All sessions will take place in Avery Hall on the campus of the University of Nebraska Lincoln (MAP)  

This information is current as of May 7, 2006
FRIDAY-MAY 12TH SATURDAY-MAY 13TH SUNDAY- MAY 14TH
1:45 PM - 2:00 PM

Opening remarks and announcements:

John Meakin, Chair, Department of Mathematics, University of Nebraska Lincoln

Twenty minute talks will be presented in parallel sessions Twenty Minute Talks will be presented in parallel sessions
 

8:00 AM - 9:00 AM

Continental Breakfast

8:00 AM - 9:00 AM

Continental Breakfast

2:00 PM - 2:45 PM 9:00 AM - 9:20 AM

9:00 AM - 9:45 AM

Ambar Sengupta, Louisiana State University

Title: Functional Integrals in Low-dimensional Gauge Theories

Abstract: Two-dimensional Yang-Mills gauge theory and three dimensional Chern-Simons theory have produced mathematical problems ranging from probability theory to topology of the moduli space of flat connections. This talk will present an overview of some of these mathematical problems and results.

 

Dong Hyun Cho, Kyonggi University

Title: A Change of Scale Formula for Conditional Wiener Integrals over Paths in Abstract Wiener Space

Abstract: Click here for the PDF file.

Bong Jin Kim, Daejin University

Title: Integration by Parts Formulas Involving Fourier-Feynman Transforms

Abstract: In this talk we investigate the Fourier-Feynman transform, the convolution product and the first variation of a functional on abstract Wiener space. We also study integration by parts formulas for the analytic Feynman integral involving Fourier-Feynman transform of those functionals.

Sarada Rajeev, University of Rochester

Title: Quantization of Contact Geometry and Quantum Thermodynamics

Abstract: In classical thermodynamics, like in classical mechanics, there is a notion of canonically conjugate observables; pressure is conjugate to
volume, temperature to entropy, magnetic field to magnetization etc. I will argue that if quantum effects are included in thermodynamics, the observables do not commute and even satisfy a kind of thermodynamic uncertainty principle. This algebra of thermodynamic observables can be determined in most cases of physical interest by analogy to the passage from classical mechanics to quantum mechanics. I will also review the geometric formulation of classical mechanics in terms of contact geometry ( the work of J. W. Gibbs). This is then generalized to the quantum theory to get a
non-commutative contact geometry. A reformulation in terms of path integrals will be discussed briefly as well.

3:00 PM - 3:45 PM 9:30 AM - 9:50 AM 10:00 AM - 10:45 AM

Tepper Gill, Howard University

Title: Elementary Introduction to the Feynman Operator Calculus, and Unification of Hyperbolic and Parabolic Evolution Equations

Abstract: In this talk, we present an elementary introduction to the Feynman operator calculus by directly extending the notion of strong continuity to the time-ordered setting. We then provide generalizations of a few generation theorems (including the Hille Yosida theorem) and use these results to unify the two distinct approaches to existence and uniqueness theorems for general hyperbolic and parabolic (time-dependent) evolution equations. We impose no domain restrictions and only assume strong continuity for the given family of operator-valued functions.

Jun Tanaka, University of California Riverside

Title: The Feynman Integral

Abstract: First, I will give an expository introduction of the Wiener Integral. From there, I will derive the heat equation from the Schrodinger equation by sending t to -i*t. This will serve as a natural
transition to the Semi Heat Equation, for which I will also give an expository outline. Finally, all of these results and the Trotter
Product Formula will be used to show how the Feynman-Kac formula is
derived.

Anna Amirdjanova, University of Michigan

Title: Physically inspired stochastic partial differential
equations.

Abstract: The talk is devoted to advances in the study of three fundamental equations of mathematical physics (heat, Burgers' and Navier-Stokes) when the classical models are perturbed by random forces. Questions of existence, uniqueness and regularity
of solutions will be discussed and some open problems presented.

Gopinath Kallianpur, University of North Carolina

Title: A Stochastic Feynman Integral

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4:00 PM - 4:45 PM 10:00 AM - 10:45 AM 11:00 AM - 11:20 AM

Brian Jefferies, University of New South Wales

Title: Feynman's Operational Calculus with Brownian Time-Ordering

Abstract: The idea of replacing a time-ordering measure in Feynman's  operational calculus by Brownian motion is quite natural and has been considered by G.W. Johnson and G. Kallianpur, who use the  term "stochastic Dyson series". Solutions to the Schroedinger equation with a stochastic perturbation are represented as a stochastic Dyson series. A similar representation of the solutions of linear stochastic partial differential equations seems to be closely related to problems in harmonic analysis concerning maximal regularity of operators acting in a Banach space.

