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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.**

**This is the 2 ^{nd} edition of my
1^{st} 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 2 ^{nd}
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 R ^{3}, 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.**

*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 P _{n}
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 P _{n}, Special functions on P_{n},
Harmonic analysis on P_{n} in polar
coordinates, Fundamental domains for P_{n}/GL(n,Z), Automorphic forms for GL(n,Z)
and harmonic analysis on P_{n}/GL(n,Z)**

**Chapter 5 **

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

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*Fourier
Analysis on Finite Groups and Applications***, Cambridge
U.
Press, Cambridge,
U.K., 1999**.

**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 **

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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,F _{q}),
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 **

**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. **

**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. **

**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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**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)**

**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**

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

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

**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 **

**and**

**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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**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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