# Past Events › Algebra / Number Theory / Combinatorics Seminar

## February 2022

### Recent trends in using representations in voting theory – committees and cyclic orders (Karl-Dieter Crisman, Gordon College)

One of the most important axioms in analyzing voting systems is that of "neutrality", which stipulates that the system should treat all candidates symmetrically. Even though this doesn't always directly apply (such as in primary systems or those with intentional incumbent protection), it is extremely important both in theory and practice.If the voting systems in question additionally are tabulated using some sort of points, we can translate the notion of neutrality into invariance under an action of the symmetric group…

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## March 2022

### Gap theorems for linear forms and for rotations on higher dimensional tori (Alan Haynes, University of Houston)

This talk is based on joint work with Jens Marklof, and with Roland Roeder. The three distance theorem states that, if x is any real number and N is any positive integer, the points x, 2x, … , Nx modulo 1 partition the unit interval into component intervals having at most 3 distinct lengths. We will present two higher dimensional analogues of this problem. In the first we consider points of the form mx+ny modulo 1, where x and y…

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### Equidistribution of norm 1 elements in cyclic number fields (Kate Petersen, University of Minnesota Duluth)

By Hilbert’s theorem 90, if K is a cyclic number field with Galois group generated by g, then any element of norm 1 can be written as a/g(a).  This gives rise to a natural height function on elements of norm 1.  I’ll discuss equidistribution problems and show that these norm 1 elements are equidistributed (in an appropriate quotient) with respect to this height.

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### Continuous extensions of Ramanujan-expandable arithmetic functions (Matthew Fox, Perimeter Institute for Theoretical Physics and Chai Karamchedu, Sandia National Labs)

We describe a natural way to continuously extend arithmetic functions that admit a Ramanujan expansion and derive the conditions under which such an extension exists. In particular, we show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization. We take the divisor function as a case study, and consider how to continuously extend it to the reals.

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### Peg solitaire in multiple colors on graphs (Tara Davis, Hawaii Pacific University and Roberto Soto, Cal State Fullerton)

Peg solitaire is a popular one person board game that has been played in many countries on various board shapes. Recently, peg solitaire has been studied extensively in two colors on mathematical graphs. We will present our rules for multiple color peg solitaire on graphs. We will present some student and faculty results classifying the solvability of the game on several graceful graphs, as well as discuss open questions.

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## April 2022

### Covering by polynomial planks (Alexey Glazyrin, University of Texas Rio Grande Valley)

In 1932, Tarski conjectured that a convex body of width 1 can be covered by planks, regions between two parallel hyperplanes, only if the total width of planks is at least 1. In 1951, Bang proved the conjecture of Tarski. In this work we study the polynomial version of Tarski's plank problem. We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and…

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### Geometrization of Markov numbers (Oleg Karpenkov, University of Liverpool)

In this talk we link discrete Markov spectrum to geometry of continued fractions. As a result of that we get a natural generalization of classical Markov tree which leads to an efficient computation of Markov minima for all elements in generalized Markov trees.

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### A conjugacy criterion for two pairs of 2 x 2 matrices over a commutative ring (Bogdan Petrenko, Eastern Illinois University)

I will explain how to apply presentations of algebras (together with some classical results from non-commutative algebra) to obtain some 5 polynomial invariants telling us when two pairs of 2x2 matrices over a commutative ring are conjugate, assuming that each of these pairs generate the matrix algebra. This talk is based on my joint paper with Marcin Mazur (Binghamton University):  Separable algebras over infinite fields are 2-generated and finitely presented, Arch. Math. 93 (2009), 521-529.

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### Bounds for nonzero Littlewood-Richardson coefficients (Müge Taskin, Boğaziçi University, Turkey)

As  $\lambda$ runs through all integer partitions, the set of   Schur functions $\{s_{\lambda}\}_\lambda$ forms a basis in the ring of symmetric functions. Hence the rule $$s_{\lambda}s_{\mu}=\sum c_{\lambda,\mu}^{\gamma} s_{\gamma}$$ makes sense and the coefficients $c_{\lambda,\mu}^{\gamma}$ are called \textit{Littlewood-Richardson (LR) coefficients}. The calculations of Littlewood-Richardson coefficients has been an important problem from the first time they were introduced, due to their important role in representation theory of symmetric groups and enumerative geometry. In this talk we will explain some of the main…

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## May 2022

### Beran’s tests of uniformity for discrete data (Michael Orrison, HMC)

Suppose you are given a data set that can be viewed as a nonnegative integer-valued function defined on a finite set. A natural question to ask is whether the data can be viewed as a sample from the uniform distribution on the set, in which case you might want to apply some sort of test of uniformity to the data. In this talk, I will share some work Anna Bargagliotti (Loyola Marymount University) and I have been doing to better…

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