Brian DeFacio, University of Missouri

Title: Feynman Functionals  for old problems: Understanding some complex systems in modern warfare

Abstract: Wars have been a  common activity among our species for longer than written records ; and even beyond oral traditions.While consulting  for the LMI Corp. in McLean , VA  for the US Navy it became clear to me that the Feynman Functional gave a formulation of  the matters our panel was discussing. Some of these issues will be presented to this workshop.

Seung Jun Chang, Dankook University

Title: Translation Theorems of Generalized Feynman Integrals with Applications

Abstract: TBA

 

 

Jean Claude Zambrini, University of Lisbon

Title: Path Integrals and Deformation of Contact Geometry

Abstract: We shall describe a deformation of contact geometry relevant to Feynman Integral strategy,and some applications

5:00 PM - 5:20 PM 11:00 AM - 11:45 AM 11:30 AM - 11:50 AM

G. W. Johnson, University of Nebraska Lincoln

Title: Remarks on Feynman's Operational Calculi for Noncommuting Operators

Abstract: The heuristic ideas. Some aspects of a rigorous approach to Feynman's operational calculi. Two 'new' results---enlarging Feynman's operational calculi and decomposing disentanglings.

Bruce Driver, University of California, San Diego

Title: Geometric Path Integrals

Abstract: Click Here (PDF file)

Lance Nielsen, Creighton University

Title: An Integral Equation for Feynman's Operational Calculus

Abstract: An integral equation for the operational calculus is presented and one or two applications will be discussed.

Brett Barwick, University of Nebraska Lincoln

Title: Using Feynman Path Integrals to Model Electron Diffraction

Abstract: This talk will relate how Feynman integrals can be used to model electron diffraction experiments and how they are used to model things in matter optics and interferometry. Theoretical along with experimental data will be presented.

5:30PM - 5:50PM 12:00 PM - 1:30 PM 12:00 PM - 12:20 PM

David Skoug, University of Nebraska Lincoln

Title: Some Favorite Theorems

Abstract: We will briefly discuss two theorems.
1. A simple formula for conditional Wiener integrals.
2. A necessary and sufficient condition that a functional F on Wiener space has an integral transform.

LUNCH BREAK

6:30 PM - 8:00 PM

Opening reception in the Arbor Room of the Cornhusker Hotel, 333 S. 13th Street, Lincoln, NE

William Boos, Iowa

Title: A Stochastic Representation of Feynman Integration

Abstract: Click Here

 

 

This information is current as of May 7, 2006
SATURDAY-MAY 13TH
Twenty minute talks will be presented in parallel sessions
1:30 PM - 1:50 PM

Kun Sik Ryu, Han Nam University

Title: The Dobrakov Integral Over Paths

Abstract: In 2002 the author introduced the definition and its properties of an analogue of Wiener measure over paths. In this article, using these concepts, we will derive an operator-valued measure over paths and will investigate the properties for integral with respect to the measure, Specifically, we will prove the Wiener integration formula for our integral and give some examples.

Mark Burgin, UCLA

Title: Feynman Integral in the Context of Hyperintegration

Abstract: Click Here for a PDF of the abstract

2:00 PM - 2:20 PM

Naoto Kumano-go, Kogakuin University
and Daisuke Fujiwara,
Gakushuin University

Title: Smooth Functional Derivatives in Feynman Path Integrals by Time Slicing Approximation

Abstract:We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, translation, real linear transformation and functional differentiation. The invariance under translation and orthogonal transformation, the integration by parts with respect to functional differentiation, the interchange of the order with Riemann-Stieltjes integrals, the interchange of the order with a limit, the semiclassical approximation and the fundamental theorem of calculus hold in Feynman path integral.

Troy Riggs, Union University

Title: Feynman’s operational calculus and the path space integral for the Dirac Equation.

Abstract: The Dirac equation in one (space) dimension has a solution in the form of a path integral.  Beginning with a formulation for this path integral given by T. Zastawniak, and using a technique of operator time-reordering, we see that this integral can be interpreted as a perturbation series.  This development provides a rigorous means of carrying out the “disentangling” process in Feynman’s time-ordered operational calculus for noncommuting operators.

