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 […]
(Joint with Ichihara, Jong, and Saito). We show that two-bridge knots admit no purely cosmetic surgeries, ie no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument is based on a recent result by Hanselman and a study of signature and finite type invariants of […]
Title: On sparse geometry of numbers Speaker: Prof. Lenny Fukshansky, Department of Mathematics, Claremont McKenna College Abstract: Geometry of Numbers is an area of mathematics pioneered by Hermann Minkowski at the end of the 19th century. He achieved stunning success introducing a novel geometric framework into the study of algebraic numbers, prompting mathematicians of later generations to compare […]
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 […]
We talk about building knots using mosaics which were as introduced as a way of modeling quantum knots by Lomonaco and Kauffman and a newer variant, hexagonal mosaics, introduced by Jennifer McLoud-Mann. In the process we find a new bound on crossing numbers for hexagonal mosaics and find an infinite family of knots which do […]
The Field Committee Meeting is our chance to socialize with our colleagues and coordinate our course offerings for the coming academic year (2022-2023). Please come to discuss course offerings and other synergistic items. Refreshments in the Shanahan sunken courtyard at HMC starting at 4:00, meeting in Shanahan B460 at 4:20. We will be back in […]
Title: Connections between Iterative Methods for Linear Systems and Consensus Dynamics on Networks Abstract: There is a well-established linear algebraic lens for studying consensus dynamics on networks, which has yielded significant theoretical results in areas like distributed computing, modeling of opinion dynamics, and ranking methods. Recently, strong connections have been made between problems of consensus […]
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 […]
The Jones polynomial is an invariant of knots in R^3. Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin-Turaev using quantum groups. Khovanov homology is a categorification of the Jones polynomial of a knot in R^3, analogously to how ordinary homology is a categorification of the Euler characteristic of a […]
Title: The 6 Cs - Covid and the 5 Claremont Colleges Speaker: Maryann E. Hohn, Department of Mathematics and Statistics, Pomona College Abstract: The Claremont Colleges' (5Cs) environment consists of students, faculty, and staff that congregate together in indoor spaces, creating a higher risk for possible COVID-19 infection. Additionally, a majority of the students live on […]
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 […]
Skein modules were introduced by Jozef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is notoriously hard, especially over the ring of Laurent polynomials. With […]
