| JANUARY 2003 | |
| Special CR Seminar | |
| Topic: | Minimal surfaces in pseudohermitian geometry and the Bernstein problem in the Heisenberg group |
| Presenter: | Jih-Hsin Cheng, Academica Sinica |
| Date: | Tuesday, January 28, 2003, Time: 4:00 p.m., Location: Fine Hall 1201 |
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| Algebraic Geometry Seminar | |
| Topic: | An Introduction to p-adic Fewnomial Theory |
| Presenter: | J. Maurice Rojas, Texas A & M |
| Date: | Tuesday, February 4, 2003, Time: 4:30 p.m., Location: Fine Hall 322 |
| Abstract: | Rene Descartes stated no later than June of 1637 that any real univariate polynomial with exactly m monomial terms has at most m-1 positive roots --- an upper bound totally independent of the degree. Finding a sharp generalization to multivariate polynomial systems would have many applications in dynamical systems and engineering, but has elluded us now for close to four centuries. We sketch an advance in a slightly different direction: an arithmetic multivariate analogue of Descartes' bound --- now for the number of geometrically isolated roots over any finite algebraic extension of the ordinary or p-adic rationals --- which is asymptotically near optimal. The upper bound is 1 + [C n (m-n)^3 log m]^n where m is the total number of distinct exponent vectors, n is the number of variables, and C is a constant depending only on the underlying field. This result generalizes and simplifies earlier work of Denef, Hendrik W. Lenstra, Jr., Lipshitz, and van den Dries, and provides the foundation for an arithmetic analogue of Khovanski's theory of fewnomials. We also highlight some connections to amoebae and complexity theory, including a variant of the P vs. NP question. |
| Ergodic Theory and Statistical Analysis Seminar | |
| Topic: | Amenable groups, symbolic dynamical systems, formal languages and their entropy |
| Presenter: | Tullio G. Ceccherini-Silberstein, Universita del Sonnio |
| Date: | Thursday, February 6, 2003, Time: 2:00 p.m., Location: Fine Hall 214 |
| Abstract: | In this talk I would like to present some recent results on CELLULAR AUTOMATA (a Garden of Eden (=Moore-Myhill) type theorem for cellular automata over amenable groups) on SUBSHIFTS (of FINITE TYPE and SOFIC: entropic inequalities and relative GOE type theorems) and FORMAL LANGUAGES (growth sensitivity of REGULAR (new proof) and CONTEXT-FREE (new result) languages). The talk will be completely self-contained and accesible to a wide audience (including first-year graduate students). |
| Joint Princeton University/IAS/Rutgers University Non-linear Analysis Seminar | |
| Topic: | Asymptotic Stability of N-solitons of NLS |
| Presenter: | Igor Rodnianski, Princeton University |
| Date: | Thursday, February 6, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Topology Seminar | |
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| Analysis Seminar | |
| Topic: | TBA |
| Presenter: | Vitali Milman, University of Tel Aviv |
| Date: | Monday, February 10, 2003, Time: 4:00 p.m., Location: Fine Hall 314 |
| PACM Colloquium | |
| Topic: | A Virtual Representation for Multi-Antenna Wireless Channels |
| Presenter: | Akbar Sayeed, University of Wisconsin at Madison |
| Date: | Monday, February 10, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: | The use of multiple-antenna arrays has emerged as a promising technology for dramatically increasing the information capacity and reliability of wireless communication systems. Accurate yet tractable channel modeling is critical to realizing the potential of antenna arrays. The underlying physical channel exhibits complex characteristics due to signal scattering over multiple paths, each path associated with a propagation delay, a Doppler shift (due to motion), directional angles, and a complex path gain. Current modeling philosophies are exemplified by two extremes: idealized but unrealistic statistical models and detailed physical (ray tracing) models that directly capture the multipath propagation but are difficult to incorporate in system design. The key premise of our work is that it is not the physical channel by itself but a fundamental understanding of its interaction with the signal space that is critical from a communication theoretic viewpoint. I will present a four-dimensional Karhunen-Loeve-like virtual representation for space-time channels that captures the essence of such interaction in time, frequency and space and exposes the intrinsic degrees of freedom in the channel. The virtual representation is essentially a Fourier series dictated by the finite array apertures, signaling duration and bandwidth and corresponds to a uniform, fixed sampling of the angle-delay-Doppler scattering space. It yields a simple geometric interpretation of the effects of scattering on channel statistics and capacity. The talk will discuss various insights into the structure of space-time channels afforded by the virtual representation as well its applications in capacity assessment, space-time code design, and wireless networks. |
| Algebraic Geometry Seminar | |
| Topic: | A Conjectural Description of the Nef Divisors on the Moduli Space of Curves |
| Presenter: | Angela Gibney, University of Michigan, Ann Arbor |
| Date: | Tuesday, February 11, 2003, Time: 4:30 p.m., Location: Fine Hall 322 |
