Graduate Number Theory Seminar


The Spring 2018 seminar will be held on Wednesdays from 10:30 to 11:30AM in Hill Center room 525.

Upcoming Seminars

Date March 28, 2018
Time 10:30-11:30AM
Place Hill 525
Speaker Yongxiao Lin, Ohio State University
Title Delta symbol methods and their applications in bounding L-functions.
Abstract We'll describe some versions of the delta symbol method, and talk about their applications, especially in the subconvexity problem for L-functions and the shifted convolution problem for Fourier coefficients. We shall highlight some recent progress of Ritabrata Munshi who used such an approach, and discuss some limitations of the method.

George Hauser

Kloosterman's Circle Method

Katie McKeon

'The Fractal Uncertainty Principle'

We'll discuss the ideas behind the fractal uncertainty principle and see how it has been used to provide spectral gaps for certain hyperbolic surfaces.

Edna Jones

'Representations by ternary quadratic forms'

How can you represent integers by ternary quadratic forms? For example, can the integer 2017 be represented as a square plus three times another square plus five times another square? A few kinds of representations over the integers (such as global representation and local representation) will be discussed. To better understand these representations, we count how many solutions there are to congruences involving ternary quadratic forms using quadratic Gauss sums and Hensel's Lemma.

George Hauser

'Number theoretic aspects of error correcting codes'

I will introduce the notion of an error correcting code, and provide the basic example of the Reed-Solomon code. Then I will state some general bounds on the efficiency of codes in general. Then I will introduce a family of codes based on curves over finite fields, and state the so called Algebraic Geometric bound, which beats the earlier known bounds. I will explain how number theoretic tools, including zeta functions, and trace formula are crucial here.

Matthew Welsh

'A Large Sieve Inequality for Roots of Cubic Congruences'

In the course of proving infinitely many primes of the form x^2 + p^2, Fouvry and Iwaniec used the parametrization (dating back to Gauss) of roots of v^2 = -1 (mod m) by representations of m = a^2 + b^2 to derive a large sieve inequality for these roots. In this talk I will explain an attempt to replicate these results for roots of v^3 = 2 (mod m).

Dave Jensen, University of Kentucky

'Introduction to Tropical Brill-Noether Theory'

In this talk, we will discuss the basic combinatorial theory of divisors on graphs and its relationship to the theory of divisors on algebraic curves. We will cover several concrete examples and applications to problems in algebraic geometry.

Surya Teja Gavva

'Primes in short intervals'

We are interested in finding primes in short intervals [x, x+y], for y =x^{\theta}, \theta as small as possible. I will try to discuss some main techniques to prove such results and go over the result of Baker, Harman, Pintz for \theta= 0.535. The best result today is for \theta =0.525.

Luochen Zhao

'The conjecture of Erdos-Szemeredi'

In this talk I will review some basics about Erdos-Szemeredi conjecture, including the motivations and major results. I'll then present Solymosi's method, which gives the best result to date. The argument will be elementary and no prerequisite is assumed.

Here is a small list of ideas (organized by subject) that may serve as inspiration.

Algorithms in Number Theory

  • Shor's Factoring Algorithm

    See also the discrete Fourier transform, quantum algorithms

Ergodic Number Theory

  • Weyl's Equidistribution Theorem

    See also Weyl's Criterion, extension to quadratic polynomials

Analytic Number Theory

  • The Number of Partitions of an Integer

    See also Hardy-Littlewood Circle Method

Coding Theory

  • The MacWilliams Identity from Theta Functions

    From 'Coding Theory and Number Theory' by Hiramatsu and Köhler