A seminar series.
Vladislav Taranchuk
(Sacramento State)
Ramsey Theory on the Integers
In 1927 van der Waerden proved that for any positive integers $r$ and $k$ there exists a positive integer $N = w(r,k)$ such that every $r$-coloring of $\{1,2,3,\dots,N\}$ contains a monochromatic arithmetic progression of length $k$. We will go over the meaning of this result and the current best known bounds on the growth rate of $w(r, k)$. Alongside the best known bounds, we will also introduce a new recursive lower bound on $w(r, k)$ proved by Blankenship, Cummings, and the speaker, that generalizes a 60 year old result by Berlekamp and generally sets a best known lower bound within a large interval. (Bring two different colored pens/pencils!)
Craig Timmons
(Sacramento State)
Error-Correcting Codes from Finite Geometries
Error-correcting codes are often used when data is transmitted over a channel in which noise can occur, thereby damaging some of the data. There are several types of error-correcting codes. In this talk, we will discuss an error-correcting code that is defined in terms of a particular finite geometry. This finite geometry comes from the incidence matrix of the so-called Wenger graphs. These graphs are well-known to those working in graph theory. The talk will begin with a brief introduction to error-correcting codes, followed by linear codes. We will then define the finite geometry, and discuss some progress on an open problem of Cioabá, Lazebnik, and Li.
Christopher O'Neill
(UC Davis)
Random numerical semigroups
A numerical semigroup is a subset of the natural numbers that is closed under addition. Consider a numerical semigroup S selected via the following random process: fix a probability p and a positive integer M, and select a generating set for S from the integers 1,...,M where each potential generator has probability p of being selected. What properties can we expect the numerical semigroup S to have? For instance, when do we expect S to contain all but finitely many positive integers, and how many minimal generators do we expect S to have? In this talk, we answer several such questions, and describe some surprisingly deep combinatorial structures that naturally arise in the process.
No familiarity with numerical semigroups or probability will be assumed for this talk.
Jessie Loucks
(Sacramento State)
Hilbert series of certain families of numerical semigroups
We say a subset $S$ of the nonnegative integers is a numerical semigroup if $S$ is closed under addition, $S$ has a finite complement in the nonnegative integers, and $0$ is an element of $S$. For a numerical semigroup $S$, the Hilbert series of $S$ is given by the power series $\mathcal{H}(S;t) = \sum_{n \in S} t^n$. This series has a known rational form which can be found by using connections between numerical semigroups and algebraic topology. In the talk, we will discuss numerical semigroups and introduce some topology with the purpose of describing the rational form of the Hilbert series. We will also discuss and characterize some particular forms that $\mathcal{H}(S;t)$ can take when $S$ falls into certain classes of numerical semigroups.
Kevin McGown
(Chico State)
Quadratic residues and primitive roots
Quadratic residues and primitive roots are of fundamental importance in elementary number theory and have applications to factorization algorithms, graph theory, cryptography, and even acoustic engineering. After introducing the necessary number theoretic background, we will discuss the distribution of quadratic residues. Character sums will appear naturally in this context. We will then turn to studying the distribution of primitive roots. In the final part, I will indicate some recent results of Trevino, Trudgian, and myself on the least primitive root. Most of this talk will be expository in nature and aimed at an undergraduate audience.
Scott Farrand
(Sacramento State)
Elementary Number Theory Redux
One of the most elementary and useful results from number theory says that if $a$ and $b$ are positive integers, then their greatest common divisor $D$ can be expressed in the form
\[
ar + bs = D
\]
for some integers (called Bezout coefficients) $r$ and $s$. We were probably all taught that Bezout coefficients are found by a standard process stemming from the Euclidean algorithm, or by some version of trial and error. We'll take a fresh look at this, by placing the number theory in two different contexts. One is graphical and the other makes use of continued fractions. These new perspectives give us a different way of viewing and solving two problems that more naturally live in the other contexts.
They also provide two very surprising and more attractive ways of finding the Bezout coefficients for a pair of integers.