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Mathematics & Statistics

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**Title**

**Speaker**

**Time**

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TBA

Caleb Springer

Penn State University

2:00pm-3:00pm

CMC 108

**Abstract**

TBA

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**Time**

**Place**

TBA

Daviel Leyva

2:00pm-3:00pm

CMC 108

**Abstract**

TBA

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**Speaker**

**Time**

**Place**

TBA

Theo Molla

2:00pm-3:00pm

CMC 108

**Abstract**

TBA

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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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**Sponsor**

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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**Sponsor**

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.