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Special events in 2011 (not part of the colloquium)
February 24, 2011
Oort building
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Abel in Holland
For details and registration click here
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April 21, 2011
Snellius
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Kloosterman lecture
Jean-Louis Colliot-Thélène (CNRS and Université Paris-Sud, Orsay)
From sums of squares in fields to motivic cohomology
and higher class field theory
Abstract.
L. Euler (1770) proved that any positive rational number is a sum of
four squares of rational numbers. E. Artin (1927) proved that any
rational function in $n$ variables with real coefficients, if positive
on ${\bf R}^n$,
is a sum of squares of such rational functions (Hilbert's 17th
problem), and A. Pfister (1967) proved
that it may then be written as a sum of at most $2^n$ squares.
Artin also showed that positive rational functions in $n$
variables with {\it rational} coefficients are sums of squares
of such functions. That they may be represented by
a bounded number of squares, more precisely $2^{n+2}$,
was predicted in 1991. This relied on two hypotheses,
both of which are now known, one by work of V. Voevodsky,
the other one by work of U. Jannsen.
I shall go through the history of sums of squares in fields.
I shall then try to give a glimpse of the various tools employed
in the proof of the $2^{n+2}$-result :
the algebraic theory of quadratic forms (as started by E. Witt),
Milnor K-Theory, Galois cohomology, motivic cohomology
and higher class field theory.
Slides [PDF ]
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Program:
All lectures are on Thursdays from 16:00-17:00,
unless otherwise stated.
March 3, 2011
Snellius
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Prof. dr. Jaap Top (Groningen)
Schoute's discriminants
Abstract.
On Saturday, May 27, 1893 the Groningen geometer P.H.
Schoute (1846-1913)
presented three string models of algebraic surfaces during the monthly
meeting in Amsterdam of
the Royal Netherlands Academy of Arts and Sciences (KNAW). In spite of
their trip to
the Trippenhuis, these models have survived to this day and can still
be admired in the
mathematics institute of the university of Groningen. In the talk we
discuss what these models
show, why Schoute designed them, and we explain some of the beautiful
mathematical properties
of the corresponding surfaces.
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May 12, 2011
Snellius
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Dr. Peter Spreij (Amsterdam)
Block Hankel confluent Vandermonde matrices
Abstract.
Vandermonde matrices are well-known. They have a number of interesting
properties and play a role in (Lagrange) interpolation problems, partial fraction
expansions, and finding solutions to linear ordinary differential equations, to
mention just a few applications. Usually, one takes these matrices square,
q × q say, in which case the i-th column is given by u(zi), where we write
u(z) = (1,z,…,zq-1)⊤. If all the z
i (i = 1,…,q) are different, the Vandermonde
matrix is non-singular, otherwise not. The latter case obviously takes
place when all zi are the same, z say, in which case one could speak of a
confluent Vandermonde matrix. Non-singularity is obtained if one considers
the matrix V (z) whose i-th column is given by the (i - 1)-th derivative
u(i-1)(z)⊤.
We will consider generalizations of the confluent Vandermonde matrix V (z) by
considering matrices obtained by using as building blocks the q × r matrices
M(z) = u(z)w(z), with u(z) as above and w(z) = (1,z,…,zr-1), together with its
derivatives M(k)(z). Specifically, we will look at the matrix whose ij-th
block is given by M(i+j)(z). This in general non-square matrix exhibits a
block-Hankel structure. We will answer a number of elementary questions for this
matrix. What is the rank? What is the null-space? Can the latter be
parametrized in a simple way? Does it depend on z? What are left or
right inverses? It turns out that answers can be obtained by factorizing
the matrix into a product of other matrix polynomials having a simple
structure. The answers depend on the size of the matrix M(z) and the
number of derivatives M(k)(z) that is involved. The talk will be completely
elementary, no specific knowledge of the theory of matrix polynomials is needed.
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