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Audrey Terras

Math. Dept., U.C.S.D., La Jolla, CA 92093-0112

email address: aterras at ucsd.edu

 

Webpage updated Oct. 16, 2013

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I have retired !!! 

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Books  

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Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

The book is can be found at SpringerLink (link.springer.com) and as an ebook.

http://images.springer.com/cda/content/image/cda_displayimage.jpg?SGWID=0-0-16-1750145-0

This is the 2nd edition of my 1st book from 1985 (the days of typewriters).  It has been much revised with many new topics, updates, and figures.  This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. It is informal - not written for experts.  The book emphasizes motivation, concrete examples, history, and applications in mathematics, statistics, physics, and engineering.  New topics include: wavelets, quasicrystals, Ramanujan graphs, elliptic curves, modular knots, triangle and quaternion groups, comparisons of the continuous theory with its finite analogues, and recent progress on Mark Kac's question "Can you hear the shape of a drum?"  There are new examples and figures (such as density plots of modular forms as well as finite analogues).

 

Chapter Contents for 2nd Edition, Volume I

Chapter 1

Distributions or generalized functions, Fourier integrals, Fourier series and the Poisson summation formula, Mellin transforms, Epstein and Dedekind zeta functions, Finite Symmetric Spaces, Wavelets, Quasicrystals, Weyl’s criterion for uniform distribution

Chapter 2

Fourier Analysis on the Sphere, Spherical Harmonics, O(3) and R3, The Radon transform

Chapter 3

Hyperbolic geometry, Harmonic analysis on H, Fundamental domains for discrete subgroups Γ of G=SL(2,R), Automorphic forms - classical and not so classical (Maass wave forms), Automorphic forms and Dirichlet series, Hecke theory and generalizations, Harmonic analysis on the fundamental domain, The Roelcke-Selberg spectral resolution of the Laplacian, and the Selberg trace formula.

 

A better version of Figure 3.36 is here  http://math.ucsd.edu/~aterras/ellipticcurve.jpg.htm.

A movie related to Volume I showing a big bang related to points in the fundamental domain of the modular group Γ=SL(2,Z), which are Γ-equivalent to points on a horocycle moving down toward the real axis.   The y-axis has been distored so that infinity is at height 10.

http://math.ucsd.edu/~aterras/bigbang.htm.

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The old books are still available at SpringerLink and at your local university library perhaps for chapter by chapter download.  I am about to start on the 2nd edition of Volume II.

Harmonic Analysis on Symmetric Spaces and Applications, Vols. I, II, Springer-Verlag, N.Y., 1985, 1988.  

 

Volume 2 concerns higher rank symmetric spaces and their fundamental domains for discrete groups of isometries. Emphasis is on the general linear group G=GL(n,R) of invertible nxn real matrices and its symmetric space G/K which we identify with the space Pn of positive definite nxn real symmetric matrices. Applications in multivariate statistics and the geometry of numbers are considered.

 

Chapter Contents for Volume II

 I am working on a 2nd edition.

Chapter 4

Geometry and analysis on Pn, Special functions on Pn, Harmonic analysis on Pn in polar coordinates, Fundamental domains for Pn/GL(n,Z), Automorphic forms for GL(n,Z) and harmonic analysis on Pn/GL(n,Z)

Chapter 5

Geometry and analysis on G/K, Geometry and analysis on G\G/K 

 

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Zeta Functions of Graphs: A Stroll through the Garden,  Cambridge U. Press, Cambridge, U.K., 2011.  Available as an ebook.

Graph theory meets number theory in this book. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann or Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and a prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta function are produced, showing you cannot “hear” the shape of a graph.  The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum chaos and random matrix theory and also with expander and Ramanujan graphs, of interest in computer science. Many well-chosen illustrations and exercises, both theoretical and computer-based, are included throughout.

