New homogenization formulas for the products of weakly converging sequences of stresses are obtained. For this case the div-curl theorem does not apply and new techniques based upon the differentiability of the G-limit are invoked. The resulting formula is applied to provide bounds for the homogenized failure surface of multi component composites. These results are used to develop a numerical scheme for the design of composite structures for maximum strength and stiffness.

November 15

## Modern Algebra and Number Theory as Applied Mathematics

William Y. VelezDepartment of Mathematics, University of Arizona

Abstract: I was not your typical first year graduate student in 1970. I had just returned from Vietnam, I had a wife and a child. I was acutely aware of the fact that my graduate studies had to provide me with a means of supporting my family. In my first semester of graduate school I took a course in an area of traditional applied mathematics. I didn't like it. I found the other courses in real analysis, algebra and algebraic number theory much more interesting. I gave up on the idea of studying applied mathematics and instead decided that I had to study an area of mathematics in which I was fascinated.

I chose algebraic number theory and by the time I received my Ph.D. I had written papers in Galois Theory, Elementary Number Theory and Algebraic Number Theory. As it turns out, these topics are applied mathematics, though I didn't know it at the time.

In this talk I will describe some work that I did dealing with communication systems for submarines. I will then describe some ideas dealing with cryptographic systems. These cryptographic systems are created for the express purpose of sending secret information. The backbone for modern cryptographic systems is number theory and algebra.

November 16

## Increasing the Number of Mathematics Majors

William Y. VelezDepartment of Mathematics, University of Arizona

Abstract: For the last ten or fifteen years I have been focusing a great deal of energy on increasing the number of minority mathematics majors in our department. In the late 1980's we graduated perhaps one minority undergraduate mathematics major every two years. We now have 300 mathematics majors in the department, of whom about 50 are minority students.

My efforts begin with minority students enrolled in our traditional three-semester calculus course. The department now allocates a small amount of funds to hire a student assistant to help me contact minority students enrolled in these courses. The student assistant sets up twenty-minute appointments. During this twenty-minute appointment I go over the student's schedule, discuss career plans, talk about the importance of resumes and internships. And one more thing: if a student comes into my office enrolled in calculus and the student does not have a major declared, I make that student into a mathematics major on the spot! This can be shocking experience for a student.

I would like to point out that my work with students has nothing to do with the fact that they are minority students. I have tried to convince our faculty that we should be as aggressive with all of our calculus students.

In this presentation, I will also present some departmental initiatives aimed at increasing the number of mathematics majors in the department.

January 10

## Modeling and Analysis of Population Evolution Data

Anand VidyashankarDepartment of Statistics, University of Georgia

Abstract: Not available

January 18

## Positivity and Symmetric Functions

Jennifer MorseDepartment of Mathematics, University of Pennsylvania

Abstract: We will define elements in the space of symmetric functions and discuss a number of their classical properties revealing the role of combinatorics in this field. A filtration for the space will then be introduced and current results regarding a refinement of classical properties will be shown. We will finish by showing how our work with the filtration plays an interesting role in a long-standing open problem regarding the positivity of Macdonald symmetric functions.

January 22

## Invariants of 3-Manifolds

Abstract: We construct infinite-dimensional affine spaces points of which correspond to oriented 3-manifolds with additional structures - framings or spin structures. Discriminant in such space is a hypersurface, corresponding to manifolds with Morse singularities. We study the topology of the chambers of these spaces, introduce axiomatics of the invariants of finite type, give examples of such invariants and discuss other problems in low-dimensional topology, related to this approach.

January 23

## Lattice-Ordered Algebraic Extensions of Totally Ordered Fields

John RedfieldDepartment of Mathematics, Hamilton College, N.Y.

Abstract: Let T be a totally ordered field (i.e., a field with a compatible total order). Many simple algebraic extensions of T have compatible lattice orders with respect to which T is the maximum totally ordered subfield. Such extensions may of course be viewed as quotient rings T[x]/(p(x)), where (p(x)) is the ideal generated by the irreducible polynomial p(x). So since the coefficientwise order on T[x] is a compatible lattice order, a natural place to start an investigation of the lattice orders on simple algebraic extensions of T would be to consider the quotient orders on the quotient rings T[x]/(p(x)). To define such an order, (p(x)) needs to be convex, and to ensure that the order is a lattice order, (p(x)) needs to be lattice-ordered as well. However, the ideal (p(x)) is convex if and only if distinct elements of (p(x)) are not comparable. So if the quotient order is to be a lattice order, it must arise not in the traditional way, from a lattice-ordered ideal, but in a very unusual way, from a trivially ordered ideal. Nevertheless, the quotient order is sometimes a lattice order. I will describe this rather curious situation in more detail and then present some general results.

