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Mathematics & Statistics

Discrete Mathematics
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Wednesday, April 24, 2019

Title
Speaker

Time
Place

TBA
Caleb Springer
Penn State University
2:00pm-3:00pm
CMC 108

Abstract

TBA

Monday, April 22, 2019

Title
Speaker
Time
Place

TBA
Daviel Leyva
2:00pm-3:00pm
CMC 108

Abstract

TBA

Monday, April 15, 2019

Title
Speaker
Time
Place

TBA
Theo Molla
2:00pm-3:00pm
CMC 108

Abstract

TBA

Monday, March 25, 2019

Title
Speaker
Time
Place

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.

Monday, March 18, 2019

Title
Speaker
Time
Place

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

Monday, March 11, 2019

No seminar this week due to Spring Break.

Monday, March 4, 2019

Title
Speaker

Time
Place

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.

Monday, February 25, 2019

No seminar this week.

Monday, February 18, 2019

Title
Speaker
Time
Place

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.

Wednesday, February 13, 2019

Title
Speaker

Time
Place
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.

Monday, February 4, 2019

No seminar this week.

Monday, January 28, 2019

Title
Speaker


Time
Place

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.

Monday, January 14, 2019

Title
Speaker



Time
Place
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.