Part I.  A quick look at various zeta functions (Riemann, Ihara, Selberg and Ruelle zetas)

Part II. Ihara zeta function and the graph theory prime number theorem

Part III. Edge and path zeta functions

Part IV.  Finite unramified Galois coverings of connected graphs

Part V.  Last look at the garden

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Fourier Analysis on Finite Groups and Applications, Cambridge U. Press, Cambridge, U.K., 1999.   Available as an ebook.

http://www.cup.cam.ac.uk/

Book Description

This book gives a friendly introduction to Fourier analysis on finite groups, both commutative and non-commutative. Aimed at students in mathematics, engineering and the physical sciences, it examines the theory of finite groups in a manner that is both accessible to the beginner and suitable for graduate research. With applications in chemistry, error-correcting codes, data analysis, graph theory, number theory and probability, the book presents a concrete approach to abstract group theory through applied examples, pictures and computer experiments. In the first part, the author parallels the development of Fourier analysis on the real line and the circle, and then moves on to analogues of higher dimensional Euclidean space. The second part emphasizes matrix groups such as the Heisenberg group of upper triangular 2x2 matrices. The book concludes with an introduction to zeta functions on finite graphs via the trace formula.

Chapter Contents

Congruences and the quotient ring of the integers mod n; The Discrete Fourier transform on the finite circle; Graphs of Z/nZ, adjacency operators, eigenvalues; Four questions about Cayley graphs; Finite Euclidean graphs and three questions about their spectra; Random walks on Cayley graphs; Applications; in geometry and analysis; The quadratic reciprocity law The fast Fourier transform; The DFT on finite Abelian groups - finite tori; Error-correcting codes; The Poisson sum formula on a finite Abelian group; Some applications in chemistry and physics; The uncertainty principle; Fourier transform and representations of finite groups; Induced representations; The finite ax+b group; Heisenberg group; Finite symmetric spaces - finite upper half planes Hq; Special functions on Hq - K-Bessel and spherical; The general linear group GL(2,Fq), Fq =finite field; Selberg’s trace formula and isospectral non-isomorphic graphs; The trace formula on finite upper half planes; The trace formula for a tree and Ihara’s zeta function.

 

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Lecture Notes on various courses – beware of typos

 

Lectures on Advanced Calculus with Applications (Math 142 a and b)

http://math.ucsd.edu/~aterras/advanced calculus lectures.pdf

 

Lectures on Applied Algebra (Math. 103 a and b) 

http://math.ucsd.edu/~aterras/applied algebra.pdf

http://math.ucsd.edu/~aterras/applied algebraII.pdf

 

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Talks.

My talk at Durham Symposium on Graph Theory and Interactions, Monday, July 15 - Thursday, July 25, 2013

http://math.ucsd.edu/~aterras/finite & Poincare uhp.pdf

 

My talk at Newton Institute, July, 2010

http://math.ucsd.edu/~aterras/2010 newton.pdf

http://math.ucsd.edu/~aterras/2010 newton.ppt

 

My talks at CRM Montreal, June-July, 2009

the pdfs

http://math.ucsd.edu/~aterras/montreal lecture1.pdf

http://math.ucsd.edu/~aterras/montreal lecture2.pdf

http://math.ucsd.edu/~aterras/montreal lecture3.pdf

http://math.ucsd.edu/~aterras/montreal lecture4.pdf

the powerpoint

http://math.ucsd.edu/~aterras/montreal lecture1.ppt

http://math.ucsd.edu/~aterras/montreal lecture2.ppt

http://math.ucsd.edu/~aterras/montreal lecture3.ppt

http://math.ucsd.edu/~aterras/montreal lecture4.ppt

 

My talk October 30,  U.C.S.D.  Math. Club

http://math.ucsd.edu/~aterras/What are primes in graphs and how many.pdf

http://math.ucsd.edu/~aterras/What are primes in graphs and how many.ppt

 

My talk from October 4, AMS Meeting in Vancouver, Canada

http://math.ucsd.edu/~aterras/ihara zeta and QC.pdf

http://math.ucsd.edu/~aterras/ihara zeta and QC.ppt

 

My talks from MSRI Graduate  Workshop, A Window  Into  Zeta  And Modular Physics, June 16-27, 2008.

the paper: http://math.ucsd.edu/~aterras/msripaper.pdf

 

the pdfs

http://math.ucsd.edu/~aterras/msri_llecture1.pdf

http://math.ucsd.edu/~aterras/msri_llecture2.pdf

http://math.ucsd.edu/~aterras/msri_llecture3.pdf

 

the powerpoint files

http://math.ucsd.edu/~aterras/msri_llecture1.ppt

http://math.ucsd.edu/~aterras/msri_llecture2.ppt

http://math.ucsd.edu/~aterras/msri_llecture3.ppt

 

