READINGS FOR PROJECTS

• General Reading
  • Hayes: The Spectrum of Riemannium: a light description of the connection between random matrix theory and number theory (there are a few minor errors in the presentation, basically to simplify the story). This is a quick read, and gives some of the history.
  • Conrey: L-functions and Random Matrix Theory: This is a high level description of the similarities between number theory and random matrix theory.
  • Katz-Sarnak: Zeros of Zeta Functions and Symmetry: Another high level article similar to the others.
  • Diaconis: Patterns in Eigenvalues: this is a bit more readable than the others, and is based on a distinguished lecture he delivered.
  • Miller and Takloo-Bighash: An Invitation to Modern Number Theory: This is the textbook I and a colleague wrote, based on years of supervising undergraduate research classes. I know several of you already have a copy -- it will be a good resource for the summer, as a lot of the background material we need is readily available here. Particularly important chapters for us are:
    • chapter15 (which discusses the connections between random matrix theory and number theory, and is available online);
    • chapter 18 (which does the explicit formula for Dirichlet characters);
    • chapter 3 (which reviews L-functions and is also online).
  • Firk and Miller: Nuclei, primes and the Random Matrix connection: a survey paper on the history of the subject, including both the nuclear physics experiments and the theoretical calculations.

   

• L-functions
  • Iwaniec-Luo-Sarnak: Low lying zeros of families of L-functions: This is a must read. This is the first major paper calculating the 1-level density for families of L-functions. We will not need much of the paper, but you must read section 1, skim the beginning of section 2 (section 2 is devoted to developing good averaging formulas for specific families; as we will be looking at different families, we won't need this), skim section 3 (same comments apply), section 4 (very important: here they prove the explicit formula relating sums over zeros to sums over primes, and while we will use the explicit formula for different families, the calculation will follow similarly), section 5 (just skim, same caveats as before).
  • Conrey-Snaith: Applications of the L-functions Ratios Conjecture: This is a very recent conjecture which is enjoying remarkable success in predicting answers. There is a lot I'll say about this during the summer. I somewhat jokingly call it the conjecture of the four lies, as there are five steps and four of the steps are provably wrong (ie, the assumptions in those steps fail); however, miraculously, all the errors seem to cancel to phenomenal level! I've become very interested in testing this conjecture as much as possible, and have written several papers in this (and have ideas for a few more which will be very accessible to us). Read through page 17.
    • Symplectic Test
    • Orthogonal Test I
    • Orthogonal Test II
    • Unitary Test
  • Rubinstein: Low lying zeros of L-functions and Random Matrix Theory: this is his dissertation, and in it he analyzes the 1-level density of the family of quadratic Dirichlet characters, and shows agreement with Random Matrix Theory. This is one of the easiest families to look at, and a great testing ground.
  • Hughes - Miller: Low-lying zeros of L-functions with orthogonal symmetry: this paper finds a more tractable version of the Katz-Sarnak determinantal expansion for the n-level density, but for restricted support.
  • Miller: A symplectic test of the L-functions Ratios Conjecture: This paper builds on those by Conrey-Snaith and Rubinstein and uses the Ratios Conjecture to predict the lower order terms up to square-root cancellation, and then shows (for suitable test functions) that this is the correct answer. An obvious project is to generalize this test for other families or to enlarge the support.
  • Miller: Extending the support for families of Dirichlet characters (work in progress): This is a project I've been working on occasionally over the past few years. I have studied the 1-level density for Dirichlet characters, and subject to some reasonable assumptions I can extend the support and smash all world records. Is it possible to do this unconditionally? I don't know, but I hope so! I would love to work on this during the summer; we will almost surely need the results in the primes in arithmetic progression section below.

   

• Graph Theory
  • McKay: Eigenvalues of Large Random Graphs: This readable paper calculates the density of eigenvalues for d-regular graphs; the answer is different than the semi-circle Wigner found for the family of all real symmetric matrices. Excellent projects are finding the density of states for special sets of matrices.
  • Jackobson-(SD)Miller-Rivin-Rudnick: Random graphs: Numerics about the neighbor spacings of d-regular graphs.
  • Womald: Models of Random Graphs: various ways to generate random graphs.

