Triangle Seminars
Monday, 23 May 2005
An informal introduction to topological string theory, part 3.
Chris Hull
(IC)
Abstract:
String Theory and Geometry Seminar
String Theory and Geometry Seminar
Posted by: KCL
The lost proof of Loewner's theorem
📍 London
Barry Simon
(Caltech)
Abstract:
(This talk is an exceptional colloquium of the Department of Mathematics.)
A real-valued function, F, on an interval (a,b) is called matrix monotone
if F(A) is less than F(B) whenever A and B are finite matrices of the same order with
eigenvalues in (a,b) and A less than B. In 1934, Loewner proved the remarkable
theorem that F is matrix monotone if and only if F is real analytic with
continuations to the upper and lower half planes so that Im F is positive in the
upper half plane.
This deep theorem has evoked enormous interest over the years and a number
of alternate proofs. There is a lovely 1954 proof that seems to have been
lost in that the proof is not mentioned in various books and review
article presentations of the subject, and I have found no references to
the proof since 1960. The proof uses continued fractions.
I'll provide background on the subject and then discuss the lost proof and
a variant of that proof which I've found, which even avoids the need for
estimates, and proves a stronger theorem.
(This talk is an exceptional colloquium of the Department of Mathematics.)
A real-valued function, F, on an interval (a,b) is called matrix monotone
if F(A) is less than F(B) whenever A and B are finite matrices of the same order with
eigenvalues in (a,b) and A less than B. In 1934, Loewner proved the remarkable
theorem that F is matrix monotone if and only if F is real analytic with
continuations to the upper and lower half planes so that Im F is positive in the
upper half plane.
This deep theorem has evoked enormous interest over the years and a number
of alternate proofs. There is a lovely 1954 proof that seems to have been
lost in that the proof is not mentioned in various books and review
article presentations of the subject, and I have found no references to
the proof since 1960. The proof uses continued fractions.
I'll provide background on the subject and then discuss the lost proof and
a variant of that proof which I've found, which even avoids the need for
estimates, and proves a stronger theorem.
Posted by: KCL
Thursday, 26 May 2005
Non-associative T-duals
Keith Hannabuss
(Oxford)
Abstract:
This seminar reviews some of the global algebraic and geometric structure present in T-duality. Bouwknegt, Evslin and Mathai have given a geometric procedure for handling T-duality for certain non-trivial principal torus bundles. Subsequent work by Mathai and Rosenberg showed that sometimes when there is no geometric T-dual there may be a non-commutative torus bundle playing the same role. This exploited a very similar duality for C-star-algebras known for a couple of decades.
This talk will review those developments and their recent extension in
collaboration with Bouwknegt and Mathai to more general situations in
which the algebraic structure becomes non-associative.
This seminar reviews some of the global algebraic and geometric structure present in T-duality. Bouwknegt, Evslin and Mathai have given a geometric procedure for handling T-duality for certain non-trivial principal torus bundles. Subsequent work by Mathai and Rosenberg showed that sometimes when there is no geometric T-dual there may be a non-commutative torus bundle playing the same role. This exploited a very similar duality for C-star-algebras known for a couple of decades.
This talk will review those developments and their recent extension in
collaboration with Bouwknegt and Mathai to more general situations in
which the algebraic structure becomes non-associative.
Posted by: IC