January 25

## Coiling and Writhing in Geometry, Physics and Biology

Dennis DeTurckDepartment of Mathematics, University of Pennsylvania

Abstract: In this talk, I'll report on the work of a number of people at Penn involving mathematical methods in plasma physics. From the physics point of view, our goal is to determine and study the persistent plasma states observed in astrophysical, solar and laboratory settings. From the mathematics point of view, our goal is to develop the tools to carry this out, and to work on a number of mathematical problems suggested by this enterprise. Key roles in the story are played by the notion of "helicity" of a vector field, which measures the extent to which the field lines wrap and coil around one another, and spectral problems for the curl operator. Because helicity of vector fields is the analogue of "writhing number" of knots, the methods we use also provide an upper bound for the writhing number of a given strand of DNA.

January 29

## Abel-Jacobi Maps and the Cubic Threefold

Abstract: The Abel-Jacobi map is a tool for studying algebraic cycles on algebraic varieties. The goal of this talk is to explain a computation of this map in the case of rational curves on a cubic threefold. The computation also involves the relatively new idea of rational connectedness, introduced by Kollar, Miyaoka, and Mori.

February 7

## An Adaptive High Resolution Recovery of Piecewise Smooth Data

Abstract: We discuss the reconstruction of piecewise smooth data from (pseudo-) spectral information. While spectral projections enjoy superior resolution with globally smooth data, the presence of jump discontinuities is responsible for spurious O(1) Gibbs oscillations and an overall deterioration to the unacceptable first-order convergence rate of spectral projections. The purpose is to regain the superior (in fact, exponential) accuracy in the piecewise smooth case. This is achieved in two complementing steps. First, a localization step using a novel detection procedure which identifies both the location and amplitudes of finitely many edges. It is followed by a second step of mollification - we present a two-parameter family of spectral mollifiers which recover the data between those edges with exponential accuracy. The ubiquitous one-parameter, finite-order mollifiers are based on dilation. In contrast, our family of two-parameter spectral mollifiers achieve their high resolution by an intricate process of high-order cancelation . Accurate recovery of piecewise smooth data is carried out in the direction of smoothness, and adaptivity is responsible for the high resolution. We conclude with examples for applications in Computational Fluid Dynamics (formation of shocks), image and geophysical data processing.

February 27

## Central Perspective in Italian Renaissance Painting

Karl HofmannDepartment of Mathematics, Tulane University (Sponsored by the Duren Professorship Program of Tulane College)

Abstract: Not available

Note: Special lecture in the History of Mathematics

March 7

## Contributions to the Theory of the Multiple Gamma Function

Victor AdamchikDepartment of Computer Science, Carnegie Mellon University

Abstract: The multiple gamma function Gn, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Keating, Sarnak the interest to the Gn function is revived. Sarnak conjectured the idea that zeros of certain "zeta functions" (L-functions) can be understood in terms of the distribution of eigenvalues from classes of random matrices. It has been shown that the G 2 function naturally appears there as a closed representation for statistical averages. In this talk I present some theoretical aspects of the multiple gamma function and their application to infinite summation and product.

March 12

Horst BehnckeDepartment of Mathematics, Universitat Osnabruck

Abstract: Periodical cicades have the longest observed period - 13, 17 years - of all periodical insects. The evolution of these insects, their synchronicity in emergence is analyzed by means of mathematical models. These are of the discrete dynamical systems type.

March 14

## Fibered 3-Manifolds and Symplectic 4-Manifolds

Abstract: Work of S. Donaldson, R. Gompf, and others has opened the way for studying symplectic 4-manifolds by analyzing Lefschetz fibrations on them. In recent joint work with R. Matveyev we have classified symplectic Lefschetz fibrations on manifolds of the form S1 x M3, the product of a circle with a 3-manifold. The result provides evidence supporting the following conjecture of C. Taubes: if the product of a three-manifold with a circle admits a symplectic structure, then the three-manifold must be fibered over a circle.

