Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Fall 2019 Talks are at noon on Monday in C630 of University Hall For more information, contact Nathan Ng < ng AT cs DOT uleth DOT ca > or Dave Morris .
 Talks in the series this semester: (Click on any title for more info, including the abstract. Then click on it again to hide the info.)

 Date Speaker Title Sept 9 everyone Open problem session at noon in C630 (University of Lethbridge) Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester. Sept 16 Gabriel Verret An update on the Polycirculant Conjecture at noon in C630 (University of Auckland, New Zealand) One version of the Polycirculant Conjecture is that every finite vertex-transitive digraph admits a non-trivial semiregular automorphism. I will give an overview of the status of this conjecture, as well as describe some recent progress with Michael Giudici, Ademir Hujdurovic and Istvan Kovacs. Sept 23 Selcuk Aygin Extensions of Ramanujan-Mordell Formula with Coefficients $1$ and $p$ at noon in C630 (University of Lethbridge) We use properties of modular forms to extend the Ramanujan-Mordell formula. Our result yields to formulas for representation numbers by the quadratic form $\sum_{i=1}^{2a}x_i^2+\sum_{i=1}^{2b}py_i^2$, for all non-negative integers $a,b$ and for all odd prime $p$. We obtain this result by computing the Fourier series expansions of modular forms at all cusps of $\Gamma_0(4p)$. Oct 7 Amir Akbary The prime number theorem for automorphic $L$-functions at noon in C630 (University of Lethbridge) We describe a theorem due to Jianya Liu and Yangbo Ye (Pure and Applied Mathematics Quarterly, Volume 3, Number 2, 481-497, 2007) concerning the prime number theorem for automorphic $L$-functions. We state the theorem and review the strategy of the proof in comparison with the classical prime number theorem. An important ingredient is a new version of Perron's formula that represents a sum of complex numbers as a contour integral plus some specific error terms. Oct 21 Khoa Dang Nguyen An analogue of Ruzsa's conjecture for polynomials over finite fields at noon in C630 (University of Calgary) In 1971, Ruzsa conjectured that if $f:\ \mathbb{N}\rightarrow\mathbb{Z}$ with $f(n+k)\equiv f(n)$ mod $k$ for every $n,k\in\mathbb{N}$ and $f(n)=O(\theta^n)$ with $\theta < e$ then $f$ is a polynomial. In this paper, we investigate the analogous problem for the ring of polynomials over a finite field. This is joint work with Jason Bell. Oct 28 Quanli Shen The fourth moment of quadratic Dirichlet $L$-functions at noon in C630 (University of Lethbridge) In this talk, I will talk about the fourth moment of quadratic Dirichlet $L$-functions. Under the generalized Riemann hypothesis, we showed an asymptotic formula for the fourth moment. Unconditionally, we established a precise lower bound. Nov 4 Peng-Jie Wong Primes in Short Intervals at noon in C630 (University of Lethbridge) Bertrand's postulate states that there is always a prime in the interval $[x,2x]$ for any $x\ge 1$. Applying the prime number theorem, one may further show that there is approximately $\int_{x}^{2x}\frac{dt}{\log t}$ primes in $[x,2x]$ for sufficiently large $x$. There is a more difficult question concerning the distribution of primes $p$ in short intervals when $[x,2x]$ is replaced by $[x, x + h]$ for some $h\le x$ and $p$ is required to be congruent to $a$ modulo $q$ for some $(a,q)=1$. In this talk, we will discuss how short $[x, x + h]$ can be. If time allows, we will sketch a proof of the Bombieri-Vinogradov theorem in short intervals, which answers such a question. Nov 18 Allysa Lumley Distribution of Values of $L$-functions over Function Fields at noon in C630 (Centre de Recherches Mathématiques, Montréal) Let $q\equiv 1 \pmod{4}$ be a prime power and $\mathbb{F}_q$ be the finite field with $q$ elements. Let $1/2<\sigma<1$ be fixed. We consider $D$ a monic square-free polynomial in $\mathbb{F}_q[T]$ and $\chi_D$ the Kronecker symbol associated with $D$. In this talk, we will discuss the distribution of large values of $\log L(\sigma,\chi_D)$ with $D$ varying over monic square-free polynomials with degree $n\to\infty$. We will highlight the expected similarities to the situation over quadratic extensions of $\mathbb{Q}$ as well as the surprising differences. Nov 25 Po-Han Hsu Large deviation principle for the divisor function at noon in C630 (Louisiana State University) Let $\omega(n)$ denote the number of distinct prime divisors of $n$. Let $W(m)$ be a random integer chosen uniformly from $\{n:n\le m, n\in \Bbb{N}\}$. Let $X(m)$ be $\omega(W(m))$. The celebrated Erdős-Kac theorem asserts that \begin{align*} \frac{X(m)-\log\log m}{\sqrt{\log\log m}}\rightarrow N(0, 1), \end{align*} where $N(0, 1)$ is the standard normal distribution. In 2016, Mehrdad and Zhu studied the large and moderate deviations for the Erdős-Kac theorem. In this talk, we will give a brief introduction to the theory. Then we will discuss how to establish the large deviation principle for $X(m)/\log\log m$. If time allows, we will discuss some generalisations. This is a joint work with Dr Peng-Jie Wong. Dec 2 Andrew Fiori Simplicity of ABV-packets for Arthur Type Parameters in GLn at noon in C630 (University of Lethbridge) In this talk we will discuss a combinatorial approach to studying the geometry of the moduli space of Langlands parameters for GLn. We will first discuss several connections between the study of partitions and the study of nilpotent conjugacy classes. We then generalize some of these ideas to describe a relationship between multi-segments and orbits in the moduli space of Langlands parameters. Finally we shall explain how these ideas lead to a proof that simple parameters have simple packets.
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