UPCOMING CONFERENCES:
Random
Processes (Holy Cross, April 2011)
Undergraduate
Research Session (Holy Cross, April 2011)
- RESEARCH INTERESTS:
Analytic Number Theory, Random Matrix Theory, Analysis and
Probability (distribution of zeros and n-level statistics for
families of L-functions, especially families of elliptic curves with rank over
Q(T), classical random matrix theory, random graphs, ranks of elliptic curves,
computational number theory, probability theory, Benford's Law, cryptography,
linear programming, multiple Dirichlet series).
Thesis
Papers
Talks
My Book (An
Invitation to Modern Number Theory)
Handouts
My
Riddles Page
ACCEPTED PAPERS
Click here for an annotated
bibliography of my work
- 1- and 2-level densities for rational families of elliptic curves:
evidence for the underlying group symmetries.
Compositio Mathematica (140
(2004),
no. 4, 952-992).
pdf
- Distribution of eigenvalues for the ensemble
of real symmetric Toeplitz matrices (with Chris Hammond).
Journal of
Theoretical Probability
(18 (2005), no. 3, 537-566).
pdf
- Benford's law, values of L-functions and the
3x+1 problem (with Alex Kontorovich),
Acta Arithmetica. (120
(2005), no. 3, 269–297).
pdf.
- Variation in the number of points on
elliptic curves and applications to excess rank,
C.
R. Math. Rep. Acad. Sci. Canada
(27 (2005), no. 4, 111–120, Expanded Version). pdf
- Incomplete exponential
sums in several variables (with Eduardo Dueñez,
Howard Straubing, and Amitahba Roy),
Journal of Number Theory
(116 (2006), no. 1, 168–199).
pdf
- Closed-form Bayesian inferences for the logit model via polynomial expansions (with Eric Bradlow and Kevin Dayaratna),
Quantitative Marketing and Economics
(4 (2006), no. 2, 173–206).
pdf
- Investigations of zeros near the central point of elliptic curve
L-functions,
Experimental Mathematics (15 (2006), no. 3,
257–279).
pdf
(data available
online).
- The low-lying zeros of a GL(4) and a GL(6) family of
L-functions (with
Eduardo Dueñez),
Compositio Mathematica
(142 (2006), no. 6,
1403–1425)
pdf.
- Low-lying zeros of L-functions with orthogonal symmetry (with Chris
Hughes),
Duke Mathematical Journal (136
(2007), no. 1, 115–172)
pdf
- Constructing one-parameter families of
elliptic curves
over Q(T) with moderate rank (with Scott Arms, Alvaro Lozano-Robledo),
Journal of Number Theory. (123
(2007), no 2, 388–402
pdf
- A derivation of James' Pythagorean
projection,
By The Numbers -- The
Newsletter of the SABR Statistical Analysis Committee (16
(February 2006), no. 1, 17–22).
pdf (expanded
version:
pdf).
Chance Magazine (20 (Winter 2007), no. 1, 40-48).
- Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and
circulant matrices (with Adam Massey and
John Sinsheimer),
Journal of
Theoretical Probability. (20 (2007), no. 3,
637--662.)
pdf
- Benford's Law applied to hydrology data - results and relevance to
other geophysical data (with Mark Nigrini),
Mathematical Geology (39
(2007), no. 5, 469--490).
pdf
- When the Cramer-Rao Inequality provides no information,
Communications in Information and Systems.
