- nkα mod 1 and Poissonian behavior
- General theory
- Examples
-
Student Projects
- One of the most challenging problems is to prove results about the
distribution of the ordered spacings of n alpha mod 1 for alpha irrational.
- Random Matrix Theory
- Historical overview and surveys
-
Nuclei, primes and the random matrix connection (Frank W. K. Firk (my
college mentor) and Steven J. Miller), invited paper to Symmetry (1, (2009),
64--105; doi:10.3390/sym1010064).
-
The spectrum of Riemannium (B.
Hayes), American Scientist 91 (2003), no. 4, 296--300.
-
Developments in Random Matrix Theory (P.
J. Forrester, N. C. Snaith, and J. J. M. Verbaarschot). In Random
matrix theory, J. Phys. A 36 (2003), no. 12, R1--R10.
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Random matrices and L-functions (J.
P. Keating and N. C. Snaith). In Random
matrix theory, J. Phys. A 36 (2003), no. 12, 2859--2881
- General theory
- Eigenvalue distribution of special ensembles of matrices
-
Spectral measure of large
random Hankel, Markov and Toeplitz matrices (W.
Bryc, A. Dembo, T. Jiang), Annals of Probability 34 (2006),
no. 1, 1--38
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Distribution of Eigenvalues
for the Ensemble of Real Symmetric Toeplitz Matrices (Chris
Hammond, Steven J. Miller), Journal of
Theoretical Probability 18 (2005), no. 3, 537--566.
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Eigenvalue spacing
distribution for the ensemble of real symmetric palindromic Toeplitz
matrices (Adam Massey, Steven
J. Miller and John. Sinsheimer), Journal of Theoretical Probability 20
(2007), no. 3, 637--662.
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Distribution of eigenvalues for highly palindromic real symmetric Toeplitz
matrices (Steven Jackson, Steven J. Miller and Thuy Pham), to appear in
the Journal
of Theoretical Probability.
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The Limiting Spectral Measure for the Ensemble of Symmetric Period
m-Circulant Matrices (with Murat Koloğlu,
Murat Kologlu and Steven J. Miller).
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A Generalization of Wigner's Law (Inna
Zakharevich),Comm. Math. Phys. 268 (2006), no. 2, 403--414.
- d-Regular graphs
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Eigenvalue spacings for
regular graphs (D. Jakobson,
S. D. Miller, I. Rivin, and Z. Rudnick). Pages 317--327 in Emerging
Applications of Number Theory (Minneapolis, 1996), The IMA Volumes in
Mathematics and its Applications, Vol. 109, Springer, New York, 1999.
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The expected eigenvalue distribution of a large regular graph (B.
McKay), Linear Algebra Appl. 40 (1981), 203--216.
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The distribution of the second largest eigenvalue in families of random
regular graphs (S. J. Miller, T. Novikoff and A. Sabelli), Experimental
Mathematics 17 (2008), no.
2, 231--244.
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Models of
random regular graphs (N. C.
Wormald). Pages 239--298 in Surveys in combinatorics, 1999 (Canterbury)
London Mathematical Society Lecture Note Series, vol. 267, Cambridge
University Press, Cambridge, 1999.
- Student Projects
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First order spacings of random matrix eigenvalues (R.
Lehman), Junior Thesis, Princeton University, Spring 2000.
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Statistical behavior of the eigenvalues of random matrices (Y.
Liu), Junior Thesis, Princeton University, Spring 2000.
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Distribution of eigenvalue spacings for band-diagonal matrices (N.
Miller), Junior Thesis, Princeton University, Spring 2003.
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Eigenvalues of weighted random graphs (R.
Qian and D. Steinhauer), Junior Thesis, Princeton University, Spring 2003.
- Classic papers
- The Circle Method
- Waring's Problem
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Waring's problem: a survey. Pages 301--340 in Number Theory for the
Millennium, III (R. C. Vaughan and T. D. Wooley) (Urbana, IL, 2000), A. K.
Peters, Natick, MA, 2002.
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Waring's problem (J.
Cisneros), Junior Thesis, Princeton University, Spring 2001.
- Goldbach's Problem
- Germain Primes
Applications
Examples
-
Benford's Law, Values of
L-Functions and the 3x+1 Problem (Alex
Kontorovich, Steven J. Miller), Acta Arithmetica 120 (2005), 269--297.
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Benford's Law for the 3x+1
function (J. Lagarias and K.
Soundararajan), J. London Math. Soc. (2) 74 (2006), no. 2, 289--303.
-
Order Statistics and Shifted
Almost Benford Behavior (Steven
J. Miller and Mark J. Nigrini),
International Journal of
Mathematics and Mathematical Sciences, Volume 2008 (2008), Article ID
382948, 19 pages, doi:10.1155/2008/382948.
