Speaker: David Jeffrey (U. Western Ontario)
Title: Twenty years of Lambert W
Abstract: The year 2016 marks 20 years since the publication of the paper
“On the Lambert W function”. As will be pointed out, the function was
studied before 1996, but this publication has proved to be the most cited reference. The talk will review some of the decisions made in defining
the function, particularly in defining its branches.
The talk will present some of the interesting and beautiful properties
of the function, including recent work that improves on the results in
the original papers. Specifically, expressions for derivatives and
series expansions, the role of differences between branches,
the convergence of series expansions.
See organizers Rufus, Alan or Robin for details.
Speaker: Asaf Hadari (UH Mānoa)
Title: Hilbert’s third problem – how to cut and paste using linear algebra
Abstract: In the year 1900 the mathematician David Hilbert famously gave a list of 21 problems that he felt were the most important challenges facing the mathematical community of the day.
The third problem, though stated differently, essentially asked whether it was necessary to use calculus to do basic geometry in three dimensions. For instance, is there a geometric way to calculate the volume of a pyramid?
This was the first of his problems that was answered, using a neat idea from linear algebra. I’ll show you how, and discuss some of the neat mathematics surrounding this problem.
Speaker: Amy Feaver (The King’s University)
Title: The Structure of Multiquadratic Number Fields
Abstract: In this talk, we will discuss the basic structure of multiquadratic number fields: that is, fields of the form $K:=\mathbb{Q}\left(\sqrt{a_1},\sqrt{a_2},…,\sqrt{a_n}\right)$ for $a_1,…,a_n\in\mathbb{Z}$ squarefree. We will begin by discussing subfields of $K$ and what we are able to know about $K$ based on its subfields. We will then define the ring of integers for these fields, and discuss several properties of these rings. These properties will include prime factorizations and the existence of a Euclidean algorithm (or lack thereof).
See organizers Rufus, Alan or Robin for details.
Speaker: Robin Deeley, UH – Manoa
Title: Local index theory: an overview
Abstract: I will review the statement of the Atiyah-Singer index theorem with the aim of discussing local index theory. The main aim of this theory is to replace the cohomology class in the statement of the index theorem with an explicitly constructed differential form. This can be done for Dirac type operators using the heat kernel. Outlining this process is the main goal of the talk, but applications will also be discussed if time permits.
Title: Algebraic Realization of Complex Equivariant Vector bundles over the 2-Sphere with actions by Rotational Symmetry Groups of the Platonic Solids
Abstract: We verify the Algebraic Realization Conjecture for complex equivariant vector bundles over the 2-sphere with effective actions by the rotational symmetries of the tetrahedron, octahedron, and icosahedron. We introduce three strongly algebraic complex line bundles over the 2-sphere by constructing classifying maps. We demonstrate how every equivariant complex line bundle is a tensor product of these three established strongly algebraic bundles, and any equivariant complex vector bundle over the 2-sphere is a Whitney sum of equivariant complex line bundles. Our classification proofs rely on equivariant CW complex constructions and the induced equivariant pointed cofibration sequences.
See organizers Rufus, Alan or Robin for details.