My talk from the Assoc. for Women in Math. Noether Lecture at the San Diego (examples of primes slide corrected to eliminate tail)

AMS meeting Jan. 7, 2008 including the parts that did not make it to the actual lecture:

http://math.ucsd.edu/~aterras/noether.pdf

http://math.ucsd.edu/~aterras/noether.ppt

 

My talk from Banff meeting on Quantum Chaos: Routes to RMT Statistics and Beyond, February 24 - 29, 2008

http://math.ucsd.edu/~aterras/audrey banff talk.pdf

http://math.ucsd.edu/~aterras/Audrey Banff Talk.ppt

 

AMS meeting in San Diego Special Session on Zeta Functions of Graphs, Ramanujan Graphs, and Related Topics

http://math.ucsd.edu/~aterras/specialsession.htm

 

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Selected Papers

1) Survey of Spectra of Laplacians on Finite Symmetric Spaces, Experimental Math., 5 (1996), 15-32.

Joint with H. Stark, Zeta Functions of Finite Graphs and Coverings, Advances in Math., 121 (1996), 124-165.

2) Joint with A. Medrano, P. Myers, H.M. Stark, Finite Euclidean graphs over rings, Proc. Amer. Math. Soc., 126 (1988), 701-710.

3) Joint with M. Martinez, H. Stark, Some Ramanujan Hypergraphs Associated to GL(n,Fq), Proc. A.M.S.,129 (2000), 1623-1629.

4) Joint with H. Stark, Zeta Functions of Finite Graphs and Coverings, Part II, Advances in Math., 154 (2000), 132-195.

5) Joint with D. Wallace, Selberg's trace formula on the k-regular tree and applications, Internatl. J. of Math. and Math. Sci., Vol. 2003, No. 8, pp. 501-526.

6) Statistics of graph spectra for some finite matrix groups: Finite quantum chaos, in Proceedings International Workshop on Special Functions - Asymptotics, Harmonic Analysis and Mathematical Physics, June 21-25, 1999, Hong Kong, Edited by Charles Dunkl, Mourad Ismail, and Roderick Wong, World Scientific, Singapore, 2000, pages 351-374.

7) Joint with H. Stark, Artin L-Functions of Graph Coverings, in Contemporary Math., Vol. 290, Dynamical, Spectral, and Arithmetic Zeta Functions - Edited by Michel L. Lapidus, and Machiel van Frankenhuysen, Amer. Math. Soc., 2001, pages 181-195.

8) Finite Quantum Chaos, a version of my AWM-MAA lecture at the MathFest, August, 2000, in Los Angeles - Amer. Math. Monthly, Vol. 109 (Feb. 2002), 121-139. To see the figures in color, go to the website http://math.ucsd.edu/~aterras/comonth.html.

9) Joint with M. DeDeo, M. Martinez, A. Medrano, M. Minai, H. Stark, Spectra of Heisenberg graphs over finite rings, 2003 Supplement Volume of Discrete and Continuous Dynamical Systems, devoted to the Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24-27, 2002, at Wilmington, NC, Edited by W. Feng, S. Hu and X. Lu, pages 213-222.

10) Joint with M. DeDeo, M. Martinez, A. Medrano, M. Minai, H. Stark, Zeta functions of Heisenberg graphs over finite rings, in Theory and Applications of Special Functions, A volume dedicated to Mizan Rahman, edited by M. Ismail and E. Koelink, Springer-Verlag, Developments in Math., Vol. 13, N.Y., 2005,  pp. 165-183.

