Audrey
Terras
Math.
Dept., U.C.S.D.,
email
address: aterras at
ucsd.edu
webpage
updated July 1, 2008
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 talks from MSRI Graduate Workshop, A Window
Into Zeta And Modular Physics, June 16-27, 2008.
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
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 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,
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.
