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Moonshine / Moonshine
(Christopher Cummins, Organizer)
- IMIN CHEN, Simon Fraser University, British Columbia
Elliptic curves with non-split mod 11 representation
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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
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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.
- TERRY GANNON, Alberta
Generalised Moonshine
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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
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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
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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
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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
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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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