2:30 PM - 3:15 PM

Michel Lapidus, University of California, Riverside

Title: From the Feynman Integral to Feynman’s Operational Calculus, and Back

Abstract: In this talk, we will discuss some of the analogies  and differences between the Feynman integral approach to quantum physics and the Feynman  operational calculus approach. In particular, by using perturbation expansions, one can see that the Feynman integral approach  naturally leads to the basic heuristic  rules of the operational calculus.  After having briefly recalled  the heuristic underlying the two approaches, we will show how, in turn,  the latter can lead to a precise definition (under suitable hypotheses) of the former. In fact, under suitable assumptions, a mathematical realization of the  operational calculus can also be viewed as an appropriate extension of and substitute for the Feynman integral in situations when the ‘integral’ itself is difficult to define directly. Conversely, the latter is a global object whose definition, in principle, does not depend on the existence of  any particular perturbation expansion. Much of the usefulness of the heuristic Feynman integral in contemporary mathematics and physics relies on the tension and cross-fertilization between these two apparently very different settings and approaches.
 
 Much of the rigorous part of the work presented in this talk is joint work with Gerald Johnson and is discussed in Chapters 16 and 18 of our joint book on the “Feynman Integral and Feynman’s Operational Calculus (Oxford Univ. Press, 2000), as well as in a joint paper in preparation.

3:30 PM - 4:00 PM
BREAK
4:00 PM - 4:45 PM

Louis Kauffman, University of Illinois Chicago

Title: Feynman Integrals and the Theory of Knots

Abstract: Witten showed how to formulate the Jones polynomial and its generalizations in terms of functional integrals and quantum field theory. This talk will review the heuristics of these constructions and how they are related to fully rigorous combinatorial models for the topological invariants. We then raise the question of new and as yet undiscovered physical interpretations for the categorifications of these invariants, particularly for Khovanov homology that generalizes the bracket polynomial state summation for the Jones polynomial.

 

 

This information is current as of May 7, 2006
SATURDAY-MAY 13TH
Twenty minute talks will be presented in parallel sessions
5:00 PM - 5:20 PM

Byoung Soo Kim, Seoul National University of Technology

Title: Extracting a Linear Factor in Feynman's Operational Calculi; Blend of Continuous and Discrete Measures 

Abstract:
Formulas which simplify the disentangling process under various conditions are important in Feynman's Operational Calculi. Extraction of a linear factor make it possible to carry out the disentangling in an iterative manner. We extend the extraction of a linear factor in the setting where the measures are allowed to have both continuous and discrete parts. 

Jan Van Casteren, University of Antwerp

Title: Backward Stochastic Differential Equations and Markov Processes

Abstract: In this lecture we endeavor to explain the notion of stochastic backward differential equations and its relationship with classical (backward) parabolic differential equations of second order. We will employ stochastic processes like Markov processes and martingale theory to treat semi-linear partial differential equations of parabolic type. Feynman-Kac formulae and heat equations fit in this set-up, and so do certain versions of the Hamilton-Jacobi-Bellmann equation.

 

5:30 PM - 5:50 PM

Adrian Lim, University of California San Diego

Title: TBA

Abstract: I will present an extension of a result by B. Driver. By defining a sequence of finite dimensional spaces of piecewise geodesics on a manifold equipped with a measure, one can write a sequence of integrals which converge to an integral over Wiener space.

Katherine Kime, University of Nebraska Kearney

Title: Finite Difference Approximation of Quantum-Mechanical Wave Packets

Abstract: If a wave packet is moving along and it reaches a place where things are a little bit different, some of it will continue onward and some of it will bounce back. It’s quite difficult to analyze such a situation using a wave packet because everything varies in time. It is much easier to work with steady-state solutions".    ---------The Feynman Lectures on Physics, Volume III

     One way that the above difficulty has been approached is through numerical approximation.  We will discuss the equations for N coupled harmonic oscillators and finite difference approximations of the wave and Schrodinger equations. We will compare the approximation of the impingement of a quantum-mechanical wave packet on a potential barrier with the approximation of the wave equation at an interface.  

7:00 PM
Banquet in honor of Jerry Johnson and Dave Skoug at the Cornhusker Hotel. For information about the banquet, go to the banquet page.