- 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, sabermetrics, ...).
Thesis
Papers
Talks
My Book (Invitation to Modern Number Theory)
Handouts
My
Riddles Page
If there are several versions of a talk, the
later version is usually the more 'complete' (ie, correct) version.
Random Matrix Theory and L-Functions
Random Matrix Theory and
Elliptic Curves
Elliptic Curves
Random Matrix
Ensembles
Analysis
&
Probability (especially Benford's Law)
Computers and Mathematics
Education
Colloquium Talks
Baseball (Sabermetrics)
- Random Matrix Theory and L-Functions
- Random Matrix Theory and L-Functions I: Analytic
Number Theory Seminar, Ohio State (10/20/03): pdf
- Random Matrix Theory and L-Functions II: Analytic
Number Theory Seminar, Ohio State (10/27/03):
pdf
- From Random Matrix Theory to L-Functions: Special
Number Theory Seminar, Tel Aviv (12/23/04):
pdf (see also the expanded talk given in Utah in 2009 below)
- Identifying Symmetry Groups of Zeros of Families of L-Functions:
Number Theory and Random Matrix Theory Workshop, CMS
Summer 2005, Waterloo (6/1/05):
ps
pdf;
tabbed version
ps
dvi
- Identifying and breaking the symmetry group of zeros of families of
L-functions: Number Theory Seminar, CUNY Lehman
(10/20/06): pdf
- A Symplectic Test of the L-Functions Ratios Conjecture:
Algebra Seminar, Brown University (9/17/07): paper.pdf
AMS Special Session on L-functions and automorphic
forms, Courant (20 minute version, 3/16/08):
pdf tabbed version
pdf Johns Hopkins University (50 minute
version, 4/4/08):
pdf (untabbed version:
pdf); Cornell University (50 minute version, 6/5/08):
pdf (untabbed version:
pdf).
-
From Random Matrix Theory to L-Functions: Graduate
Workshop on Zeta Functions, L-Functions and their Applications, Utah Valley
University, June 2, 2009: pdf I've created a
webpage with links to talks and papers related to this conference; click
here for these papers and links.
-
Random Matrix Theory and Number Theory: Progress report from 2009 SMALL REU
at Williams College (final presentation of my students John Goes, Steven
Jackson, David Montague, Eve Ninsuwan, Ryan Peckner and Vincent Pham):
Williams College (8/11/09):
pdf
-
Low Lying Zeros of Number Field L-Functions (presented by Ryan Peckner).
Young Mathematicians Conference, Ohio State
(8/29/09):
pdf
-
Tests of the L-Functions Ratios Conjecture. Maine - Québec
Number Theory Conference (10/3/09):
pdf
- Random Matrix Theory and Elliptic Curves
- 1 and 2 Level Density Functions for Families of Elliptic Curves:
Evidence for the Underlying Group Symmetries.
(Thesis defense) Princeton (5/26/02), Number Theory
Seminar Ohio State (5/30/02):
pdf Boston University (2/10/03):
pdf
- Evidence for a Spectral Interpretation of the Zeros of Families of
Elliptic Curves. Joint meeting of the AMS and UMI Pisa
(6/13/02):
pdf
title Shorter version (20 minutes): AMS
Sectional, Salt Lake City (10/27/02):
dvi
pdf
- Random Matrix Theory and Elliptic Curves: Evidence for the Underlying
Group Symmetries (with three appendices). Johns
Hopkins University (3/3/04):
pdf An alternate version: Five
College Number Theory Seminar, Amherst, MA, (4/20/04):
pdf
- Random Matrix Theory models for zeros near the central point (and
applications to elliptic curves) (45 minutes) Workshop
on Spectral Theory and Automorphic Forms, Montréal,
May 2004.
ps. An alternate version (60 minutes):
Boston University (2/10/03):
ps Brown University (9/13/04 - expanded):
dvi
pdf Another alternate version: (120 minutes)
Fellowship of the Ring Seminar, Brandeis University
(4/1/05):
ps
pdf; tabbed version
ps
dvi. Another alternate version (30 minutes),
Advances in Number Theory and Random Matrix Theory,
Rochester, NY (6/7/06):
pdf; tabbed version
pdf; paper
the talk is based on (to appear in Experimental Mathematics)
- How the Manhattan Project helps us understand primes and elliptic curves
(with appendices on Dirichlet L-Fns, Random Graphs and a bibliography).
Colloquium, University of Connecticut (3/24/05):
ps
pdf; tabbed version
ps
dvi.
- From the Manhattan Project to Number Theory: How nuclear physics helped
us understand primes: Theoretical Physics Seminar,
Brown University (4/12/06):
pdf; relevant papers: Investigations of Zeros Near the Central
Point of Elliptic Curve L-Functions (to appear in
Experimental
Mathematics)
pdf (data available
online).