April 4

## An Introduction to Geophysical Flows

Abstract: The atmosphere and the ocean are the largest fluid bodies on our planet; they display an incredibly rich variety of flow structures over a large span of length and time scales. The theoretical and observational study of the atmosphere and the ocean has been rapidly increasing since the second half of the 20th century. Advances in the observational, theoretical and computational fronts have opened possibilities to include more complex physics, formulating global circulation models, exploring weather forecasting at longer times scales, and gaining better understanding of the energetic transfers taking place over the whole spectrum of length scales, from the planetary scale down to the diffusive scale. Studies of this kind are relevant to the understanding of phenomena with profound impact in human activities such as "El Nino", the hole in the ozone layer in Antartica, and the transport of contaminants by the winds and currents. In recent years a renewed interest in these problems has emerged in the mathematical community. This talk aims to give an introduction of geophysical flows and its idiosyncrasies for the young mathematical audience, followed by some of the recent mathematical work done in the study of simplified equations for these flows in the limiting regimes of high stratification and/or high rotation.

April 11

## Babylonian Astronomy

Jim CarlsonDepartment of Mathematics, University of Utah

Abstract: We all know that the Greeks developed elaborate earth-centered geometrical models of the heavens. We will discuss much earlier Babylonian models based on a kind of empirical curve fitting, and will show how these models contributed to Greek astronomy in the hellenistic period.

Note: Special Lecture in the History of Mathematics

April 12

## Moduli Spaces of Cubic Hypersurfaces and Hyperbolic Geometry

Abstract: It has long been known that the totality of cubic equations in two variables, up to the natural equivalence relation, can be understood using real hyperbolic geometry. We will show how the same problem, for cubics in three and four variables, can be understood using complex hyperbolic geometry.

April 18

## Infinite Energy Solutions to the Navier-Stokes Equations

Abstract: In the analysis of the incompressible Navier-Stokes and Euler equations, an interesting question concerns the possible appearance of finite-time singularities in solutions. We will show that this may take place for certain forms of solution of "stagnation" type, which possess both bounded and unbounded components when defined over a semi-infinite strip.

April 25

## Image Segmentation and Energy Dissipation

Abstract: In this talk I will present some work in progress in vision research. We consider the problem of recognizing what parts of an image are perceived as being in the foreground. We use a variant of the Pao-Geiger-Rubin model, which uses an energy dissipation approach to this problem. The model is surface-based, rather than contour-based. Specifically, the edges in the image are not viewed as isolated contours, but are viewed as bounding a surface. Each local edge has a local hypothesis; for example, a north-south edge might think "the region immediately to the left of me is part of the figure". The model then uses energy dissipation methods to seek assignments of local hypotheses that are mutually agreeable, yielding a segmentation of the image that might be perceived. We test the model on various images to address questions like: does the model "perceive" smaller objects to be in the foreground (the way we do)? convex objects to be in the foreground (the way we do)? how does it perform on optical illusions that viewers report to have two different segmentations? This is joint work with Nava Rubin of the Center for Neural Science, NYU. I thank Anita Disney (CNS, NYU), Davi Geiger (Courant, NYU), Bob Shapley (CNS, NYU), and Dave McLaughlin (Courant, NYU) for useful discussions.

May 6

## Parabolic PDEs and Large Stiff Systems Solved by ROCK Method

Abstract: ROCK methods are a new class of so-called Runge-Kutta-Chebyshev methods which combines and generalize (to higher order) the Chebyshev methods of Van der Houwen, Sommeijer, Shampine and Verwer and those of Lebedev, Finogenov and Medovikov. They are intended for large mildly stiff problems, originating mainly from parabolic PDEs converted into ODEs by space discretization. These methods are of high order, explicit, and as easy to use as the forward Euler method, but much more efficient due to the extended stability domain along the negative real axis. They are based on a three-term recurrence relation of a family of orthogonal polynomials. We present here these new methods and give some examples of their efficiency.

May 9

## Noncommutative Groebner Bases: A Tool for Symbolic Computation and Research

Abstract: In this talk I will contrast the theories of commutative and noncommutative Groebner bases and discuss applications of the theory. In particular, I will touch on both theoretical and practical applications.

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