11) Joint with H. Stark, Zeta functions of graph coverings, in DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, Volume: 64, edited by M. Nathanson, Amer. Math. Soc., 2004, pp. 199-212.   http://www.ams.org/bookstore?fn=20&arg1=dimacsseries&item=DIMACS-64

Comparison of Selberg's Trace Formula with its Discrete Analogues," in DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, Volume: 64, edited by M. Nathanson, Amer. Math. Soc., 2004, pp. 213-225.   http://www.ams.org/bookstore?fn=20&arg1=dimacsseries&item=DIMACS-64

12) Finite models for quantum chaos, IAS/Park City Mathematics Series, Vol. 12 (2007), Automorphic Forms and Applications; Edited by: Peter Sarnak and Freydoon Shahidi. pages 333-375.

                      http://www.ams.org/bookstore/pcmsseries

13) Joint with H. Stark, Zeta Functions of Finite Graphs and Coverings, Part III, Advances in Mathematics 208 (2007) 467–489.

14) Joint with M. D. Horton and D. Newland, The Contest between the Kernels in the Selberg Trace Formula for the (q+1)-regular Tree, in Contemporary Mathematics, Volume 398 (2006), The Ubiquitous Heat Kernel, Edited by Jay Jorgenson and Lynne Walling, pages 265-294.                       http://www.ams.org/bookstore/conmseries

15) Joint with M. D. Horton and H. M. Stark, What are Zeta Functions of Graphs and What are They Good For?,  Contemporary Mathematics, Volume 415 (2006), Quantum Graphs and Their Applications; Edited by Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, and Peter Kuchment, pages 173-190. 

                     http://www.ams.org/bookstore/conmseries

16) Joint with Anthony Shaheen, Fourier expansions of complex-valued Eisenstein series on finite upper half planes, International Journal of Mathematics and Mathematical Sciences, Volume 2006, Article ID 63918, Pages 1–17.

17) Joint with M. D. Horton and H. M. Stark, Zeta Functions of Weighted Graphs and Covering Graphs, in Proc. Symp. Pure Math., Vol. 77, Analysis on Graphs, Edited by Exner, Keating, Kuchment, Sunada and Teplyaev, AMS, 2008.

18) Zeta functions and Chaos, submitted for a chapter in the volume from the MSRI conference organized by Floyd Williams, titled Window into Zeta and Modular Physics. http://www.math.ucsd.edu/~aterras/msripaper.pdf

19) Looking into a Graph Theory Mirror of Number Theoretic Zetas, submitted to the volume from the Banff Women in Numbers Conference Proceedings. http://www.math.ucsd.edu/~aterras/mywinpaper.pdf

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preliminary versions of some papers with color pictures

Joint with D. Wallace, Selberg's trace formula on the k-regular tree and applications

              http://math.ucsd.edu/~aterras/treetrace.pdf

 

Joint with M. DeDeo, M. Martinez, A. Medrano, M. Minai, H. Stark, Spectra of Heisenberg graphs over finite rings: Histograms, Zeta Functions, and Butterflies

              http://math.ucsd.edu/~aterras/heis.pdf

 

Joint with H. Stark, Zeta Functions of Finite Graphs and Coverings, Part III, Advances in Mathematics 208 (2007) 467–489

              http://math.ucsd.edu/~aterras/newbrauer.pdf

 

Joint with M. D. Horton and D. Newland, The Contest between the Kernels in the Selberg Trace Formula for the (q+1)-regular Tree.   

              http://math.ucsd.edu/~aterras/heatblasted.pdf

 

Joint with M. D. Horton and H. M. Stark, What are Zeta Functions of Graphs and What are They Good For?        

             http://math.ucsd.edu/~aterras/snowbird.pdf

 

Joint with Anthony Shaheen, Fourier expansions of complex-valued Eisenstein series on finite upper half planes,

              http://math.ucsd.edu/~aterras/finite fourier expansions.pdf

 

Joint with M. D. Horton and H. M. Stark, Zeta Functions of Weighted Graphs and Covering Graphs, preprint;      http://www.math.ucsd.edu/~aterras/cambridge.pdf

 

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SOME OF MY EARLIER TALKS

1) talk given in the Analysis on Graphs and its Applications Program at Newton Institute, Cambridge, England,  March, 2007 ;   (examples of primes slide corrected to eliminate tail)

        newton.pdf

        newton.ppt

2) a stroll through the graph zeta garden (given at IAS women & math. program, may, 2006)  zeta stroll.pdf

3) What are zeta functions of graphs and what are they good for?  (given at Snowbird, Aachen and Princeton in 2005)       what are zetas.pdf