- Finite conductor models for zeros near the central point of elliptic
curve L-functions (with Eduardo Dueñez, Duc Khiem Huynh and Jon P. Keating).
L-functions, ranks of elliptic curves and random
matrix theory workshop, Banff (7/12/07): slides:
pdf (my part);
pdf (Dueñez);
ppt (Keating);
pdf (Huynh)
pdf (combined). AMS Special Session on
Number Theory, Wesleyan University, Middletown, CT (10/11/08):
pdf
- Finite conductor models for zeros near the central point of elliptic
curve L-functions (with Eduardo Dueñez, Duc Khiem Huynh and Jon P. Keating).
(This is an expanded version, and includes results towards an average
version of the Birch and Swinnerton-Dyer Conjecture, as well as the results
from Duc Khiem's thesis on modeling the first zero of quadratic twists by
Jacobi ensembles with discretization and lower order terms included.)
University of Rochester:
pdf
Five College Number Theory Seminar:
pdf
- Elliptic Curves
- Ranks of One-Parameter Families of Elliptic Curves Over Q(T) and
Thoughts on the Excess Rank Question (with five appendices).
Boston College (3/10/03):
dvi
pdf
- Constructing 1-Parameter Families of Elliptic Curves over Q(T) with
Moderate Rank. AMS Sectional, Boulder, CO (10/4/03):
pdf
- The effect of zeros of elliptic curve L-functions at the central point
on nearby zeros. AMS Sectional, Lawrenceville, NJ
(4/18/04):
pdf Expanded Version: Algebra Seminar, Brown University (10/24/05):
pdf tabbed version
dvi
- Towards an ``Average'' Version of the Birch and Swinnerton-Dyer
Conjecture (presented by John Goes). Young
Mathematicians Conference, Ohio State
(8/29/09): pdf
- Random Matrix Ensembles
- Eigenvalue Statistics for Ensembles of Random Matrices (especially
Toeplitz Ensembles & Diophantine Obstructions).
Probability and Ergodic Theory Seminar, Ohio State (10/30/03):
pdf Boston University (6/7/04 expanded):
pdf Brown University (9/15/04
expanded):
pdf
- On the probability that random graphs are Ramanujan.
Expanders and Ramanujan Graphs: Construction and
Applications, AMS National Meeting, San Diego (1/9/08):
pdf
-
Benford's Law
- Benford's Law, Values of L-functions and the 3x+1 Problem.
Boston College (10/19/04), University of Michigan
(11/15/04), University of Arizona (1/11/06), Brown University (3/20/06): slides
paper.pdf. Number Theory Special Session, AMS Sectional
Meeting, WPI, Worcester, MA (4/25/09, 20min version):
pdf Mathematica programs:
Just the 3x+1 Map
Benford Investigations Some Numerical Simulations:
Data
- Benford's Law and Order Statistics. Brown
University (2/1/06)
paper.pdf
- Poisson Summation and Benford's Law: From values of L-functions to the
3x+1 problem to products of random variables.
University of Bristol (12/12/07); Workshop on the Theory and Applications of
Benford's Law, Santa Fe, NM (12/18/07):
pdf
- The logarithmic link between economic, hydrologic, and seismic
statistics (with Mark Nigrini), Workshop on the Theory
and Applications of Benford's Law, Santa Fe, NM (12/17/07):
- Chains of distributions and Benford's Law (with Dennis Jang, Jung Uk
Kang, Alex Kruckman and Jun Kudo), Workshop on the
Theory and Applications of Benford's Law, Santa Fe, NM (12/17/07):
pdf
- Benford's Law: Why the IRS might care about the 3x+1 problem and
ζ(s) (general introduction to the
subject), Western New England College (2/11/08):
pdf (untabbed version
pdf); Smith College (10/7/08):
pdf;
Williams College, Bronfman Science Lunch (10/21/08, 20min version):
pdf
Central Connecticut State University (9/25/09):
pdf
- Theory and applications of Benford's law to fraud detection, or: Why the
IRS should care about number theory! IRS (Boston
Offices, 3/28/08):
pdf (untabbed version
pdf).
-
Additive Number Theory (especially More Sum Than Difference
Sets)
- When almost all sets are difference dominated.
Analysis Seminar, Brown University (9/12/07)
paper.pdf; Williams College (9/12/08):
pdf
Wesleyan University (11/20/08):
pdf
Workshop on Combinatorial and Additive Number Theory (CANT 2009), CUNY
Graduate Center, New York, May 2009:
pdf
- Explicit constructions of infinite families of MSTD sets
(presented by Dan Scheinerman), Number Theory Session,
Joint Mathematical Meetings, Washington, DC (1/05/09):
pdf
- Computers and Mathematics Education
- Computers in Undergraduate Education and Zeros of Elliptic Curves.
