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Mathematics & Statistics

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Computing the endomorphism ring of an ordinary abelian surface over a finite field

Caleb Springer

Penn State University

2:00pm-3:00pm

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.

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On Semantic Games for Post's Systems

Daviel Leyva

2:00pm-3:00pm

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.

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Transitive tournament tilings in oriented graphs with large total degree

Theo Molla

2:00pm-3:00pm

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.

No seminar this week.

No seminar this week.

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Error detecting codes for bases not equal to \(4n+2\)

Larry Dunning

2:00pm-3:00pm

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.

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\(\mathrm{PGL}\left(2,\mathbf{F}_q\right)\) acting on \(\mathbf{F}_q(x)\)

Xiang-dong Hou

2:00pm-3:00pm

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 \(H <\mathrm{PGL}\left(2,\mathbf{F}_q\right)\).

No seminar this week due to Spring Break.

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On Transfer Properties of Monoid Algebras

Felix Gotti

University of California at Berkeley

2:00pm-3:00pm

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.

No seminar this week.

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Self-Distributivity, A Categorical View

Emanuele Zappala

2:00pm-3:00pm

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.

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On fibered 2-knots with circle actions

Mizuki Fukuda

Tohoku University

2:00pm-3:00pm

CMC 108

Masahiko Saito

**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.

No seminar this week.

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Quandles, groups and universal algebra

David Stanovský

Charles University

Prague, Czech Republic

2:00pm-3:00pm

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.

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Applications of Chebotarev Density Theory to Computer Science

Giacomo Micheli

Institute of Mathematics

École Polytechnique Fédérale de Lausanne

Lausanne, Switzerland

2:00pm-3:00pm

CMC 108

Kaiqi Xiong

**Abstract**

In this talk I first describe two problems arising from cryptography and coding theory, and then tackle them using the Chebotarev density theorem. In fact, we show how to transform a class of problems over finite fields into Galois theoretical questions over global function fields, which then can be attacked using advanced machinery from number theory, group theory, and algebraic geometry.