Time: 4:30-5:30 p.m. Place: Jeffery Hall 422
Speaker: Francesco Cellarosi (Queen's University)
Title: Smooth arithmetical sums over k-free integers.
Abstract: We use partial zeta functions to analyse the asymptotic behaviour of certain smooth arithmetical sums over smooth $k$-free integers. This is joint work with Ram Murty.
Time: 4:30-5:30 p.m. Place: Jeffery Hall 422
Speaker: Anup Dixit (Queen's University)
Title: On Picard-type theorems involving $L$-functions.
Abstract: The little Picard's theorem states that any non-constant entire function takes all complex values or all complex values except one point. In a similar flavour, suppose $f$ is an entire function such that for complex values $a$ and $b$, the set of zeros of $f$ is same as the set where $f'$ takes values $a$ and $b$, then it is possible to show that $f$ is a constant function. Such results are called Picard-type theorems. In this talk, we will discuss similar questions for $L$-functions, where it is possible to prove much stronger results.
Time: 4:30-5:30 p.m. Place: Jeffery Hall 422
Speaker: Seoyoung Kim (Queen's University)
Title: Artin's primitive root conjecture for function fields without Riemann Hypothesis.
Abstract: Artin's primitive root conjecture for function fields is known by Bilharz in his thesis in 1937, which was conditional on the proof of the Riemann hypothesis for global function fields, which was proved by Weil in 1948. In this talk, we suggest a simple proof of Artin's primitive root conjecture for function fields unconditional on the Riemann hypothesis for global function fields by using the technique from the proof of the prime number theorem by Hadamard and de la Vall/'ee Poussin. This is joint work with M. Ram Murty.
Time: 4:30-5:30 p.m. Place: Jeffery Hall 422
Speaker: Richard Leyland
Title: Isogenies of Elliptic Curves with Complex Multiplication.
Abstract: In my thesis work I seek to answer Mazur's Question which asks if there exists any isomorphisms of mod $N$ Galois representations attached to elliptic curves that are not induced by isogenies. The first step in answering this question is determining which isogenies of elliptic curves are defined over a field $F$. In this talk, I will show how to construct isogenies between CM elliptic curves by using ideals of the endomorphism rings. In particular, we will see that if the field of definition $F$ does not contain the CM field, then we can reduce the problem to finding cyclic isogenies.
Time: 4:30-5:30 p.m. Place: Jeffery Hall 422
Speaker: Mike Roth (Queen's University)
Title: A measure of positivity of a line bundle along a subschemes, and a simpler proof of the Ru-Vojta arithmetic theorem.
Abstract: Diophantine geometry seeks to link properties of rational solutions of a set of equations to the geometric properties of the variety they define. One of the main tools in Diophantine geometry is Diophantine approximation — results bounding how the complexity of a rational point must grow as it approaches a subvariety. In this talk I will discuss a somewhat recent new measure of the positivity of an ample line bundle along a subscheme, and show how its formal properties give a simple proof of a theorem of Ru-Vojta on Diophantine approximation. This is joint work with David McKinnon at Waterloo.
Time: 4:30-5:30 p.m. Place: Jeffery Hall 422
Speaker: Anup Dixit (Queen's University)
Title: On the distribution of certain sequences and a prime number theorem.
Abstract: The fourth problem of Landau conjectures that there are infinitely many primes of the form $n^2+1$. Inspired by this, we consider sequences of the form $\{[n^{\alpha}] + 1\}$, where $[x]$ denotes the greatest integer less than or equal to $x$. In this talk, we will discuss how often the elements of such a sequence lie in a given arithmetic progression for $\alpha<2$ and also establish an analogue of prime number theorem for $\alpha<1$. This is joint work with Prof. M. Ram Murty.
Time: 4:30-5:30 p.m. Place: Jeffery Hall 422
Speaker: Arpita Kar (Queen's University)
Title: On the distribution of certain sums of Random multiplicative functions.
Abstract: In this talk, I will discuss a result of Adam Harper regarding sums of Rademacher multiplicative functions $f(n)$, over those $n \leq x$ with $k$ distinct prime factors where $k$ is a function of $x$. In particular, we will discuss the Martingale central limit theorem due to Mcleish, and see how it establishes Harper's result.
Time: 4:30-5:30 p.m. Place: Jeffery Hall 422
Speaker: Brad Rodgers (Queen's University)
Title: Moments and pseudomoments of the Riemann zeta-function, pt. 2.
Abstract: In a previous talk we discussed moments of the Riemann zeta function and "pseudomoments", in which the zeta-function is replaced by a finite Dirichlet polynomial. In this second talk we will further discuss random multiplicative functions and a variant of conjecture of Helson, proved with averaging weights by Bondarenko, Heap, and Seip. I hope to reintroduce the fundamental notions, so that this talk can be followed by audience members who missed the previous talk.
Time: 4:30-5:30 p.m. Place: Jeffery Hall 422
Speaker: M. Ram Murty (Queen's University)
Title: THE PALEY GRAPH CONJECTURE AND DIOPHANTINE TUPLES.
Abstract: Let $n$ be a fixed natural number. An $m$-tuple $(a(1), ..., a(m))$ is said to be a Diophantine $m$-tuple with property $D(n)$ if $a(i)a(j)+n$ is a perfect square for $i, j$ distinct and less than or equal to $m$. We will show that the Paley graph conjecture in graph theory implies that the number of such tuples is $O((log n)^c)$ for any $c>0$. This is joint work with Ahmet Guloglu.
Time: 4:30-5:30 p.m. Place: Jeffery Hall 422
Speaker: Brad Rodgers (Queen's University)
Title: Differential equations and arithmetic.
Abstract: I will describe some applications of differential equations and differential operators to number theory.
