Audrey
Terras
Math. Dept., U.C.S.D.,
email address: aterras
at ucsd.edu
webpage updated March 27
Courses
Spring Quarter 2008
Math
142B - Advanced Calculus: MWF 12:00p -
12:50p WLH 2111
http://math.ucsd.edu/~aterras/ma142.htm
Math
20D - Intro/Differential Equations: MWF 1:00p - 1:50p CENTR 119
http://math.ucsd.edu/~aterras/ma20d.htm
Research
Interests
Spectra of Laplacians and Adjacency
Operators of Cayley Graphs, Ramanujan
graphs;
Selberg Trace
Formula; Fourier analysis on finite and infinite
groups;
Zeta Functions of Graphs; Automorphic
forms
My talk from the Assoc. for Women in
Math. Noether Lecture at the
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
http://math.ucsd.edu/~aterras/audrey banff
talk.pdf
http://math.ucsd.edu/~aterras/Audrey Banff Talk.ppt
AMS meeting in
Zeta Functions of Graphs, Ramanujan
Graphs, and Related Topics
http://math.ucsd.edu/~aterras/specialsession.htm
Draft of a Book
on Zeta Functions of Graphs:
http://math.ucsd.edu/~aterras/newbook.pdf
Beware
of typos! Please tell me about them.
Selected Papers
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.
Joint with A. Medrano, P. Myers, H.M. Stark, Finite Euclidean graphs
over rings, Proc. Amer. Math. Soc., 126 (1988), 701-710.
Joint with M. Martinez, H. Stark, Some Ramanujan
Hypergraphs Associated to GL(n,Fq),
Proc. A.M.S.,129 (2000), 1623-1629.
Joint with H. Stark, Zeta Functions of Finite Graphs and
Coverings, Part II, Advances in Math., 154 (2000), 132-195.
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.
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.
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.
Finite Quantum Chaos, a version of my AWM-MAA lecture at the MathFest, August, 2000, in
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.
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.
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
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.
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, 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
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
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.
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
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
Books
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.
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
Fourier
Analysis on Finite Groups and Applications,
Book
Description
Chapter
Contents
Congruences and the quotient ring of the
integers mod n;
SOME
OF MY 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,
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,
http://msri.org/publications/ln/msri/1999/random/terras/1/index.html
http://math.ucsd.edu/~aterras/euclid.gif
http://math.ucsd.edu/~aterras/chaos.gif
CONFERENCES
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)
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
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
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 –
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
15) I was one of the many lecturers in the
16) The 19th Algebraic Combinatorics
Symposium, July 1-3, 2002,
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 zeta 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
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.
