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Series: Other Talks

Knowledge of the distribution of class groups is elusive -- it is not
even known if there are infinitely many number fields with trivial
class group. Cohen and Lenstra noticed a strange pattern --
experimentally, the group \mathbb{Z}/(9) appears more often than
\mathbb{Z{/(3) x \mathbb{Z}/(3) as the 3-part of the class
group of a real quadratic field \Q(\sqrt{d}) - and refined this
observation into concise conjectures on the manner in which class
groups behave randomly. Their heuristic says roughly that p-parts of
class groups behave like random finite abelian p-groups, rather than
like random numbers; in particular, when counting one should weight by
the size of the automorphism group, which explains why
\mathbb{Z}/(3) x \mathbb{Z}/(3) appears much less often than \mathbb{Z}/(9)
(in addition to many other experimental observations).
While proof of the Cohen-Lenstra conjectures remains inaccessible, the
function field analogue -- e.g., distribution of class groups of
quadratic extensions of \mathbb{F}_p(t) -- is more tractable.
Friedman and Washington modeled the \el$-power part (with \ell
\neq p) of such class groups as random matrices and derived heuristics
which agree with experiment. Later, Achter refined these heuristics,
and many cases have been proved (Achter, Ellenberg and Venkatesh).
When $\ell = p$, the $\ell$-power torsion of abelian varieties, and
thus the random matrix model, goes haywire. I will explain the correct
linear algebraic model -- Dieudone\'e modules. Our main result is an
analogue of the Cohen-Lenstra/Friedman-Washington heuristics -- a
theorem about the distributions of class numbers of Dieudone\'e
modules (and other invariants particular to \ell = p). Finally, I'll
present experimental evidence which mostly agrees with our heuristics
and explain the connection with rational points on varieties.

Series: Other Talks

This is joint work with Mitya Boyarchenko. We construct a
special hypersurface X over a finite field, which has the property of
"maximality", meaning that it has the maximum number of rational
points relative to its topology. Our variety is derived from a
certain unipotent algebraic group, in an analogous manner as
Deligne-Lusztig varieties are derived from reductive algebraic groups.
As a consequence, the cohomology of X can be shown to realize a piece
of the local Langlands correspondence for certain wild Weil parameters
of low conductor.

Series: Other Talks

A discussion of the Moulton et all (2000) paper "Metrics on RNA Secondary Structures."

Series: Other Talks

Joint colloquium between the School of Physics & the School of Earth and Atmospheric Sciences

hosted by Predrag Cvitanovi.

<a href="https://docs.google.com/spreadsheet/ccc?key=0Avrez5uyvwE7dERQQkV1eElNRUd...

To schedule a meeting with the speaker</a>.

Computational models of the Earth system lie at the heart of modern climate
science. Concerns about their predictions have been illegitimately used to
undercut the case that the climate is changing and this has put dynamical
systems in an awkward position. I will discuss ways that we, as a community,
can contribute by highlighting some of the major outstanding questions that
drive climate science, and I will outline their mathematical dimensions. I
will put a particular focus on the issue of simultaneously handling the
information coming from data and models. I will argue that this balancing
act will impact the way in which we formulate problems in dynamical systems.

Series: Other Talks

A discussion of the Allali and Sagot (2005) paper "A New Distance for High Level RNA Secondary Structure Comparison."

Series: Other Talks

Continued discussion of the Ding, Chan, and Lawrence paper (2005) "RNA secondary structure prediction by centroids in a Boltzmann weighted ensemble."

Series: Other Talks

Series: Other Talks

A discussion of the Ding & Lawrence (2003) paper "A statistical sampling algorithm for RNA secondary structure prediction."

Series: Other Talks

A discussion of the Chan & Ding (2008) paper "Boltzmann ensemble features of RNA secondary structures: a comparative analysis of biological RNA sequences and random shuffles."

Series: Other Talks

A discussion of the Smith and Waterman (1987) and Nussinov, Pieczenik, Griggs, and Kleitman (1978) papers. See http://people.math.gatech.edu/~heitsch/dmbws.html for more details. Please note new time.