at noon
in M1040

(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 noon
in M1040

A power series $f(x_1,\ldots,x_m)\in \mathbb{C}[[x_1,\ldots,x_m]]$ is said to be D-finite if all the partial derivatives of $f$ span a finite dimensional vector space over the field $\mathbb{C}(x_1,\ldots,x_m)$. For the univariate series $f(x)=\sum a_nx^n$, this is equivalent to the condition that the sequence $(a_n)$ is P-recursive meaning a non-trivial linear recurrence relation of the form: $$P_d(n)a_{n+d}+\cdots+P_0(n)a_n=0$$ where the $P_i$'s are polynomials. In this talk, we consider D-finite power series with algebraic coefficients and discuss the growth of the Weil height of these coefficients.

This is from a joint work with Jason Bell and Umberto Zannier in 2019 and a more recent work in June 2022.

at noon
in M1040

In 1900, Hilbert posed the following problem: “Given a Diophantine equation with integer coefficients: to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in (rational) integers.”

Building on the work of several mathematicians, in 1970, Matiyasevich proved that this problem has a negative answer, i.e., such a general ‘process’ (algorithm) does not exist.

In the late 1970's, Denef–Lipshitz formulated an analogue of Hilbert's 10th problem for rings of integers of number fields.

In recent years, techniques from arithmetic geometry have been used extensively to attack this problem. One such instance is the work of García-Fritz and Pasten (from 2019) which showed that the analogue of Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. In joint work with Lei and Sprung, we improve their proportions and extend their results in several directions. We achieve this by using multiple elliptic curves, and by replacing their Iwasawa theory arguments by a more direct method.

at noon
in M1040

(University of Lethbridge)

The Fibonacci sequence $(F_n)_{n \geq 0}$ is the binary recurrence sequence defined by $F_0 = F_1 = 1$ and $$ F_{n+2} = F_{n+1} + F_n \text{ for all } n \geq 0. $$ There is a broad literature on the Diophantine equations involving the Fibonacci numbers. In this talk, we will study the Diophantine inequality $$ | F_n + F_m - 2^a | < 2^{a/2} $$ in positive integers $n, m$ and $a$ with $n \geq m$. The main tools used are lower bounds for linear forms in logarithms due to Matveev and Dujella-Pethö version of the Baker-Davenport reduction method in diophantine approximation.

at noon
in M1040

(University of Lethbridge)

We will discuss graphs that have a unique hamiltonian cycle and are vertex-transitive, which means there is an automorphism that takes any vertex to any other vertex. Cycles are the only examples with finitely many vertices, but the situation is more interesting for infinite graphs. (Infinite graphs do not have "hamiltonian cycles," but there are natural analogues.) The case where the graph has only finitely many ends is not difficult, but we do not know whether there are examples with infinitely many ends. This is joint work in progress with Bobby Miraftab.

at noon

In this talk we will present a method to study the Galois group of certain polynomials defined over $\mathbb{Q}$. Our approach is similar in spirit to some previous work of F. Hajir, who studied, more than a decade ago, the generalized Laguerre polynomials using a similar approach. For example this method seems to be well suited to study the Galois groups of Jacobi polynomials (a classical family of orthogonal polynomials with two parameters — three if we include the degree). Given a polynomial $f(x)$ with rational coefficients of degree $N$ over $\mathbb{Q}$, the idea consists in finding a good prime $p$ and look at the Newton polygon of $f$ at $p$. Then combining the Galois theory of local field over $\mathbb{Q}_p$ and some classical results of the theory of permutation of groups we sometimes succeed in showing that the Galois group of $f$ is not solvable or even isomorphic to $A_N$ or $S_N$ ($N\geq 5$).

The existence of a good prime $p$ is subtle. In order to get useful results one would need to have some “effective prime existence results”. As an illustration, we would like to have an explicit constant $C$ (not too big) such that for any $N>C$, there exists a prime $p$ in the range $N < p < \frac{3N}{2}$ such that gcd$(p-1,N)= 1 \text{ or } 2$ (depending on the parity of $N$). Such a result is not so easy to get when $N$ is divisible by many distinct and small primes. We hope that such effective prime existence results are within the reach of the current techniques used in analytic number theory.

at 2pm

The blow up of the anticanonical base point on $X$, a del Pezzo surface of degree 1, gives rise to a rational elliptic surface $E$ with only irreducible fibers. The sections of minimal height of $E$ are in correspondence with the 240 exceptional curves on $X$. A natural question arises when studying the configuration of those curves:

If a point of $X$ is contained in “many” exceptional curves, is it torsion on its fiber on $E$?

In 2005, Kuwata proved for del Pezzo surfaces of degree 2 (where there are 56 exceptional curves) that if “many” equals 4 or more, then yes. In a joint paper with Rosa Winter, we prove that for del Pezzo surfaces of degree 1, if “many” equals 9 or more, then yes. Moreover, we find counterexamples where a torsion point lies at the intersection of 7 exceptional curves.

at noon
in M1040

(University of Lethbridge)

The quality of the triplet $(a,b,c)$, where $\gcd(a,b,c) = 1$, satisfying $a + b = c$ is defined as $$ q(a,b,c) = \frac{\max\{\log |a|, \log |b|, \log |c|\}}{\log \mathrm{rad}(|abc|)}, $$ where $\mathrm{rad}(|abc|)$ is the product of distinct prime factors of $|abc|$. We call such a triplet an $ABC$-solution. The $ABC$-conjecture states that given $\epsilon > 0$ the number of $ABC$-solutions $(a,b,c)$ with $q(a,b,c) \geq 1 + \epsilon$ is finite.

In the first part of this talk, under the $ABC$-conjecture, we explore the quality of certain families of $ABC$-solutions formed by terms in Lucas and associated Lucas sequences. We also introduce, unconditionally, a new family of $ABC$-solutions that has quality $> 1$.

In the remainder of the talk, we prove a conjecture of Erdős on the solutions of the Brocard-Ramanujan equation $$ n! + 1 = m^2 $$ by assuming an explicit version of the $ABC$-conjecture proposed by Baker.

at noon
in M1040

In the finite-dimensional situation, Lie's third theorem provides a correspondence between Lie groups and Lie algebras. Going from the latter to the former is the more complicated construction, requiring a suitable representation, and taking exponentials of the endomorphisms induced by elements of the group.

As shown by Garland, this construction can be adapted for some Kac-Moody algebras, obtained as (central extensions of) loop algebras. The resulting group is called a loop group. One also obtains a relevant infinite-rank Chevalley lattice, endowed with a metric. Recent work by Bost and Charles provide a natural setting, that of pro-Hermitian vector bundles and theta invariants, in which to study these objects related to loop groups. More precisely, we will see in this talk how to define theta-finite pro-Hermitian vector bundles from elements in a loop group. Similar constructions are expected, in the future, to be useful to study loop Eisenstein series for number fields.

This is joint work with Manish M. Patnaik.

at noon

I will talk about recent work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta-function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how we can obtain non-trivial upper bounds for smaller shifts. I will also discuss some applications to the question of obtaining cancellation of averages of the Mobius function. Joint work with H. Bui.