4) Introduction to Artin L-Functions of Graph Coverings, Winter, 2004 at IPAM, UCLA:    pdf version (new ucla talk.pdf);     powerpoint version (fun zeta and L fns.ppt)

5) Introductory lectures on finite quantum chaos (newchaos.pdf)  

6) Artin L-Functions of Graph Coverings, Part I (Summer, 2002) artin1.pdf

   Artin L-Functions of Graph Coverings, Part II (Summer, 2002) artin2.pdf

7) "Artin L-functions of Graph Coverings" given at Math. Sciences Research Institute, Berkeley, CA - June 7-11, 1999:  Random Matrices and Their Applications:   Quantum Chaos, GUE Conjecture for Zeros of Zeta Functions, Combinatorics, and All That.       http://msri.org/publications/ln/msri/1999/random/terras/1/index.html

 

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SOME ANIMATIONS   http://math.ucsd.edu/~aterras/euclid.gif

http://math.ucsd.edu/~aterras/chaos.gif

 

image005.gif

 

 

 

 

 

 

 

 

 

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CONFERENCES

-3) Durham Symposium on Graph Theory and Interactions, Monday, July 15 - Thursday, July 25, 2013

-2) MSRI - 2011- Arithmetic Statistics,  January 10, 2011 – May 20, 2011

-1) SMS - 2009 Summer School on Automorphic Forms and L-Functions: Computational Aspects, June 22 - July 3, 2009

0) 08w5112 WIN: Women in Numbers, November 2 -7, 2008. http://www.birs.ca/birspages.php?task=displayevent&event_id=08w5112

1) MSRI Graduate  Workshop, A Window  Into  Zeta  And Modular Physics, June 16-27, 2008.

http://www.msri.org/calendar/sgw/WorkshopInfo/449/show_sgw

2) AIM, Workshop on Computing arithmetic spectra, March 10 - 14, 2008

http://www.aimath.org/ARCC/workshops/arithspectra.html

3) Banff meeting on Quantum Chaos: Routes to RMT Statistics and Beyond, February 24 - 29, 2008

http://www.math.tamu.edu/~berko/banff/

4) IPAM meeting on Expanders in Pure and Applied Mathematics, February 11 - 15, 2008

http://www.ipam.ucla.edu/programs/eg2008/

5) AMS meeting in San Diego, Noether Lecture: Monday January 7, 2008, 10:05 a.m.-10:55 a.m.

Special Sessions:  Zeta Functions of Graphs, Ramanujan Graphs, and Related Topics, Sunday January 6, 2008, 8:00-10:50 a.m., 2:15- 6:05 p.m

Expanders and Ramanujan Graphs: Constructions and Applications,  Tuesday Jan. 8, 1:00 p.m.-5:50 p.m., Wednesday January 9, 2008, 8:00 a.m.-10:50 a.m., 1:00 p.m.-5:50 p.m.

6) Southern California Number Theory, UC Irvine, October 27, 2007

           http://math.uci.edu/~krubin/scntd/

7) Isaac Newton Institute for Mathematical Sciences, Analysis on Graphs and its Applications, 8 January - 29 June 2007;     http://www.newton.cam.ac.uk/programmes/AGA/

8)  IAS Program for Women in Math., May 16-27, 2996   http://www.math.ias.edu/womensprogram    or   http://www.math.ucsd.edu/~aterras/ias women.pdf

9) Conference on Lie Groups, Representations and Discrete Mathematics, IAS Princeton, February 6 - 10, 2006

10)   Seminar Aachen-Köln-Lille-Siegen on Automorphic Forms, June 29, 2005     

     http://www.matha.rwth-aachen.de/seminar-akls/Automorphic_Forms__Aachen-2005-06-29.pdf

11)   The AMS – IMS – SIAM Joint Summer Research Conference on Quantum Graphs and Their Applications; Sunday, June 19 to Thursday, June 23;  http://www.math.tamu.edu/~kuchment/src05_graphs.htm

12)  Number Theory Conference in Honor of Harold Stark, Aug. 5-7, 2004;  (http://math.ucsd.edu/~aterras/Birthday.ppt)