(7 (2007), no. 3, 265--272).
pdf
- The Modulo 1 Central Limit Theorem and Benford's Law for Products
(with Mark Nigrini),
International Journal of Algebra. (2 (2008), no. 3,
119--130).
pdf
- A Symplectic Test of the L-Functions Ratios Conjecture,
IMRN (2008). (Vol. 2008, article ID rnm146, 36 pages, doi:10.1093/imrn/rnm146).
pdf
- An identity for sums of polylogarithm functions,
Integers (electronic journal of combinatorial number theory). (8
(2008), A15)
pdf
- A probabilistic proof of Wallis' formula for
π,
American Mathematical Monthly. (115 (2008), no. 8, 740--745)
pdf
- The distribution of the second largest eigenvalue in families of
random regular graphs
(with Tim Novikoff and Anthony Sabelli),
Experimental Mathematics. (17 (2008), no. 2,
231--244.)
pdf
- Order statistics and Benford's law
(with Mark Nigrini),
International Journal of Mathematics and Mathematical Sciences
(Volume 2008 (2008), Article ID 382948, 19 pages,
doi:10.1155/2008/382948)
pdf
- Lower order terms in the 1-level density for families of holomorphic
cuspidal newforms,
Acta Arithmetica.
(137 (2009), 51-98)
pdf
- Chains of distributions, hierarchical Bayesian models and Benford's
Law (with D. Jang, J. U. Kang, A. Kruckman and J. Kudo),
Journal of
Algebra, Number Theory: Advances and Applications. (volume 1, number
1 (March 2009), 37--60)
pdf
- An orthogonal test of the L-functions Ratios Conjecture,
London Mathematical Society. (2009,
doi:10.1112/plms/pdp009)
pdf
- Silver Scheduler:
Demand-Driven Scheduling of
Movies in a Multiplex (with Jehoshua Eliashberg, Quintus Hegie, Jason Ho, Dennis Huisman, Sanjeev
Swami, Charles B. Weinberg and Berend Wierenga), lead article
International Journal of Market Research
and won award for best paper of 2009
(doi:10.1016/j.ijresmar.2008.09.004)
pdf
- The effect of convolving families of
L-functions on the underlying group symmetries (with Eduardo Dueñez),
Proceedings of the London Mathematical Society. (2009; doi: 10.1112/plms/pdp018)
pdf
- When almost all sets are difference dominated (with Peter Hegarty),
Random Structures and Algorithms. (35, 2009, no. 1, 118-136)
pdf
- Nuclei, primes and the random matrix connection (with Frank W. K.
Firk), invited paper to
Symmetry
(1, (2009), 64--105; doi:10.3390/sym1010064)
pdf
- Data diagnostics using second order tests of Benford's Law (with Mark Nigrini),
Auditing: A Journal of Practice and Theory. (28 (2009),
no. 2, 305--324. doi: 10.2308/aud.2009.28.2.305)
MSWord file
- Explicit constructions of infinite families of MSTD sets (with
Brooke Orosz and Dan Scheinerman), Journal
of Number Theory (130 (2010) 1221--1233)
pdf
- A unitary test of the L-functions Ratios Conjecture (with John Goes,
Steven Jackson, David Montague, Kesinee Ninsuwan, Ryan Peckner and Thuy
Pham),
Journal of Number Theory.
(130 (2010) 2238--2258)
pdf
- Towards an `average' version of the Birch and Swinnerton-Dyer
Conjecture (with John Goes),
Journal of Number Theory (130
(2010), no. 10, 2341--2358)
pdf
- An Orthogonal Test of the L-functions Ratios Conjecture, II (with
David Montague), Acta Arithmetica
(146 (2011), 53--90.)
pdf
- Rational irrational proofs (with David Montague), to appear in
Mathematics Magazine.
pdf
(expanded version:
pdf;
letter from Nerode and Tennenbaum on history of proof:
pdf)
- Explicit constructions of infinite families of MSTD sets (with Dan
Scheinerman),
Additive Number Theory: Festschrift In Honor of the Sixtieth Birthday of
Melvyn B. Nathanson, David Chudnovsky and Gregory Chudnovsky
(Editors), Springer-Verlag, 2010.
pdf
- Demand-driven scheduling of movies in a multiplex (with Jehoshua
Eliashberg and Charles B. Weinberg),
newsletter of the European Marketing Academy, October 2010
(requested summary of
Silver-Scheduler paper in honor of it receiving the
International Journal of Market Research Best Paper Award for 2009).
