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Moonshine / Moonshine
(Christopher Cummins, Organizer)

IMIN CHEN, Simon Fraser University, British Columbia
Elliptic curves with non-split mod 11 representation

We calculate explicitly the j-invariants of the elliptic curves classified by the modular curve Xns+(11) by giving an expression defined over Q of the j-function in terms of the function field generators x and y of the elliptic curve Xns+(11). As a result we exhibit infinitely-many elliptic curves over Q with non-split mod 11 representation. This is joint work with Chris Cummins.

CHRIS CUMMINS, Concordia University
Modular equations and moonshine

In this talk I will review the relationship between modular equations, moonshine and replicable functions. Generalistions to higher genus and non-monstrous cases will be discussed.

Generalised Moonshine

We'll quickly review the philosophy of ``Generalised Moonshine''. Using it, we'll sketch some new proofs of old results from classical number theory, and the generalisations which these proofs yield.

JOHN MCKAY, Concordia University, Montreal
Two hundred years of Moonshine-from Gauss to the present day

Starting with Gauss, we give the history of the functions which arise in Moonshine and provide a parametrization of the representations of M, and their axiomatization. The prototypical function is Hermite's (1859) j-function and our other functions derive from its generalization through axiomatization. Dedekind provides one characterization through his Schwarzian differential equation. Another is provided by the action of the standard Hecke operator. These characterizations are entwined by a beautiful identity.

Much mathematics arises from the study of moonshine including some suggestive numerology relating data from E8,E7, and E6 to the three sporadic finite simple groups M, B, F24' respectively.

A lot of magic remains unexplained.

SIMON NORTON, University of Cambridge, Centre for Mathematical Sciences, Cambridge  CB3 0WB, England
Counting nets in the Monster

The concept of a ``net'' in the Monster will be defined, and the relation with Generalized Moonshine explored. It will be shown how, by means of character theory and subject to a specific conjecture, the total number of (conjugacy classes of) nets in the Monster can be counted exactly.

ABDELLAH SEBBAR, Department of Mathematics and Statistics, University of Ottawa Ottawa, Ontario  K1N 6N5
On the Moonshine congruence groups

In this talk we discuss the properties that distinguish the invariance groups of Monstrous Moonshine Hauptmoduls from similar Hauptmoduls (e.g. replicable functions). In particular we list the properties that would characterize them completely.

NORIKO YUI, Queen's University, Ontario
Mirror moonshine phenomenon: examples

Lian and S-T. Yau first observed that mirror maps of certain families of Calabi-Yau hypersurfaces are related to McKay-Thompson series arising from the Monster. This phenomenon is called ``Mirror Moonshine Phenomenon''. In this talk, I would like to give more examples of families of Calabi-Yau hypersurfaces in support of Mirror Moonshine Phenomenon. These include Calabi-Yau varieties with K3 fibrations.

This is a preliminary report on joint work in progress with Ling Long at Penn State.

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