The Statistics of Gaps Between Values Represented by Quadratic Forms

We will present the outline of a program extending the work of Gallagher on gaps between primes to binary quadratic forms. The core ideas is to formulate for each form an analogue of the Hardy-Littlewood prime k-tuple conjecture. We will discuss results of Connors and Keating, Smilansky, and Freiberg, Kurlberg, and Rosenzweig, as well as our own direction of investigation.

Stationary Phase and Stirling's formula    

Laplace's Method or Stationary Phase is a method for finding approximate values of exponential integrals that usually have no solution in terms of basic functions. I'll cover a classic proof and apply the method to obtain Stirling's formula. 

Statistical distribution of 2-regular integer sequences   

We generalize a recent result of Bettin, Drappeau & Spiegelhofer on the statistical distribution of Stern's diatomic sequence to show that a large family of 2-regular integer sequences obey a log-normal statistical distribution. This generalization is made possible by viewing the distribution of k-regular sequences from the perspective of products of random matrices.   

Playing Games on Fractals: Dynamical & Diophantine   

We will present sketches of a program extending the Schmidt-Summerer-Roy parametric geometry of numbers to Diophantine approximation for systems of m linear forms in n variables, together with a sampling of open questions that await exploration. Our variational principle (arXiv:1901.06602) provides a unified framework to compute fractal dimensions of a variety of sets of number-theoretic interest, as well as their dynamical counterparts via the Dani correspondence. Highlights include the introduction of certain combinatorial objects that we call «templates», which arise from a dynamical study of Minkowski’s successive minima in the geometry of numbers; and a new variant of Schmidt’s game designed to compute the Hausdorff and packing dimensions of any set in a doubling metric space.  

The transcendence of Certain infinite sums    

The arithmetic nature of special values of the Riemann zeta function is one of the central problems in number theory. In this talk, we will discuss the transcendence nature of infinite series \sum_{n=1}^{\infty} 1/(an^2 + bn +c), where a, b, c are integers and a is non-zero. This is a joint work with Prof. M. Ram Murty. 

A class of deformed Archimedean CFTs via Tate's thesis    

Perhaps the best-known example of a conformal field theory is the free field theory, which is defined in terms of the Laplacian operator on the complex projective line and describes the physics of a massless boson. In this talk, we will give a tourists' introduction to CFT before turning to a deformed free field theory recently introduced by Huang, Stoica, and Zhong. This deformed action is given in terms of a nonlocal operator and can be defined over both Archimedean and non-Archimedean local fields. In the non-Archimedean setting, this theory gives a new interpretation of quadratic reciprocity. Finally, we will touch upon a new contribution to this story which leverages the local functional equation in Tate's thesis to understand the conformal invariance of the deformed Archimedean action. This is a joint work in progress with An Huang.  

The torsion subgroup of rational elliptic curves with complex multiplication   

A rational elliptic curve E is an affine curve defined as E: y^2 = x^3 + Ax + B for rational numbers A and B such that 4A^3 + 27B^2 is non-zero. The set of points with rational coordinates on E can be made into an abelian group. A celebrated theorem of L. J. Mordell states that the group of such points is finitely generated, i.e., is isomorphic to T(E) x Z^r for some finite torsion group T(E) and r copies of the integers Z. An incredibly deep (and quite difficult) theorem of B. Mazur provides a complete classification of groups that can occur as T(E). After recalling some of the basic theory about elliptic curves with complex multiplication (CM), I will prove that for rational CM elliptic curves, there are six possible torsion groups that can occur and that each of these six groups is realized by some CM elliptic curve. Afterwards, I will discuss some open problems related to T(E). 

The Unreasonable Effectiveness of Cramér’s Model  

The aim of this talk is to discuss the strengths and limitations of Cramér’s probabilistic model of primes. We will first introduce the model and briefly discuss how it predicts the distribution of gaps between primes. Next we will indicate some of its obvious shortcomings and show how addressing them could lead one to rediscover the Hardy-Littlewood Conjecture. In the last part of the talk we will discuss a simplified proof, due to Ford, of Gallagher’s result which says that, despite its limitations, Cramér’s model seems to be right “on average”. 

Approximating arithmetic functions and random matrix theory 

Random matrix integrals frequently appear in connection to problems related to the distribution of the Riemann zeta-function and other L-functions. In this talk I hope to review the connection between the GUE Hypothesis for zeta zeros on the one hand and short-interval sums / random matrix integrals on the other. I will also describe an interpretation for related random matrix integrals in terms of a best-approximation problem for permutations and discuss ongoing work related to best-approximation problems over the integers. 

Coprimality of consecutive elements in regular sequences

The study of finding blocks of primes in certain arithmetic sequences is one of the classical problems in number theory. It is also very interesting to study blocks of consecutive elements in such sequences that are pairwise coprime. In this talk, we will discuss the existence of arbitrarily long blocks of pairwise coprime consecutive elements in the sequence ([f(n)])_n, where f is a twice continuously differentiable real-valued function satisfying certain conditions. This is a joint work with Jean-Marc Deshouillers.

Transcendental values of the Lambert W-function

The Lambert W-function W(z) is defined for complex numbers z as the solution x of the equation xexp(x)=z. Like the logarithm function, it has countably many branches. It was introduced by J. H. Lambert in 1758 and was studied later by Euler in 1783 in the context of solutions of certain differential equations. Recently, it re-emerged in the theory of differential equations of the SIR model for the spread of infectious diseases. However, no one has studied the function from a number theoretic perspective. So, we begin such a study and prove some new theorems regarding transcendental values of the Lambert W-function. This is joint work (in progress) with Sonika Dhillon (ISI, Delhi).