- Effective equidistribution and the Sato-Tate law for families of
elliptic curves (with M. Ram Murty),
Journal of Number Theory (131
(2011), no. 1, 25-44)
pdf
- The lowest eigenvalue in Jacobi ensembles and Painlevé VI (with
Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina
Snaith),
Journal
of Physics A: Mathematical and Theoretical (43 (2010),
405204, 27pages)
pdf
- Isoperimetric Sequences (with Frank Morgan, Edward Newkirk and Lori
Pedersen),
Mathematics Magazine (84 (Feb. 2011), 37--42).
pdf
- Distribution of eigenvalues for highly palindromic real symmetric
Toeplitz matrices (with Steven Jackson and Thuy Pham), to appear in the
Journal of
Theoretical Probability.
pdf
- A combinatorial identity for studying Sato-Tate type problems (with
M. Ram Murty and F. Strauch), to appear in
Rendiconti del Seminario Matematico.
pdf
- Moments of the rank of elliptic curves (with Siman Wong), to appear
in the Canadian Journal of Mathematics.
pdf
- The `Real' Schwarz lemma (with David Thompson): to appear in the
American Mathematical Monthly.
pdf
- An elliptic curve family test of the Ratios Conjecture (with Duc
Khiem Huynh and Ralph Morrison),
Journal of Number Theory
(131 (2011), 1117--1147)
pdf
- On the number of summands in Zeckendorf decompositions (with Gene S. Kopp, Murat Koloğlu and Yinghui
Wang), Fibonacci
Quarterly. (49 (2011), no 2, 116--130)
pdf
- The Limiting Spectral Measure for Ensembles of Symmetric Block
Circulant Matrices (with
Gene S. Kopp Murat Koloğlu,
Frederick Strauch, Wentao Xiong). to appear in the Journal of
Theoretical Probability pdf
- Gaussian Behavior in Generalized Zeckendorf Decompositions (with
Yinghui Wang). Submitted to Conference Proceedings of
CANT 2011.
pdf
- Finding and Counting MSTD sets (with Geoffrey Iyer, Oleg Lazarev and Liyang Zhang). Submitted to Conference Proceedings of
CANT 2011.
pdf
- Generalized Ramanujan Primes (with Nadine Amersi, Olivia Beckwith, Ryan Ronan and Jonathan Sondow). Submitted to Conference Proceedings of
CANT 2011.
pdf
- Generalized More Sums Than Differences Sets (with Geoffrey Iyer, Oleg Lazarev and Liyang Zhang). Submitted to the Journal of Number Theory.
pdf
SUBMITTED PAPERS
Click here for an annotated
bibliography of my work
- From Fibonacci Numbers to Central Limit Type Theorems (with Yinghui
Wang), submitted to
Journal of Combinatorial Theory, Series A.
pdf (expanded arxiv version)
- Quadratic fields with cyclic 2-class groups (with Carlos Dominguez
and Siman Wong): submitted to the
Journal of Number Theory.
pdf
- The Weibull distribution and Benford's Law (with Victoria Cuff and
Allie Lewis), submitted to
The
American Statistician
pdf
- Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets (with Luc Robinson and Sean Pegado). Submitted to
INTEGERS.
pdf
- Low-lying zeros of number field L-functions (with Ryan Peckner), submitted to Acta Arithmetica. pdf
- Virus Propagation in Certain Types of Networks (with Thealexa Becker, Alec Greaves-Tunnell, Leo Kontorovich, Amitabha Roy, Karen Shen).
Submitted to
SIAM Journal on Applied Dynamical Systems,
pdf
- Benford's Law and Continuous Dependent Random Variables (with Thealexa Becker, Ryan Ronan,
Frederick Strauch and Alec Greaves-Tunnell), Submitted to the
Journal of Applied Probability,
pdf.
