in M1060

In principle, the zeros of two distinct Dirichlet $L$-functions ought to be unrelated. In fact, they turn out to be far more entangled than one expects, though not always. My talk will survey some recent results in this circle of ideas, including Linnik–Sprind\v{z}uk-type equivalences that reduce GRH to the vertical distribution of the zeros of a single $L$-function. I will also describe in greater depth some joint work with Kyle Loftus evaluating the twisted moment $\sum_{\rho} x^{\rho} L(\rho,\chi_1)$ over the zeros $\rho$ of a second $L$-function $L(s,\chi_2)$. Our results are unconditional and hold in short windows, and they imply that no nontrivial linear combination of Dirichlet $L$-functions vanishes on the entire zero set of another.

at 12:00pm
in M1060

(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.

at 12:00pm
in M1060

(University of Lethbridge)

In recent work with Mikko Jaskari we study the problem of finding tight explicit bounds on sums over zeros of $L$-functions near the $1$-line with a variety of zero-free regions and zero-density estimates. As an application to demonstrate the techniques, we apply them to the case of the Riemann zeta function and the prime number theorem to obtain new best estimates on the error terms in the prime number theorem for large $x.$ We show that for $\log(x)\geq 3$ we have \[|\psi(x)-x| < 0.239 \, x \exp\left(-0.1982767\left(1+\frac{\log\log\log(x)}{15\log\log(x)}\right) \frac{\log(x)^{3/5}}{\log\log(x)^{1/5}}\right)\] and for $\log(x)>10^{15}$ we have \[|\psi(x)-x| < (1.1485 {\times} 10^{-1154556} ) \, x \exp\left(\! -0.1982767\left(\! 1+\frac{\log\log\log(x)}{15\log\log(x)} \!\right) \! \frac{\log(x)^{3/5}}{\log\log(x)^{1/5}} \!\right) . \] In this talk I will explain how we obtain this result.

at 12:00pm
in M1060

(University of Lethbridge)

The Erdős-Ko-Rado (EKR) theorem states that if $n$ and $k$ are two positive integers such that $n\geq 2k$ and $\mathcal{F}$ is a family of $k$-subsets of $\{1,2,\ldots,n\}$ such that $A\cap B \neq \varnothing$, for all $A,B\in \mathcal{F}$, then $ |\mathcal{F}| \leqslant \binom{n-1}{k-1}. $ Moreover, if $n\geqslant 2k+1$, then equality holds if and only if there exists $a\in \{1,2,\ldots,n\}$ such that $\mathcal{F} = \left\{A \subset \{1,2,\ldots,n\}: |A| = k \mbox{ and }a \in A\right\}$.

The Erdős-Ko-Rado theorem can sometimes be extended to other combinatorical objects, provided that a well-defined notion of intersection between objects exists. In this talk, I will give some examples where the EKR theorem can be generalized to other objects. Then, I will present some recent EKR type results on transitive permutation groups.

at 12:00pm
in M1060

Erdős and Moser asked whether every even integer $n$ can be written as $n = \varphi(a) + \varphi(b)$, where $\varphi$ is Euler's totient function. If $n + 2 = p_1 + p_2$ with $p_1, p_2$ primes, then $n = \varphi(p_1) + \varphi(p_2)$. Hence any bound for the exceptional set in Goldbach's problem carries over to this question, which currently gives $O(X^{0.72})$.

We prove that all but $O(X^{3/5} (\log X)^9)$ even integers up to $X$ are sums of two totients. In fact, we prove the stronger statement that all but $O(X^{3/5} (\log X)^9)$ odd integers up to $X$ are the sum of a prime and a totient. The gain over the Goldbach case comes from replacing one binary problem by a family of them, $p_1 + 2^j p_2 = n + 2^j$, of which only a bounded number can fail simultaneously. We follow Pintz's treatment of sums of three primes with a bounded third prime. This is joint work with Alia Hamieh and Stanley Yao Xiao.

at 12:00pm
in M1060

(University of Lethbridge)

Many games are played on hypergraphs. It is often infeasible to understand the game on all hypergraphs, so we narrow the focus to a structure with specific properties. Combinatorial designs are highly structured hypergraphs. Constructing graphs from combinatorial designs can produce an interesting family of graphs, and playing games on these structures often leads to illuminating game analysis.

In this talk, we will explore a few instances where games are played either on designs or on graphs constructed from designs. The goal of the talk is to demonstrate the elegance of these structures and their utility in game analysis.

at 12:00pm
in M1060

Let $f$ be a rational map of degree at least two. The solutions of $f^n(x)=\alpha$, as $n$ varies, form an infinite rooted tree on which the absolute Galois group acts. A central question in arithmetic dynamics is when the resulting arboreal Galois representation has finite index in the generic iterated Galois group.

In this talk, I will discuss a connection between this finite-index problem and eventual stability, which asks whether the number of irreducible factors of $f^n(x)=\alpha$ remains bounded. Under suitable hypotheses on the critical orbits and geometric monodromy, we establish an equivalence between eventual stability and finite index. The result is conditional on the $abc$ conjecture over number fields and unconditional for non-isotrivial rational maps over characteristic-zero function fields.

The proof combines primitive prime divisors with adelic intersection theory on projective curves. I will also discuss applications to polynomials of arbitrary degree and an explicit family of quartic rational maps whose Galois groups exhibit unusual behavior.

No prior familiarity with arboreal Galois representations will be assumed.

at 12:00pm
in M1060

(University of Lethbridge)

Abstract TBA

at 12:00pm
in M1060

Abstract TBA

at 12:00pm
in M1060

(University of Lethbridge)

Abstract TBA