| Abstract: | The moduli space $\overline{M}_{g,n}$ of stable, n-pointed curves of genus g is an important object of study in many areas of mathematics. This is largely because many questions about curves can be translated into questions about the birational geometry of the moduli space. One very effective way to learn about the birational geometry of a variety is to study its nef divisors. A divisor in the moduli space of curves is conjecturally nef if and only if it nonnegatively intersects a class of smooth, rational curves called F-curves. I will describe the F-curves and explain why they are thought to specify all effective curves on \overline{M}_{g,n}. I will also mention the surprising fact that if the conjecture is true for \M_{0,g+n} then it is true for \overline{M}_{g,n} as well as the current state of knowledge about the conjecture including new numerical criteria that guarantee that a divisor is nef. |
| Department Colloquium | |
| Topic: | Quasi-periodic localization and related |
| Presenter: | Jean Bourgain, Institute for Advanced Study |
| Date: | Wednesday, February 12, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
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| Algebraic Geometry Seminar | |
| Topic: | TBA |
| Presenter: | Gabriele LaNave, New York University |
| Date: | Tuesday, February 18, 2003, Time: 4:30 p.m., Location: Fine Hall 322 |
| Department Colloquium | |
| Topic: | Arnold's Diffusion |
| Presenter: | John Mather, Princeton University |
| Date: | Wednesday, February 19, 2003, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: | For a small perturbation of an integrable convex Hamiltonian system in two and a half or three degrees of freedom, there exist orbits that that wander over most of phase space. In this talk, I will provide a precise statement of this result and provide a very brief indication of the methods of proof. I will also briefly discuss related results. |
| Special Geometric Analysis Seminar *** Note special date | |
| Topic: | TBA |
| Presenter: | Ben Chow, UC San Diego |
| Date: | Thursday, February 20, 2003, Time: TBA, Location: TBA |
| Geometric Analysis Seminar | |
| Topic: | Geometric Paneitz-Branson operator: concentration phenomena and fourth order pde's |
| Presenter: | Frederic Robert, ETH |
| Date: | Friday, February 21, 2003, Time: 3:00 p.m., Location: Fine Hall 314 |
| PACM Colloquium | |
| Topic: |
Numerical experiments on the interaction between the large- and small-scale motion of the Navier-Stokes Equations |
| Presenter: | Heinz Kreiss, University of California, Los Angeles |
| Date: | Monday, February 24, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: | The problem we want to discuss is motivated by weather prediction. To start a numerical forcast one needs initial data which must be provided by observations. Unfortunately, the observational net is too sparse to determine the small-scale of the initial data. We ask the following question: Using the time history of the large-scale data, can one reconstruct the small-scale of the data? As a model problem, we consider solutions to the unforced incompressible Navier-Stokes equations in a $2\pi$-periodic box. We split the solution into two parts representing the large-scale and small-scale motions. We define the large-scale as the sum of the first $k_c$ Fourier modes in each direction, and the small-scale as the sum of the remaining modes. We attempt to reconstruct the small-scale by incorporating the large-scale solution as known forcing into the equations governing the evolution of the small-scale. We want to find the smallest value of $k_c$ for which the time evolution of the large-scale sets up the dissipative structures so that the small-scale is determined to a significant degree. Existing theory based on energy estimates gives a pessimistic estimate for $k_c$ that is inversely proportional to the smallest length-scale of the flow. At this value of $k_c$ the energy in the small-scale is exponentially small. In contrast, numerical calculations indicate that $k_c$ can often be chosen remarkably small. We attempt to explain why the time evolution of a relatively few number of large-scale modes can be used to reconstruct the small-scale modes in many situations. We also show that similar behavior is found in solutions to Burgers' equation. |
| Algebraic Geometry Seminar | |
| Topic: | Cohomology of local systems on M_2 and A_2 |
| Presenter: | Carel Faber |
| Date: | Tuesday, February 25, 2003, Time: 4:30 p.m., Location: Fine Hall 322 |
| Geometric Analysis Seminar | |
| Topic: | TBA |
| Presenter: | David Holcman, UC San Francisco |
| Date: | Friday, February 28, 2003, Time: 3:00 p.m., Location: Fine Hall 314 |
| PACM Colloquium | |
| Topic: | TBA |
| Presenter: | Luminata Vese, University of California, Los Angeles |
| Date: | Monday, March 3, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |
| Algebraic Geometry Seminar | |
| Topic: | Volume of the Space of Real Cubic Surfaces |
| Presenter: | James Carlson, University of Utah |
| Date: | Tuesday, March 4, 2003, Time: 4:30 p.m., Location: Fine Hall 322 |
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We show that the moduli space of real cubic surfaces has, in a natural way, the structure of real hyperbolic orbifold of dimension four. We discuss the structure of this space, its fundamental group, and we compute its exact hyperbolic volume. As a result we can, for instance, show that real cubics with twenty-seven real lines comprise less than two percent of the full space. |
| PACM Colloquium | |
| Topic: | TBA |
| Presenter: | Andrea Bertozzi, Duke University |
| Date: | Monday, March 10, 2003, Time: 4:00 p.m., Location: Fine Hall 214 |