13)  Workshop on Automorphic Forms, Group Theory and Graph Expansion, Feb. 9-13, 2004, Institute for Pure and Applied Math. at UCLA. Website (http://www.ipam.ucla.edu/programs/agg2004/ )

14)  Computational Number Theory Workshop at the Foundations of Computational Mathematics 2002 Meeting at the University of Minnesota, Aug. 8-10, 2002. The website is: http://www.ima.umn.edu/geoscience/summer/FoCM02/index.html

15)  I was one of the many lecturers in the Park City summer research session which took place in Park City, Utah from June 30 to July 20, 2002. The topic was Automorphic Forms. I was there for the segment on quantum chaos. For information about this program you can go to the website http://www.ias.edu/parkcity.

16)  The 19th Algebraic Combinatorics Symposium, July 1-3, 2002, Kumamoto University Kumamoto, Japan, http://www.kumamoto-u.ac.jp/univ-e.html

17)  I organized a Special Session on Zeta Functions of Graphs and Related Topics at the Fourth International Conference on Dynamical Systems and Differential Equations to be held May 24-27, 2002 in Wilmington, North Carolina. The aim of the session was to discuss current work on the Ihara-Selberg functions attached to graphs and related topics such as Ramanujan graphs, the trace formula on trees. The hope was to emphasize connections between various fields such as graph theory, topology, mathematical physics, number theory, dynamical systems. One example is the connection between graph zeta functions and Jones polynomials of knots found by Lin and Wang. The conference website is http://www.uncwil.edu/mathconf/.  Special session abstracts can be found at abstracts.htm.   Proceedings appeared in 2003 Supplement Volume of Discrete and Continuous Dynamical Systems, devoted to the Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24-27, 2002, at Wilmington, NC, Edited by W. Feng, S. Hu and X. Lu

 

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Some of my Pictures can be found at

pictures.pdf,

more pictures.pdf,

and

11 tessellation.pdf .

The last one is a tessellation of the finite upper half plane for the field with 11*11 elements coming from the group of non-singular 2x2 matrices from the field with 11 elements.  Explanations can be found in

http://www.math.ucsd.edu/%7Eaterras/newchaos.pdf .  

 

An alternative picture of that tessellation follows.

 

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11 tess.bmp

 

Short Biography

 

AUDREY TERRAS received her B.S. degree in Mathematics from the University of Maryland, College Park in 1964, where she was inspired by the lectures of Sigekatu Kuroda to become a number theorist. She was particularly impressed by the use of analysis (in particular using zeta functions and multiple integrals) to derive algebraic results. She received her M.A. (1966) and Ph.D. (1970) from Yale University. In 1972 she became an assistant professor of mathematics at the University of California, San Diego. She became a full professor at U.C.S.D. in 1983. She retired in 2010. She has had 25 Ph.D. students. She is a fellow of the American Mathematical Society and of the American Association for the Advancement of Science, has served on the Council of the American Mathematical Society, gave the 2008 Noether lecture of the Association for Women in Mathematics. She has published 4 books, helped to edit another, and published lots of research papers. Her research interests include number theory, harmonic analysis on symmetric spaces and finite groups (including applications), special functions, algebraic graph theory, especially zeta functions of graphs, arithmetical quantum chaos, and Selberg’s trace formula. When lecturing on mathematics, she believes it is important to give examples, applications and colorful pictures.

 

Ph.D. STUDENTS WITH COMPLETED DEGREES   (U.C.S.D.)    25 students 

        2010      Thomas Petrillo   

        2006      Matthew Horton

        2005      Derek Newland, Anthony Shaheen

        2001      F. Javier Marquez

        2000     Marvin Minei

        1998      Maria Martinez, Michelle DeDeo, Archie Medrano

        1995      Perla Myers

        1993      Jeff Angel, Cindy Trimble

        1991      Nancy Celniker, Steven Poulos, Elinor Velasquez    

        1989      Maria Zack

        1988      Jason Rush

        1986      Daniel Gordon, Douglas Grenier, Dennis Healy

        1985      Michael Berg

        1982      Dorothy Wallace Andreoli, John Hunter

        1981      Thomas Bengtson

        1979      Kaori Imai Ohta

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