|
Monday, September 27, 1600-1700 |
Speaker :
Alexander Stolin, Göteborg.
Title : Fermat's Last Theorem and
the Kervaire-Murthy conjectures
Abstract
:In my talk I
will explain how one and the same method from algebraic number theory can be
used in the proofs both of some cases of Fermat's Last Theorem and of the
Kervaire-Murthy conjecture in algebraic K-theory
|
Monday, October 11, 1600-1700 |
Speaker :
Iurii Drozd, Kiev (currently at Uppsala).
Title : Vector bundles over singular projective curves.
Abstract (preliminary 06/10/04)
:I present the results on the classification of vector bundles over singular projective curves, mostly obtained by Greuel and me. In particular, the tame-wild dichotomy for this problem is established and all vector bundles are classified in the tame case. I also present a classification of stable vector bundles over a cuspidal cubic (a wild curve with respect to the classification of all vector bundles).
.
|
Monday, October 25, 1600-1700 |
Speaker :
Douglas Rogers, University of Hawaii.
Title : Dissecting the Pythagorean
proposition.
Abstract
:The Pythagorean proposition seems to exercise a hold on the popular appreciation of mathematics perhaps beyond any other single piece of mathematics. Yet, it seems that there is still much to be said on the subject.
Euclid himself gives the Pythagorean proposition a double take, in Book I of
The Elements in terms of congruent triangles, and again in Book VI in terms of
Eudoxus' doctrine of proportionality. But it has been suggested that the
prototypical proof was rather by dissection, and certainly there are some very
striking proofs in this manner.
This talk focuses on proofs by dissection, attempting to suggest a
mathematical morphology - how one proof changes into another - that might be
helpful as a foundation to more purely historical investigations.
In fact, the ancient Chinese proof is presumed to have been by dissection, but, since it is no longer extant, it is a matter of lively debate as to how it went (as can be seen from Don Wagner's paper). This contains a contribution from
Jöran Friberg, whose work has concentrated more on the Babylonian
tradition, as
reported here.
However, it is known that the Chinese did favour the use of the trysquare (carpenter's square or gnomon). So the talk explores what can be done after this fashion, drawing on the speaker's
article
(this paper is about to appear in Elemente der Mathematik, but a version in Norwegian rendered by Christoph Kirfel has just appeared in NORMAT).
Liu Hui had a very elegant demonstration by dissection that the area of two
rectangles of sides a and b is equal to the area of the rectangle with one
side the perimeter of the right-triangle with legs a and b, and the other
the diameter of the inscribed circle of this right triangle - in effect, Liu
Hui cuts up the two rectangles and reassembles the pieces into one long
rectangle.
A highlight of the talk is to turn this into a demonstration of the
Pythagorean proposition.
|
Monday, November 1, 1600-1700 |
Speaker :
Natan Kruglyak, Luleå.
Title : Calderon-Zygmund type Decompositions, Covering Theorems and Applications.
Abstract
:
In 1952 A. Calderon and A. Zygmund, in their fundamental paper 'On the Existence of Certain Singular Integrals' used a geometric construction (which goes back to F. Riesz "rising sun" lemma) to define a family of decompositions of a given function into "good" and "bad" parts. This construction happens to be a cornerstone of modern harmonic analysis and is used in such fundamental results as the John-Nirenberg characterisation of BMO, the Fefferman-Stein maximal theorem,
characterisation of A_p weights and others.
It is possible to see that Calderon-Zygmund decompositions correspond to a special couple of functional spaces, namely the couple (L_1, L_{\infty}). In the talk I plan to discuss
(a) possibilities and difficulties of an extension of this construction to other couples
(b) a covering theorem which appears in connection with this extension
(c) applications to the theory of interpolation and singular integrals.
|
Monday, November 22, 1600-1700 |
Speaker :
Alan Rendall, Albert-Einstein Institut, Golm
(Deutschland).
Title : Mathematics of cosmic
acceleration.
Abstract
:Recent
astronomical observations show that the expansion of the universe is
accelerating. I will explain how attempts to understand this
theoretically lead to the concepts of the cosmological constant and dark
energy. Mathematical problems which arise naturally in this context involve the
study of the global dynamics of certain classes of solutions of the
Einstein equations. It appears that a particular solution (the
de Sitter solution) acts as an attractor for the dynamics. That this is so
can be proved by using a combination of tools from the theory of
partial differential equations (symmetric hyperbolic systems) and
differential geometry (conformal invariants).