- Low-lying zeros for Dirichlet
L-functions (with Daniel Fiorilli), Submitted to
Duke Mathematical Journal
pdf
- Models for zeros near the central point of families of elliptic
curves (with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina
Snaith), Submitted to
Journal
of Physics A: Mathematical and Theoretical
pdf
- Distribution of eigenvalues of weighted, structured matrix ensembles (with Olivia Beckwith, Karen Shen),
submitted to the
Journal of
Theoretical Probability
pdf
- The low-lying zeros of cuspidal Maass forms on SL2(Z) (with Nadine Amersi, Geoffrey Iyer, Oleg Lazarev and Liyang Zhang),
submitted to Acta Arithmetica.
pdf
PREPRINTS / IN PREPARATION (available upon
request)
Click here
for an annotated bibliography of my work
- Missing Sums (title TBD, with Oleg Lazarev)
- Moment Formulas for Ensembles of Classical Compact Groups (with Geoffrey Iyer)
- Gaps between summands in generalized Zeckendorf decompositions (with Olivia Beckwith)
- The n-level density of zeros of quadratic Dirichlet L-functions (with Jake Levinson)
- Applying Benford's Law to Large Natural Data Sets (with Victoria
Cuff and Allie Lewis).
- m-Paths and the 3x+1 Problem (with Bruce Adcock, Carlos Dominguez
and Sucheta
Soundararajan).
- Leading, learning and herding (with Daniel Stone), pdf
- Using the 2-Level Density to Improve Bounds on Excess Rank in
Families.
- The expected eigenvalue distribution of large, weighted d-regular
graphs (with Leo Goldmakher and Kesinee Ninsuwan).
- The Pythagorean Won-Loss Formula and Hockey (with Kevin Dayaratna). pdf
- If a prime divides a product (with Cesar Silva).
pdf
BOOKS
Click here for an annotated
bibliography of my work
-
An Invitation to Modern Number Theory (with Ramin Takloo-Bighash),
Princeton University Press, 2006, 519 pages.
- Theory and Applications of Benford's law (senior editor for book,
duties include writing chapters and coordinating contributions): In
preparation.
- The Probability Lifesaver (study guide for probability, senior
author for the book, duties include writing chapters and coordinating
contributions): In preparation.
- Cryptography book (title TBD) (with Midge Cozzens and Wes Pegden): In preparation.
Thesis
Papers
Talks
My Book (Invitation to Modern Number Theory)
Handouts
My
Riddles Page
- COURSES:
Welcome letter
to Williams Students
Possible
thesis / colloquium projects
Green Chicken contest
- 2010 - 2011:
Math 302 (Complex Analysis, Fall)
Math 308/406 (Analysis and
Number Theory, Fall)
Putnam Indep Study
(Fall)
Math 105
(Multivariable Calculus, Spring)
- Previous classes (Williams):
Math 103
(Calculus I), Math
105 (Multivariable Calculus, 2010 iteration),
Math 209 (Differential Equations
and Vector Calculus),
Math 341
(Probability, Fall),
Math 399 (Sabermetrics,
Fall),
Math 406
(Analysis and Number Theory); Winter Study:
Math 10
(cryptography)
- Previous classes
(Brown):
Math
1 (freshman seminar: riddles to modern mathematics), Math 90 (calculus,
summer
calculus)
-
Math 35 (honors
multivariable calculus), Math
52 (linear algebra)
Math 54 (honors linear algebra),
Math 153 (abstract
algebra),
-
Math 162
(mathematical statistics), Math 197 (Benford's
law,
sabermetrics)
- Previous classes
(Ohio State)
Math 104 (Basic College
Mathematics),
Math 148 (Algebra,
Trigonometry and their Applications)
-
Math 187/487 (Problem Solving / Advanced
Problem Solving),
Math 683L (Topics in
Linear Algebra)
- Working groups (Euler,
fractals)
-
LINKS:
photos of Cam
Pictures
Twistie
Art
Arxiv
MathSciNet
Mathlabs (Princeton
, NYU,
OSU,
AIM)
Williams Math
Williams College
Spencer Neighborhood
