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Audrey Terras

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

Webpage updated January, 2019

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Books

Abstract Algebra with Applications, Cambridge University Press, 2018

Abstract Algebra with Applications provides a friendly and concise introduction to algebra, with an emphasis on its uses in the modern world. The first part of this book covers groups, after some preliminaries on sets, functions, relations, and induction, and features applications such as public-key cryptography, Sudoku, the finite Fourier transform, and symmetry in chemistry and physics. The second part of this book covers rings and fields, and features applications such as random number generators, error correcting codes, the Google page rank algorithm, communication networks, and elliptic curve cryptography. The book's masterful use of colorful figures and images helps illustrate the applications and concepts in the text. Real-world examples and exercises will help students contextualize the information. Meant for a year-long undergraduate course in algebra for mathematics, engineering, and computer science majors, the only prerequisites are calculus and a bit of courage when asked to do a short proof.

Harmonic Analysis on Symmetric Spaces—Higher Rank Spaces, Positive Definite Matrix Space and Generalizations, 2nd Edition, Springer, 2016.

This book gives an introduction to harmonic analysis on symmetric spaces, focusing on advanced topics such as higher rank spaces, positive definite matrix space and generalizations. It is intended for beginning graduate students in mathematics or researchers in physics or engineering. As with the earlier book entitled "Harmonic Analysis on Symmetric SpaceEuclidean Space, the Sphere, and the Poincaré Upper Half Plane, the style is informal with an emphasis on motivation, concrete examples, history, and applications. The symmetric spaces considered here are quotients X=G/K, where G is a non-compact real Lie group, such as the general linear group GL(n,R) of all n x n non-singular real matrices, and K=O(n), the maximal compact subgroup of orthogonal matrices. Other examples are Siegel's upper half "plane" and the quaternionic upper half "plane". In the case of the general linear group, one can identify X with the space Pn of n x n positive definite symmetric matrices.

Many corrections and updates are included in this new edition. Updates include discussions of random matrix theory and quantum chaos, as well as recent research on modular forms and their corresponding L-functions in higher rank.  Many applications have been added, such as the solution of the heat equation on Pn, the central limit theorem of Donald St. P. Richards for Pn, results on densest lattice packing of spheres in Euclidean space, and GL(n)-analogs of the Weyl law for eigenvalues of the Laplacian in plane domains, as well as computations of analogues of Maass waveforms for GL(3).

Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, fundamental domains in X for discrete groups Γ (such as the modular group GL(n,Z) of n x n matrices with integer entries and determinant ±1), connections with the problem of finding densest lattice packings of spheres in Euclidean space, automorphic forms, Hecke operators, L-functions, and the Selberg trace formula and its applications in spectral theory as well as number theory.

A movie related to this book showing the projection of (t,v,x1,x2,x3) onto the x-coordinates in the Grenier fundamental domain (see page 151 of the old edition) for GL(3,Z) acting on the determinant one surface in positive 3x3 matrix space as the coordinates (t,v) travel along t=v  from .9 to 1.03.

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane, 2nd Edition, Springer, NY, 2013

The 2nd edition includes corrections, new topics, and updates.  It is intended for beginning graduate students in mathematics and statistics, or researchers in physics or engineering.  The prerequisites are minimized and the style is informal, with emphasis on motivation, concrete examples, history and applications in mathematics, statistics, physics, and engineering.  Topics include: inversion formulas for Fourier transforms, the Radon transform, NonEuclidean geometry, Poisson's summation formula, fundamental domains for discrete groups, tessellations of symmetric spaces, special functions, modular forms, Maass wave forms, the Selberg trace formula, finite analogues of symmetric spaces.  Applications include: the central limit theorem, CAT scans, microwave engineering, the hydrogen atom, expander graphs, crystals and quasicrystals, wavelets, modular knots, L-functions, zeta functions, spectral theory of the Laplacian.

The new and old editions of these books are available through SpringerLink and as ebooks. http://www.springer.com

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.

The Old Editions.

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

Volume 1 gives an introduction to harmonic analysis on the simplest symmetric spaces - Euclidean space, the sphere, and the Poincaré upper half plane H and fundamental domains for discrete groups of isometries such as SL(2,Z) in the case of H. The emphasis is on examples, applications, history. The intention is to be a friendly introduction for non-experts.

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

Volume I

Chapter 1

Distributions or generalized functions Fourier integrals Fourier series and the Poisson summation formula Mellin transforms, Epstein and Dedekind zeta functions

Chapter 2

Spherical Harmonics O(3) and R3. The Radon transform

Chapter 3

Hyperbolic geometry Harmonic analysis on H Fundamental domains for discrete subgroups G of G=SL(2,R) Automorphic forms - classical Automorphic forms- 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.

Chapter Contents

Volume II

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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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.

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)

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

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Talks.

My talk at Durham Symposium on Graph Theory and Interactions

Monday 15 July - Thursday 25 July 2013

My talk at Newton Institute, July, 2010

My talks at CRM Montreal, June-July, 2009

the pdfs

the powerpoint

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

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

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

the pdfs

the powerpoint files

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:

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

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

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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

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.

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.

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.

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.

19) Looking into a Graph Theory Mirror of Number Theoretic Zetas, submitted to the volume from the Banff Women in Numbers Conference Proceedings.

20) Finite Analogs of Maass Wave Forms, preprint.  http://www.math.ucsd.edu/~aterras/finite analogs of maass forms.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

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

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

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

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

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

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)

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

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CONFERENCES

-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.

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

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

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

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

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

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

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

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 Association for Women in Mathematics, the American Mathematical Society, and 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 5 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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