The Spring 2018 seminar will be held on Wednesdays from 10:30 to 11:30AM in Hill Center room 525.
Date | March 28, 2018 |
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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. |
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.
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.
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.
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).
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.
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.
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.
See also the discrete Fourier transform, quantum algorithms
See also Weyl's Criterion, extension to quadratic polynomials
See also Hardy-Littlewood Circle Method
From 'Coding Theory and Number Theory' by Hiramatsu and Köhler