| Jean-Marc Fontaine | Galois representations over a p-adic field. | 14, 20, 28 February 2001 | |
| Joost van Hamel | Lichtenbaum-Tate duality for varieties over p-adic fields. | 7 March 2001 | |
| Tzee-Char Kuo | Polar curves. | 14 March 2001 | |
| Gavin Brown | What is a flip of a complex threefold? | 21 March 2001 | |
| David Kohel | On Shimura curves and their invariants. | 28 March 2001 | |
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I will address the question: What are Shimura curves
and what can we say (computationally) about them? Shimura curves
X0D(N) generalize the more well-known modular curves X0(N)
which parametrize elliptic curves together with a cyclic isogeny
of degree N (roughly, a dominant morphism of degree N mapping
base point to base point). These played a central role in the proof
of Fermat's last theorem. Both classes of curves can be defined in
terms of a covering of the hyperbolic upper half complex plane H.
Whereas modular curves are "easy" to compute via Fourier expansions
of certain analytic functions on H, Shimura curves are much harder
to compute -- the analogous functions have no Fourier expansions.
However in both cases the data of the singularities of the reduction
modulo the "bad primes" dividing N, together with a collection of
commuting operators on them, determine various invariants of the
curve and largely determine the curve itself.
Time permitting, I will discuss the computational methods, using the arithmetic of quaternion algebras over Q, by which one can describe the data of the singularities of reduction and their associated operators. In the spirit of number theory, all varieties will be of dimensions zero or one, except those abelian varieties which are necessary to understand the former, and base rings are not assumed to be complete with respect to an infinite prime.
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| Ben Martin | Étale slices in geometric invariant theory | 4 April 2001 | |
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Let G
be a reductive algebraic group acting on an affine variety
X. It is natural to ask whether there exists a slice for the
G-action.
Very roughly, this means a closed subvariety C such that the map
G × C -> X, (g,c) |-> g·c
is an isomorphism (at least locally).
The powerful Étale Slice Theorem of Luna asserts the existence of a
slice under very general hypotheses, when the ground field has
characteristic zero. Bardsley and Richardson gave a partial generalisation
of Luna's theorem to characteristic p.
I will give a careful statement of Luna's theorem, covering the necessary geometric invariant theory along the way, and then mention a couple of applications. In the second half of the talk, I will discuss the obstructions to extending the theorem to characteristic p, including a mistake in Bardsley's and Richardson's paper. The main problem is that various maps associated to the G-action may fail to be separable.
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| Laurentiu Paunescu | Metric properties of complex polynomials | 11 April 2001 | |
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We study the effect of changing coordinates (algebraically)
on some important class of complex polynomials. In particular
we show that the property of being M-tame is not preserved
under such change (n > 3 variables).
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| Wednesday 18th | Stephen Donkin | Some cohomology of line bundles on flag manifolds in characteristic p. | |
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Thursday 26th
3:05-4:00 Eastern Ave. 404 |
Claus Fieker | Class field theory. | |
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May |
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| Wednesday 2nd |
Gavin Brown
(now in Warwick) |
K3 surfaces and graded rings | |
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| Wednesday 9th | Ben Martin | Moduli spaces | |
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| Wednesday 16th | William Stein | Visibility of Mordell-Weil Groups of Abelian Varieties | |
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| Wednesday 23rd | Helena Verrill | The geometry of higher weight modular symbols for subgroups of SL_2(Z). | |
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Thursday 31st
3:05-4:00 Eastern Ave. 404 |
John Coates | Euler characteristics of p-adic Lie groups and related arithmetic questions. (This seminar is joint with the Computational Algebra Seminar.) | |
June |
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| Wednesday 6th | Nils Bruin | Solving generalised Fermat equations | |
| Wednesday 13th | Joost van Hamel | Cohomological obstructions to the local-global principle | |
| Wednesday 27th | Anatol Kirillov | Introduction to Schubert Calculus and the Saturation Conjecture | |
July |
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Tuesday 24th 1 - 2.30pm |
Miles Reid | More about graded rings, K3 surfaces and Fano threefolds | |
August |
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| Thursday 2nd | Anthony Henderson |
A Personal Introduction to Perverse Sheaves I
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| Thursday 9th | Ruibin Zhang | Quantum Groups and Knots | |
| Thursday 16th | Anthony Henderson | A Personal Introduction to Perverse Sheaves II | |
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October |
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Special series:
Affine Hecke Algebras and Algebraic Geometry
(AHAAG)
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| Thursday 11th | Anthony Henderson | AHAAG I: Overview | |
| Thursday 18th | Anthony Henderson | AHAAG II: Overview II | |
| Thursday 25th | John Graham | AHAAG III: Introduction to affine Hecke algebras | |
November |
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| Friday 2nd | King Lai | AHAAG IV: Affine hecke algebras and groups over local fields | |
| Friday 9th | Martine Girard | Group of Weierstrass points of a plane quartic with a fixed number of hyperflexes. | |
| Friday 16th | Joost van Hamel | AHAAG V: Equivariant K-theory | |
| Friday 23rd | Joost van Hamel | AHAAG VI: Equivariant K-theory 2 | |
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Friday 30th Carslaw 709 |
Ilknur Tulunay | AHAAG VII: The Steinberg variety | |
December |
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| Friday 7th | - | (No Seminar) | |
| Friday 14th | Andrew Mathas | AHAAG VIII: Obtaining the Hecke algebra for SL2 | |
October 2002 |
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| Friday 25 | Daniel Chan | Noncommutative coordinate rings and stacks | |
November 2002 |
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Friday 15 |
Joost van Hamel | Towards an intersection homology for real algebraic varieties | |
| Friday 22 | David Kohel | $p$-Adic point counting algorithms for elliptic curves | |
January 2003 |
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| Friday 31 | Pierre Berthelot | P-adic cohomology and rational points over finite fields | |