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.

Speaker: David Yuen (Lake Forest College)

Title: Modularity in Degree Two

Abstract: Modularity in degree one involves weight two elliptic modular forms. Modularity in degree two involves weight two Siegel paramodular forms.

The Paramodular Conjecture of Brumer and Kramer tells us where to look for Siegel modular forms that correspond to certain abelian surfaces defined over the rationals.

This talk focuses on computations, and we consider several ways of computing Siegel modular forms over the paramodular group in degree two. We compile, in conjunction with Brumer and Kramer’s data on abelian surfaces, substantial evidence for the Paramodular Conjecture. We discuss strong evidence for existence and nonexistence of nonlift paramodular eigenforms of weight two in general levels N up to N < 1000.

We will show how to construct a nonlift paramodular eigenform of level 277 that conjecturally corresponds to the abelian surface of conductor 277. We discuss recent work that proves that, indeed, these two objects have the same L-function, and thus the abelian surface of conductor 277 is modular.

Speaker: Sita Benedict, UH-Manoa

Title: The modulus of curve families in R^n and its properties (Part 1 of 2)

Abstract: This talk will be an introduction to what is the modulus of a curve family, introduced in 1950 by Ahlfors and Beurling and generalized to its current form by Fuglede in 1957. We will define the modulus for curve families in R^n, but the definition makes sense in a general metric measure space. Although the definition is a bit cumbersome, the modulus has some very nice and useful properties, which lead naturally to the definition of quasiconformal mappings and also Newtonian spaces. These are, respectively, generalizations of conformal mappings and Sobelev spaces. We will prove these nice properties of the modulus and also calculate the modulus of certain families of curves.

See Thomas for details

See organizers Rufus, Alan or Robin for details.