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| Department Colloquium | ||
| Topic: | The lost proof of Loewner's theorem | |
| Presenter: | Barry Simon, Caltech | |
| Date: | Wednesday, May 7, 2003, Time: 4:30 p.m., Location: Fine Hall 314 | |
| Abstract:
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A real-valued function, F, on an interval (a,b) is called matrix monotone if F(A) < F(B) whenever A and B are finite matrices of the same order with eigenvalues in (a,b) and A < 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 > 0 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 avoids the need for estimates, and proves a stronger theorem. | |
| Mathematical Physics Seminar *** Please note special date and time | ||
| Topic: | A Canonical Factorization for Meromorphic Herglotz Functions on the Disk and a Proof of the Jacobi Matrix P2 Sum Rule on One Foot | |
| Presenter: | Barry Simon, Caltech | |
| Date: | Thursday, May 8, 2003, Time: 1:30 p.m., Location: Jadwin A06 | |
| Abstract: | Last year Killip and Simon provided a complete description of the spectral measure associated to Jacobi matrices with L2 potentials. I will present a simple proof of their result that relies on two elements: an analysis of the general form of meromorphic Herglotz functions on the disk and the upper semicontinuity of the entropy. I'll begin by describing the general issue of the spectral and inverse spectral problems for Jacobi matrices, the significance of the P2 sum rule, then the canonical factorization and then the proof of the P2 sum rule. | |
| Ergodic Theory and Statistical Analysis Seminar *** Please note special time | ||
| Topic: | Aggregation of inertial particles in turbulent flows | |
| Presenter: | Jérémie Bec, Institute for Advanced Study | |
| Date: | Thursday, May 8, 2003, Time: 2:30 p.m., Location: Fine Hall 214 | |
| Abstract: | The clustering properties of inertial (finite-size) particle suspensions in an incompressible turbulent flow play an essential role in the understanding of many natural and industrial problems, such as optimization of combustion processes, the growth of rain droplets in turbulent clouds, the formation of planetesimals of the Solar System, co-existence between several species of plankton. We consider the motion of collisionless inertial particles embedded in a d-dimensional smooth incompressible flow. This system is governed by a 2d-dimensional dissipative dynamical system in the position-velocity phase space, so that the phase-space density becomes singular in the statistical steady state. I will demonstrate that there exists a threshold in Stokes number (non-dimensional viscous friction time) for the condensation of the particles onto dynamical fractal clusters in the physical space. This result was confirmed by numerical studies which gave also some hints on the scaling properties of the multifractal distribution for the mass of particles. Finally, I will show how these properties can be related to the large deviations of the finite-time Lyapunov exponents. | |
| Joint Institute for Advanced Study /Princeton University/ Rutgers University Number Theory Seminar | ||
| Topic: | Small gaps between primes | |
| Presenter: | Dan Goldston, San Jose State University | |
| Date: | CANCELLEDThursday, May 8, 2003, Time: 4:15 p.m., Location: Fine Hall 322 | |
| Abstract: | This talk will discuss the main new idea behind the proof that there are infinitely many primes much closer together than the average spacing between primes, and how this idea was discovered. A sketch of the proof will be given. This is joint work with C. Yalcin Yildirim. | |
| Joint Institute for Advanced Study /Princeton University/ Rutgers University Number Theory Seminar | ||
| *** Please note special day and time *** | ||
| Topic: | Recent Developments Related to Prime Gaps | |
| Presenter: | Dan Goldston, San Jose State University | |
| Date: | CANCELLED Friday, May 9, 2003, Time: 2:00 p.m., Location: Fine Hall 322 | |
| Abstract: | This talk will discuss improvements in the method for detecting small gaps between primes made by a number of people in the last two months, and problems that still need to be examined. | |
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| Algebraic Geometry Seminar | ||
| Topic: | Invariants of Finite Groups on Small Tensor Powers | |
| Presenter: | Robert Guralnick, University of Southern California | |
| Date: | Tuesday, May 13, 2003, Time: 4:30 p.m., Location: Fine Hall 322 | |
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MAY 26- MAY 30, 2003 |
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| Topology Seminar *** Please note special time *** | ||
| Topic: | Manifolds with planar presentation and the width of satellite knots | |
| Presenter: | Marty Scharlemann, University California at Santa Barbara | |
| Date: | Thursday, May 29, 2003, Time: 1:00 p.m., Location: Fine Hall 314 | |
| Abstract: | We consider compact 3-manifolds M
having a submersion h to R in which each generic point inverse is a planar
surface. The standard height function on a submanifold of the
3-sphere is a motivating example. To (M, h) we associate a
connectivity graph G. For M in the 3-sphere, G is a tree if
and only if there is a Fox re-imbedding of M which carries horizontal circles to
a complete collection of complementary meridian circles. On the other
hand, if the connectivity graph of the complement of M is a tree, then
there is a level preserving reimbedding of M so that its complement is a
connected sum of handlebodies. Corollary: The width of a satellite knot is no less than the width of its pattern knot. In particular, the width of K_1 # K_2 is no less than the maximum of the widths of K_1 and K_2. |
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