Minnesota (8/10/02):
pdf Expanded Version:
pdf
- Statistical Investigations as a Tool in Undergraduate Mathematics
Research: Poster and Workshop Sessions
International Conference on Statistics Honolulu,
Hawaii (6/5/03 and 6/7/03):
pdf
- NSF Workshop on Computation in Algebra, Number Theory and Combinatorics.
Washington DC (9/21/02):
ps and
ps
- The Pythagorean Won-Loss Theorem: An introduction to modeling.
Great Activities for an Introductory Statistics Class,
AMS National Meeting, San Diego, (1/7/08):
pdf
- Workshop on Mathematicians in Mathematics Education, participant,
Institute for Mathematics Education,
Tucson, Arizona, March 20 - 22, 2008.
- Colloquium / General Talks
(mostly Random Matrix Theory and Number Theory)
- How the Manhattan Project helped us understand primes.
Colloquium, University of Cincinatti (10/16/03) and
Ohio State (10/1/03):
pdf. An alternate version (with appendices on Dirichlet L-Fns, Elliptic
Curves and a bibliography): Colloquium, Colby College
(3/8/05):
ps
pdf; tabbed version
ps
dvi. Another alternate version (with appendices on Dirichlet
L-Fns, Random Graphs and a bibliography). Colloquium,
University of Connecticut (3/24/05):
ps
pdf; tabbed version
ps
dvi
- From Nuclear Physics to Number Theory: How the Manhattan Project helped
us understand Primes: SUMS Conference, Brown
University (2/12/05):
ps
pdf; tabbed version
ps
dvi This is similar to the above colloquia, but aimed more at
undergraduates.
- L-functions and Random Matrix Theory. Brown
University (10/6/06). See
pdf for a more extensive version.
- From the Manhattan Project to Number Theory: How Nuclear Physics Helps
Us Understand Primes (20 minutes). Bronfman Science
Lunch, Williams College (6/23/09):
pdf (20
minute version)
pdf (expanded
version)
- Heuristics and Ballpark Estimates: From the 3x+1 problem to counting
primes and birthdays: PROMYS, Boston University
(7/28/09)
- Pythagoras at the Bat: an Introduction to Statistics and Mathematical
Modeling: Wellesley (9/21/09):
pdf
- How low can we go? Lower order terms in CLTs from Benford's Law to
Elliptic Curves (25 minute version). NES MAA Fall 2009 Meeting (11/21/09):
pdf (no
tab pdf)
- The Riemann Hypothesis at 150: From Primes to Nuclei and Many
Things Between. Williams College (11/18/09) Blackboard
talk, but see the following page for
relevant papers on the subject.
- Baseball Talks
- The Pythagorean Won-Loss Formula in Baseball (An Introduction to
Statistics and Modeling) (60 minute version) Brown
University (9/28/05) and (12/7/05):
slides(tab)
slide(no tab)
paper_pdf. Shorter 15 minute version: 2006
Hudson River Undergraduate Mathematics Conference (4/8/06):
slides(tab)
slides(no tab)
- Pythagoras at the Bat: An Introduction to
Statistics and Modeling (20 minute version): Society
For American Baseball Research, Boston Meeting, Boston, MA (5/20/06):
slides(tab)
slides(no tab)
paper_pdf. Williams College (40 minute
version, 1/15/08):
pdf. Holy Cross (50 minute version, 2/7/08):
pdf. Western New England College (40 minute version, 1/15/08):
pdf. Connecticut Smoky Joe Wood SABR Chapter, Hamden, CT (30 minute
version, 2/16/08)
pdf. PROMYS Program, Boston University (40 minute
version, 7/25/08):
pdf.
Bennington College (2/27/09):
pdf
Hampshire College (7/22/09):
pdf
Wellesley (9/21/09):
pdf
- See also talks on Computers and Mathematics Education.
Random Matrix Theory and L-Functions
Random Matrix Theory and
Elliptic Curves
Elliptic Curves
Random Matrix
Ensembles
Analysis
&
Probability (especially Benford's Law)
Computers and Mathematics
Education
Colloquium Talks
Baseball (Sabermetrics)
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
- 2009 - 2010:
Math 341
(Probability, Fall)
Math 399 (Sabermetrics,
Fall) Winter Study:
Math 10
(cryptography)
Math 105
(Multivariable Calculus, Spring)
- Previous classes (Williams):
Math 103
(Calculus I)
Math 209 (Differential Equations
and Vector Calculus)
Math 406
(Analysis and Number Theory)
- 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