   

• Random Matrix Theory
  • Odlyzko: Distribution of spacings between zeros of the zeta function: One of the classics in the field of the phenomenal agreement between the zeros of the Riemann zeta function and the eigenvalues of matrices.
  • Mezzadri: How to generate random matrices from the classical compact groups: if we choose to numerically explore eigenvalues of the classical compact groups, this is a great place to begin.
  • Lehman: Survey of the Semi-Circle Law (former student of mine, written at a research class at Princeton).
  • Zakharevich: Paper on Generalization of the Semi-Circle Law (former student of mine at AIM).
  • Toeplitz Ensemble Papers:
    • Hammond - Miller: Eigenvalue spacing distribution for real symmetric Toeplitz matrices (former student of mine at AIM and OSU).
    • Miller - Massey - Sinsheimer: Eigenvalue spacing distribution for palindromic Toeplitz matrices (former students at OSU and Brown).
    • Jackson - Pham - Miller: Eigenvalues of highly palindromic Toeplitz matrices (former SMALL students).
    • Kologlu - Kopp - Miller: Eigenvalues of symmetric period m-circulant matrices (former SMALL students)
  • We made some nice progress last summer with various weighted ensembles -- email me for the working papers, see especially:
    • Random Matrix Ensembles with Split Limiting Behavior (with Paula Burkhardt, Peter Cohen, Jonathan Dewitt, Max Hlavacek, Carsten Sprunger, Yen Nhi Truong Vu, Roger Van Peski, and Kevin Yang, and an appendix joint with  Manuel Fernandez and Nicholas Sieger), Random Matrices: Theory and Applications (7 (2018), no. 3, 39 pages, DOI: 10.1142/S2010326318500065) pdf
    • Spectral Statistics of Non-Hermitian Random Matrix Ensembles (with Ryan Chen, Yujin Kim, Jared Lichtman, Shannon Sweitzer, and Eric Winsor), to appear in Random Matrices: Theory and Applications  pdf

   

• Benford's law of digit bias
  • Hill (general theory)
  • Raimi (survey)
  • Kontorovich-Miller (connection to L-functions, random matrix theory and 3x+1).
  • Miller-Nigrini:
    • Products and the Mod 1 CLT (products of random variables tend to Benford)    
    • Almost Benford Behavior
    • Benford's law applied to Hydrology data
    • Useful paper: handout on order statistics.
  • Brown University Independent Study Group: Chains (applications to Bayesian theory, related to products) (very important read!)
  • Fewster (simple explanation of Benford's law)
  • Working notes for an introductory chapter on a Benford's law book that I'm editing -- DO NOT DISTRIBUTE FURTHER!!!
  • Wikipedia page on copulas
  • See also:
    • Benford's Law and Continuous Dependent Random Variables (with Thealexa Becker, David Burt, Taylor C. Corcoran, Alec Greaves-Tunnell, Joseph R. Iafrate, Joy Jing, Jaclyn D. Porfilio, Ryan Ronan, Jirapat Samranvedhya, Blaine Talbut and Frederick W. Strauch), (388 (2018), 350--381) Annals of Physics. pdf.
    • Benford's Law Beyond Independence: Tracking Benford Behavior in Copula Models (with Becky Durst). pdf

   

• Additive number theory (MSTD sets)
  • Nathanson: Problems in additive number theory: great introduction to the subject and problems.
  • Martin-O'Bryant: Many sets have more sums than differences: first paper to prove a positive percentage of sets are sum-dominated (proof if probabilistic).
  • Hegarty-Miller: When almost all sets are difference domianted: almost all sets are difference dominated if we change the binomial model.
  • Miller-Orosz-Scheinerman: Constructing MSTD sets: elementary construction of a very big family of sum-dominated (or MSTD) sets.
  • Email me for the works in progress from last summer, including
    • Generalizations of a Curious Family of MSTD Sets Hidden By Interior Blocks (with Hung Chu, Noah Luntzlara and Lily Shao), submitted to INTEGERS.   pdf
    • Infinite Families of Partitions into MSTD Subsets (with Hung Chu, Noah Luntzlara and Lily Shao), submitted to INTEGERS.   pdf

       

• Additive number theory (Generalized Zeckendorf Decompositions)
  • On the number of summands in Zeckendorf decompositions
  • From Fibonacci Numbers to Central Limit Type Theorems
  • Email me for the works in progress or recently concluded, including:
    • The Zeckendorf Game (with Paul Baird-Smith, Alyssa Epstein and Kristen Flint), submitted to Proceedings of CANT.  pdf
    • The Generalized Zeckendorf Game (with Paul Baird-Smith, Alyssa Epstein and Kristen Flint), submitted to the Fibonacci Quarterly.  pdf
    • Gaussian Behavior in Zeckendorf Decompositions From Lattices (with Eric Chen, Robin Chen, Lucy Guo, Cindy Jiang, Joshua M. Siktar, Peter Yu), submitted to the Fibonacci Quarterly.   pdf
    • The bidirectional ballot polytope (with Carsten Peterson, Carsten Sprunger and Roger Van Peski), to appear in INTEGERS. pdf
    • On Summand Minimality of Generalized Zeckendorf Decompositions (with Katherine Cordwell, Max Hlavacek, Chi Huynh, Carsten Peterson, and Yen Nhi Truong Vu), to appear in Research in Number Theory pdf