at 1:30pm
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 1:30pm
in M1060

(University of Lethbridge)

In this talk, we first introduce the Brumer-Rosen-Zantema exact sequence (BRZ), a four-term sequence related to strongly ambiguous ideal classes in finite Galois extensions of number fields. Then, using BRZ, we obtain some known cohomological results in the literature concerning Hilbert's Theorem 94, the capitulation map, and the Principal Ideal Theorem. This is a joint work with Ali Rajaei (Tarbiat Modares University) and Ehsan Shahoseini (Institute for Research in Fundamental Sciences).

at 1:30pm
in M1060

A vanishing sum of roots of unity (VSRU) is a finite list $z_1,\ldots,z_K$ of $N$-th complex roots of unity whose sum is zero. While there are many simple examples—including the famous "beautiful equation" of Euler, $e^{i \pi} + 1 = 0$—such sums become extremely complex as the parameter $N$ attains more complex prime power divisors (and we will see several classical examples illustrating this idea, as well as new examples from my work).

One fruitful line of inquiry is to seek a quantitative relationship between the prime divisors of $N$, their associated exponents, and the cardinality parameter $K$. A theorem of T.Y. Lam and K.H. Leung from the early '90's states: $K$ must always be (at least) as large as the smallest prime dividing $N$. This generalizes the well known observation that that sum of all $p$-th roots of unity (where $p$ is any prime number) must vanish; and, one notices that Euler's equation is one example of this fact.

In this talk, we will discuss two significant strengthenings of this result (one due to myself and I. Łaba, another due to myself, G. Kiss, I. Łaba and G. Somlai), which are derived from complexity measurements for polynomials with integer coefficients which have many cyclotomic polynomial divisors. As applications, we give connections in two other areas of mathematics. The first is in the study of integer tilings: additive decompositions of the integers $Z = A+B$ as a sum set, where each integer is represented uniquely. The second application is to the Favard length problem in fractal geometry, which asks for bounds upon the average length of the projections of certain dynamically-defined fractals onto lines.

This talk is based upon my individual work, as well as my joint work with I. Łaba, as well as my joint work with G. Kiss, I. Łaba and G. Somlai. All are welcome, and the first 15-20 minutes will include introductory ideas and examples for all results discussed in the latter portion of the talk.

at 1:30pm
in M1060

Fix a positive integer $n$ and a graph $F$. A graph $G$ with $n$ vertices is called $F$-saturated if $G$ contains no subgraph isomorphic to $F$ but each graph obtained from $G$ by joining a pair of nonadjacent vertices contains at least one copy of $F$ as a subgraph. The saturation function of $F$, denoted $\mathrm{sat}(n, F)$, is the minimum number of edges in an $F$-saturated graph on $n$ vertices. This parameter along with its counterpart, i.e. Turan number, have been investigated for quite a long time.

We review known results on $\mathrm{sat}(n, F)$ for various graphs $F$. We also present new results when $F$ is a complete multipartite graph or a cycle graph. The problem of saturation in the Erdos-Renyi random graph $G(n, p)$ was introduced by Korandi and Sudakov in 2017. We survey the results for random case and present our latest results on saturation numbers of bipartite graphs in random graphs.

at 1:30pm
in M1060

(University of Lethbridge)

Fix a positive integer $X$ and multi-sets of complex numbers $\mathcal{I}$ and $\mathcal{J}$. We study the shifted convolution sum \[ D_{\mathcal{I},\mathcal{J}}(X,1) = \sum_{n\leq X} \tau_{\mathcal{I}}(n)\tau_{\mathcal{J}}(n+1), \] where $\tau_{\mathcal{I}}$ and $\tau_{\mathcal{J}}$ are shifted divisor functions. These sums naturally appear in the study of higher moments of the Riemann zeta function and additive problems in number theory. We review known results on $2k$-th moment of the Riemann zeta function and correlation sums associated with generalized divisor function. Assuming a conjectural bound on the averaged level of distribution of $\tau_{\mathcal{J}}(n)$ in arithmetic progressions, we present an asymptotic formula for $D_{\mathcal{I},\mathcal{J}}(X,1)$ with explicit main terms and power-saving error estimates.

at 1:30pm
in M1060

Drinfeld modules are the analogues of elliptic curves in positive characteristic. They are essential objects in number theory for studying function fields. They do not have points, in the traditional sense—we're going to count them anyway! The first methods achieving this were inspired by classical elliptic curve results; we will instead explore an algorithm based on so-called Anderson motives that achieves greater generality. Joint work with Xavier Caruso.

at 1:30pm
in M1060

(University of Lethbridge)

We investigate the sums of reciprocals to an arithmetic progression taken modulo one, that is sums of fractional parts $1/\{na-y\}$, where $a,y$ are real parameters. Such upper bounds are known in the literature, with many applications in number theory. For instance, they related to exponential sums and are also used in counting lattice points in polygons. We present an alternate and novel method to obtain new efficient and fully explicit results. The technique uses the Three Distance Theorem and the theory of continued fractions. This is based on joint work with Victor Beresnevich. If time permits, we will also briefly discuss Poissonian pair correlation of gaps.