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Thursday March 22, 2012
Stewart Library
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2:00-3:00pm
Pantelis E. Eleftheriou, University of Waterloo
Definable quotients of locally definable groups
A locally definable group in an o-minimal structure is
a group whose domain is a countable union of definable sets
U_i and whose multiplication is definable when restricted
to each U_i x U_j. An important example is the universal cover
of a definable group. In recent work with Y. Peterzil, we
examined the following converse:
Conjecture. Let U be a connected abelian locally definable
group which is generated by a definable set. Then U is the
cover of some definable group.
We will report progress on this conjecture, mentioning joint
work with Y. Peterzil, as well as work by other authors.
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3:30-4:30pm
Martin Bays, McMaster University
Abelian integrals and categoricity
An integral of the form $\Int_{P_0}^P f(x,y)dx$, where
$f$ is a rational function and $x$ and $y$ satisfy a polynomial
dependence $p(x,y)=0$, is known as an Abelian integral. Fixing
one endpoint $P_0$ and allowing the other to vary on the Riemann
surface $p(x,y)=0$, we obtain a multifunction whose value
depends on the (homology class of) the path along which we
integrate.
We consider the model-theoretic status of such multifunctions,
and in particular the problem of giving categorical elementary
descriptions of structures incorporating them and their interactions
with the complex field. Following work by Zilber and Gavrilovich,
we will find that the classical theory of Abelian varieties,
along with Faltings' work and some model theoretic ideas due
to Shelah, allow us to give partially satisfactory answers
in some special cases.
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August 29, 2011
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Martin Bays, McMaster
Some definability results in abstract Kummer theory
Classically, Kummer theory for a commutative algebraic group
reduces Galois theoretic properties of points of the group to
"linear", group theoretic properties. We observe that
a geometric part of this theory goes through in arbitrary commutative
groups of finite Morley Rank, and deduce from the proof a useful
definability property in the original case of algebraic groups.
(This is joint work with Misha Gavrilovich and Martin Hils.)
Anand Pillay, University of Leeds
Nash groups
The category of Nash manifolds lies in between the real
algebraic and real analytic categories. I revisit the category
of Nash groups, giving a correct account of the virtual algebraicity
of affine Nash groups, a generalization to real closed fields
as well as some new examples of nonaffine Nash groups. (Joint
with E. Hrushovski.)
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| June 30, 2011 |
Chris Miller
Oscillatory trajectories, Minkowski dimension and definability.
In recent joint work with A. Fornasiero and P. Hieronymi,
we showed that an expansion of the real field (in the sense
of model theory) avoids defining the set of all integers if
and only if every compact definable subset of the line has
either nonempty interior or Minkowski dimension zero. I will
give an outline of the proof, and discuss consequences for
expansions of the real field by oscillatory trajectories of
planar vector fields.
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Geometry
and Model Theory Seminar 2011-12