at noon
in UHall W565

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.

at noon
in UHall W565

Let $L(s)$ be the $L$-function associated to a modular form of weight $k$ for the full modular group. Using spectral theory, Anton Good proved that $L(k/2+it) \ll t^{1/3} (\log t)^{5/6}$. Matti Jutila discovered a simpler proof which makes use of the Voronoi summation formula, exponential integrals, and Farey fractions. We shall present Jutila's argument.

at noon
in UHall W565

This is a continuation of last week's talk. I will sketch Matti Jutila's proof of a subconvex bound for a modular $L$-function on the critical line. The main ideas are an approximate functional equation, the use of Farey fractions, Voronoi's summation formula, and exponential integral and sum estimates.

at noon
in UHall W565

In this talk we will show how to exhibit large and small gaps between the zeros of the Riemann zeta function, assuming the Riemann hypothesis. This is based on a technique of Montgomery and Odlyzko. The problem of finding small gaps between the zeros leads to a very interesting optimization problem.

at 9am
in UHall D632

no abstract available

Wednesday
at 9am
in UHall D632

In 20th century, one of the most striking discoveries in number theory is Montgomery's pair-correlation. It says that pair-correlation of zeros of the Riemann zeta function is the same with that of eigenvalues of unitary matrices. In 1990's, Rudnick, Katz and Sarnak studied the zeros of $L$-functions more systematically. Moreover, Katz and Sarnak proposed the $n$-level density conjecture which claims that distributions of low-lying zeros of $L$-functions in a family is predicted by one of compact matrix groups, which are $U(N)$, $SO(\text{even})$, $SO(\text{odd})$, $O(N)$, and $Sp(2N)$. At the end of the talk, I will state an $n$-level density theorem for some families of Artin $L$-functions and talk about counting number fields with local conditions. I will start with a friendly definition of $L$-functions and give some examples. No background or knowledge for $L$-functions are required for this talk.

Friday
at 9am
in UHall C620

In the early 1970s, Harold Stark formulated a conjecture about the first non-vanishing Taylor coefficient at zero of Artin $L$-functions. About 10 years later, he refined his conjecture for abelian $L$-functions having order of vanishing one at zero, under certain hypotheses. In 1996, Karl Rubin extended this last refinement of Stark to the higher order of vanishing setting. In this expository talk for a general audience, we will give a survey of this area of research and present a more general conjecture, which we formulated in the past few years. At the end, we will present evidence for our conjecture and indicate one possible direction for further research.

at noon
in UHall W565

A well-known conjecture asserts that there are infinitely many primes $p$ for which $p - 1$ is a perfect square. We obtain upper and lower bounds of matching order on the number of pairs of distinct primes $p,q \le x$ for which $(p-1)(q-1)$ is a perfect square. This is joint work with Carl Pomerance (Dartmouth College).

Friday
at 10:30am
in UHall C640

A number $n$ is called squarefull if for every prime $p$ dividing $n$, we have $p^2$ also dividing $n$. Erdos conjectured that the number of pairs of consecutive squarefull numbers $(n, n+1)$ with $n < N$ is at most $(log N)^A$ for some $A >0$. This conjecture is still open. We will show that the abc conjecture implies this number is at most $N^e$ for any $e>0$. We will also discuss a related conjecture of Ankeny, Artin and Chowla on fundamental units of certain real quadratic fields and discuss its connection with the Erdos conjecture. This is joint work with Kevser Aktas.

at 11:00am
in UHall C630

Let $a \in \mathbb{Z} \setminus \{0,\pm 1\}$, and let $f_{a}(p)$ denote the order of $a$ modulo $p$, where $p \nmid a$ is prime. There are many results that suggest $p-1$ and $f_{a}(p)$ are close. For example, Artin's conjecture and Hooley's subsequent proof upon the Generalized Riemann Hypothesis. We will examine questions related to the relationship between $p-1$ and $f_{a}(p)$.

at noon
in C630

A transversal of a latin square of order $n$ is a subset of entries picked in such a way that each row, each column and each symbol is present at most once. In many latin squares, you can find a full transversal by selecting $n$ entries which do not duplicate any row, column or symbol. But what about when you can't find a full transversal? Brualdi has conjectured that a transversal of length $n-1$ is always present in any latin square. In this talk, we will discuss recent work which shows that for small orders, Brualdi's conjecture holds. Moreover, we show that his conjecture also holds for small generalized latin squares as well.

at 11am
in C620

In this talk we discuss the distribution of low-lying zeros of certain families of Artin L-functions attached to geometric parametrizations of number fields. We describe several explicit examples of such families and in each case we present the symmetry type of the distribution of low-lying zeros. This is joint work with Arul Shankar and Nicolas Templier.

at 1pm
in C620

I will discuss Vaughan's approximation to the number of primes in arithmetic progressions. In particular, I will show that on average over large moduli, it is far superior to the usual approximation.