# Research

## Discrete Mathematics

### Wednesday, April 24, 2019

Title: Computing the endomorphism ring of an ordinary abelian surface over a finite field
Speaker: Caleb Springer, Penn State University
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

The endomorphism ring of an ordinary abelian variety over a finite field is an order in a CM field. Knowing the endomorphism ring of a given abelian variety is useful for various problems, such as understanding isogeny graphs and computing class polynomials. In this talk, we present a subexponential algorithm for computing the endomorphism ring by utilizing class groups.

### Monday, April 22, 2019

Title: On Semantic Games for Post's Systems
Speaker: Daviel Leyva
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

In the mid-20th century, Jaakko Hintikka developed an alternative approach to semantics for predicate logic using notions of game theory. The approach, known today as game-theoretical semantics, associates a formula of predicate logic with a (semantic) game played between two players in such a way that the outcome of the game tells us whether the formula is true or false in a particular model.

Much has happened since Hintikka's approach was formally brought to light, including research on games for different systems of many-valued logic; however, games for the systems of Post have remained in the dark. In this talk, we shall define games for Post's propositional systems of logic and, hopefully, show that the games do capture these systems.

### Monday, April 15, 2019

Title: Transitive tournament tilings in oriented graphs with large total degree
Speaker: heo Molla
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

An orientation of a simple graph is called an oriented graph, and an orientation of the complete graph is called a transitive tournament if it does not contain a directed cycle. In this talk, we will investigate the minimum degree threshold for oriented graphs on $$n=mk$$ vertices to contain a collection of $$m$$ vertex-disjoint copies of the transitive tournament on $$k$$ vertices.

As observed by Yuster, for $$k=3$$, the Hajnal–Szemeredi Theorem implies that $$5n/6$$ is the correct minimum degree threshold. For $$k=4$$, we will show that the asymptotically correct minimum degree threshold is $$11n/12$$. We will also discuss a number of related conjectures and results.

This is joint work with Louis DeBiasio, Allan Lo, and Andrew Treglown.

### Monday, April 8, 2019

No seminar this week.

### Monday, April 1, 2019

No seminar this week.

### Monday, March 25, 2019

Title: Error detecting codes for bases not equal to $$4n+2$$
Speaker: Larry Dunning
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

The problem of constructing a decimal error detecting code, base 10, that handles transcription(single), transposition, and twin errors using a single check digit for arbitrary lengths remains unresolved. A three-digit code based on a block design does, however, exist. Codes have been constructed that do handle transcription and transposition errors (e.g., Damm) for all bases $$b\ge 4$$ except $$b=6$$. Our main result will be a construction that yields codes for all bases $$b\ge 4$$ where $$b\ne 4n+2$$ detecting transcription, transposition and twin errors.

### Monday, March 18, 2019

Title: $$\mathrm{PGL}\left(2,\mathbf{F}_q\right)$$ acting on $$\mathbf{F}_q(x)$$
Speaker: Xiang-dong Hou
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

Let $$\mathbf{F}_q(x)$$ be the field of rational functions over $$\mathbf{F}_q$$ and treat $$\mathrm{PGL}\left(2,\mathbf{F}_q\right)$$ as the group of degree one rational functions in $$\mathbf{F}_q(x)$$ equipped with composition. $$\mathrm{PGL}\left(2,\mathbf{F}_q\right)$$ acts on $$\mathbf{F}_q(x)$$ from the right through composition. The Galois correspondence and Lüroth's theorem imply that every subgroup $$H$$ of $$\mathrm{PGL}\left(2,\mathbf{F}_q\right)$$ is the stabilizer of some rational function $$\pi_H(x)\,2\,F_q(x)$$ with $$\deg \pi_H=|H|$$ under this action, where $$\pi_H(x)$$ is uniquely determined by $$H$$ up to a left composition by an element of $$\mathrm{PGL}\left(2,\mathbf{F}_q\right)$$. In this paper, we determine the rational function $$\pi_H(x)$$ explicitly for every $$H <\mathrm{PGL}\left(2,\mathbf{F}_q\right)$$.

### Monday, March 11, 2019

No seminar this week due to Spring Break.

### Monday, March 4, 2019

Title: On Transfer Properties of Monoid Algebras
Speaker: Felix Gotti, University of California at Berkeley
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

For a field $$F$$ and a (commutative) monoid $$M$$, let $$F[x;M]$$ denote the monoid algebra of polynomial expressions with coefficients in $$F$$ and exponents in $$M$$. We say that a property $$P$$ (satisfied by certain monoids) is transfer on monoid algebras if, whenever a monoid $$M$$ satisfies $$P$$ the (multiplicative monoid of) the integral domain $$F[x;M]$$ also satisfies $$P$$ for any field $$F$$. I will discuss several algebraic transfer properties, including being a GCD-monoid and satisfying the ACCP (i.e., ascending chain condition on principal ideals). Robert Gilmer in the 1980's posed the question of whether being atomic is a transfer property. To answer Gilmer's question, I will provide a class of atomic monoids having non-atomic monoid algebras.

### Monday, February 25, 2019

No seminar this week.

### Monday, February 18, 2019

Title: Self-Distributivity, A Categorical View
Speaker: Emanuele Zappala
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

In this talk I will introduce the concept of self-distributivity for operations of arbitrary arity and their (co)homology theory. In particular, I will focus on ternary operations and their relation to families of binary operations satisfying a mutual distributivity condition, in terms of chain complex maps. I will then address the issue of internalizing these structures in categories with finite products and produce examples in Hopf algebras and Lie algebras. A natural question will arise: Is there any operad governing self-distributivity? I will outline a proof that this is not the case.

### Wednesday, February 13, 2019

Title: On fibered 2-knots with circle actions
Speaker: Mizuki Fukuda, Tohoku University
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

A 2-knot is an embedded 2-sphere in the 4-sphere. A branched twist spin is a 2-knot constructed by using a circle action and a classical knot. It is known by Hillman and Plotnick that a 2-knot is a branched twist spin if and only if the 2-knot is fibered and its monodromy is periodic. Roughly speaking, a 2-knot is called fibered if its complement admits a fibration structure over the circle. In this talk, I introduce branched twist spins and show sufficient conditions to distinguish branched twist spins by using elementary ideals and Gluck twists.

### Monday, February 4, 2019

No seminar this week.

### Monday, January 28, 2019

Title: Quandles, groups and universal algebra
Speaker: David Stanovský, Charles University
Prague, Czech Republic
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

The main motivation behind the theory of quandles is their application in knot theory, for construction of coloring invariants. Nevertheless, these objects are exciting on their own. I will present some of the attempts to understand what quandles are, using more established branches of algebra. Group theory is a particularly powerful tool: algebraically connected quandles can be represented as certain configurations in transitive groups, and subsequently, one can use deep group-theoretical results to prove theorems about quandles. Universal algebra offers a general framework to study classes of abstract algebraic structures. We were thrilled to find out that some quandle-theoretic concepts have their universal algebraic counterparts (for example, extensions by constant cocycles correspond to uniform strongly abelian congruences), and that some universal algebraic concepts have a neat interpretation in quandles (for example, solvability and nilpotence). The goal of my talk is to present the main ideas behind the interplay of these seemingly distant subjects.

### Monday, January 14, 2019

Title: Applications of Chebotarev Density Theory to Computer Science
Speaker: Giacomo Micheli, Institute of Mathematics
École Polytechnique Fédérale de Lausanne
Lausanne, Switzerland
Time: 2:00pm‐3:00pm
Place: CMC 108