Numerics
I'm currently editing a book on Benford's law -- if you are interested
in this topic, let me know and I can share chapters.
- Riemann Zeta Function and L-functions
- Historical overview
- General surveys
-
L-Functions and random
matrices (J. B. Conrey).
Pages 331--352 in Mathematics unlimited --- 2001 and Beyond, Springer-Verlag,
Berlin, 2001.
-
The Riemann
hypothesis (J. B. Conrey),
Notices of the AMS, 50 (2003), no. 3, 341--353.
-
Zeros of
zeta functions and symmetries (N.
Katz and P. Sarnak), Bull. AMS 36 (1999), 1--26.
-
Random matrices and L-functions (J.
P. Keating and N. C. Snaith). In Random
matrix theory, J. Phys. A 36 (2003), no. 12, 2859--2881
- Numerical computations
-
Investigations of zeros near
the central point of elliptic curve L-functions (Steven
J. Miller), Experimental Mathematics 15 (2006), no. 3, 257--279.
-
Beyond pair correlation (H.
Montgomery and K. Soundararajan). Pages 507--514 in Paul Erdös and His
Mathematics, I (Budapest, 1999), Bolyai Society Mathematical Studies, Vol.
11, János Bolyai Math. Soc., Budapest, 2002.
-
The 1013 first zeros
of the Riemann zeta function, and zeros computation at very large height,
A. Odlyzko, preprint.
-
On the
distribution of spacings between zeros of the zeta function (A.
Odlyzko), Math. Comp. 48 (1987), no. 177, 273--308.
-
The 10^22-nd
zero of the Riemann zeta function (A.
Odlyzko). Pages 139--144 in Proceedings of the Conference on Dynamical,
Spectral and Arithmetic Zeta Functions, ed. M. van Frankenhuysen and M. L.
Lapidus, Contemporary Mathematics Series, AMS, Providence, RI, 2001.
- Additive Number
Theory (especially More Sums Than
Differences Sets)
-
Gist: given a finite set of integers A, form A+A and A-A, where A+A = {a1 +
a2: a_i in A} and A-A = {a1 - a2: a_i in A}. As addition is commutative but
subtraction isn't, a generic pair gives two differences and only one sum.
Thus expect A+A to be smaller than A-A. Amazingly, Martin and O'Bryant
showed that a positive percent of the time A+A is larger! If you use a
difference model of choosing sets randomly, however, Hegarty and I showed
almost all sets are difference dominated. In the other direction, I and some
students constructed the world record for densest families of sum-dominated
sets.
-
Miller-Roman-Sinnott: y^2= n(n+1)(n+2)(n+3): A nice proof when four
consecutive integers can and cannot be a square; can you generalize to
products of six consecutive integers? This could make a really nice Monthly
article.
The 3x+1 Problem
-
The
3x+1 problem and its generalizations (J.
Lagarias). Pages 305-334 in Organic mathematics (Burnaby, BC, 1995), CMS
Conf. Proc., vol. 20, AMS, Providence, RI, 1997.
-
The 3x+1 problem: An
annotated bibliography (J.
Lagarias), preprint.
-
Benford's Law, Values of
L-Functions and the 3x+1 Problem (Alex
Kontorovich, Steven J. Miller), Acta Arithmetica 120 (2005), 269--297.
-
Benford's Law for the 3x+1
function (J. Lagarias and K.
Soundararajan), J. London Math. Soc. (2) 74 (2006), no. 2, 289--303.
-
Stochastic models of the 3x+1 and
5x+1 Problems (Alex Kontorovich and Jeff Lagarias). Jeff Lagarias is
writing a book on the 3x+1 Problem; if you are interested I can get you
chapters to read.
HANDOUTS:
- REVIEW MATERIAL: Numerous worked
out calculus problems (differentiation, integration, statement of topics you
should know)
-
Notes on Induction, Calculus,
Convergence, the Pigeon Hole Principle and Lengths of Sets (from the first
appendix to
An Invitation to
Modern Number Theory, by myself and Ramin Takloo-Bighash, Princeton
University Press 2006). We will not need all the material there for this course,
but it is easier to just post the entire chapter.
-
Handout on Types of Proofs (from a
handout I wrote for math review sessions at Princeton, 1996-1997; this was
written for students from calculus to linear algebra).
-
Intermediate and Mean
Value Theorems and Taylor Series (you should know this material already; the
main results are stated and mostly proved, subject to some technical results
from analysis which we need to rigorously prove the IVT).
-
Free textbooks:
- Isaac Asimov: The
Relativity of Wrong
INTERESTING VIDEOS (these are not required
readings, but are posted for your enjoyment)
FUN MATH
(these are not required readings, but are
posted for your enjoyment)