Colloquium - Spring 2013
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Mathematics Department Colloquia take place on Friday afternoons from 4:00-5:00PM in the Hill Center, Room 705, on Busch Campus. Also, due to recent construction on Route 18, most on-line maps and driving instructions are out of date. Here are updated driving directions. If you need information on public transportation , you may want to check the New Jersey Transit page for information on fares and schedules for the Northeast Corridor Line. Taxis are available at the New Brunswick train station (fare about $7) and can take you to and from the Hill Center (Victory Cabs, (732) 545-6666). The Rutgers Campus Bus System provides free inter-campus transportation, with the A and H buses taking passengers between Busch Campus and College Avenue, with the A providing a faster ride from College Avenue and the H providing a faster ride from the Hill Center : please visit their website for bus schedules and maps, including real-time tracking of campus buses.
Unfortunately, colloquium cancellations do occur from time to time. Please feel free to call our department (732)-445-3921 before embarking on your journey.
Colloquium participants and hosts may wish to also consult the Rutgers University academic calendar, as well as its calendars of religious holidays and of weather emergencies and university closings.
Organizer(s) | Lisa Carbone, Konstantin Mischaikow | Archive | |
Website | http://www.math.rutgers.edu/~carbonel/html/colloquium.html |
Past Talks
Friday, May 3rd |
Shige Peng, Shandong University and Princeton University |
" BSDE, PDE, Nonlinear Expectation and Model Uncertainty" |
| Time: 4:00 PM |
| Location: Hill 705 |
| Abstract: Nonlinear Feynman-Kac formula tells us that, when the coefficients depend only on the state of the path of a Brownian, then a Backward Stochastic Differential Equation (BSDE) becomes a quasilinear PDE of parabolic type. This reveals that, in general, a BSDE is in fact a new type of PDE called path dependent PDE, in which the continuous path plays the role of state variable x. The nonlinear semigroup associated to this PDE is a nonlinear expectation. We have also established the fully nonlinear parabolic PDE and corresponding paths of G-Brownian motion which is continuous paths under a fully nonlinear G-expectation, an important tool to measure risk for under probability and/or distribution uncertainties. |
Friday, April 26th |
Fioralba Cakoni, University of Delaware |
"Transmission Eigenvalues in Inverse Scattering Theory" |
| Time: 4:00 PM |
| Location: Hill 705 |
| Abstract: The transmission eigenvalue problem is a new class of eigenvalue
problems that has recently appeared in inverse scattering theory for inhomogeneous media. This is a
non-selfadjoint nonlinear eigenvalue problem which makes its mathematical investigation challenging and interesting.
Transmission eigenvalues are related to the so-called "non-scattering"
frequencies for which one can construct an incident wave that does not
scatter by a given inhomogeneity. Such eigenvalues provide information
about material properties of scattering media and can be determined from
scattering data, hence can play an important role in a variety of problems in target identification.
In this lecture we will survey the state-of-the-art of the transmission eigenvalue problem. In particular, we will describe how this problem arises in scattering theory, how transmission eigenvalues can be computed from scattering data and what is known mathematically about these eigenvalues. Our discussion will include the investigation of the transmission eigenvalue problem for anisotropic media, as well Faber-Krahn type inequalities and monotonicity properties for the real transmission eigenvalues. |
Friday, April 12th |
Marc Chamberland , Grinnell College |
"The 3x+1 Problem: Status and Recent Work" |
| Time: 4:00 PM |
| Location: Hill 705 |
| Abstract: The 3x+1 Problem is a long-standing conjecture. Let T be a map from the positive integers into itself, where T(x)=x/2 if x is even and T(x) = (3x+1)/2 if x is odd. The conjecture asks whether, under iteration of the map T, any positive integer eventually reaches the value one. This talk gives a survey of the various approaches and results, intersecting areas such as number theory, dynamical systems, and functional equations. |
Friday, March 8th |
Michael Saks, Rutgers University |
"Probabilistically verifiable proofs" |
| Time: 4:00 PM |
| Location: Hill 705 |
| Abstract: In the usual way we think about proofs, a proof� is a step-by-step
demonstration that an assertion is true, whose correctness can be checked
by checking each step of the proof.
Over the past 30 years, theoretical computer scientists proposed and extensively investigated the following twist on the notion of proof: A probabilistically verifiable proof is one in which the process of verification involves some random choices. We allow the possibility that we incorrectly accept an invalid proof provided that we can make the probability of this happening very small. This seemingly small twist has turned out to be amazingly powerful, allowing for the construction of systems of proof with previously unthinkable properties: (1) Zero-knowledge proofs: A prover can convince a verifier that a mathematical existence statement is true without revealing **any** information (in a mathematically precisely sense) about the object whose existence is being demonstrated. (2) Efficient interactive proofs of winning strategies for two player games: For example, it is possible for a knowledgeable enough prover to provide a relatively efficient proof that a given chess position is a winning position for white. (3) Probabilistically checkable proofs: It is possible to write any mathematical proof in a format which can be reliably spot checked: A skeptical verifier can examine the proof in a small number of randomly selected places and confidently accept the proof if it passes all of the spot checks. In this talk I'll give a glimpse at the ideas underlying these systems and some of their applications to the theory and practice of computer science. |
Friday, February 22nd |
Samuel Grushevsky, Stony Brook University |
"Curves, abelian varieties, and the Schottky problem" |
| Time: 4:00 PM |
| Location: Hill 705 |
| Abstract: To any complex algebraic curve (aka Riemann surface) one can associate its Jacobian, which is an algebraic complex torus. However, not all such tori (called abelian varieties) arise as Jacobians of Riemann surfaces, and the Schottky problem asks for a characterization of those that do. We will survey the geometric approaches to the Schottky problem, focusing on the recent progress in the approach using integrable systems. |
Friday, February 1st |
Claus Sorensen, Princeton University |
"Integral structures in Steinberg representations and p-adic Langlands" |
| Time: 4:00 PM |
| Location: Hill 705 |
| Abstract: As a vast generalization of quadratic reciprocity, class field theory describes all abelian extensions of a number field. Over Q, they are precisely those contained in cyclotomic fields.However, there are a lot more non-abelian extensions, which arise naturally. The Langlands program attempts to systematize them, by relating Galois representations and automorphic forms; mathematical objects of rather disparate nature. We will illustrate the basic plot for GL(2) through the example of elliptic curves and modular forms - the context of Wiles' proof of Fermat's Last Theorem. The main goal of the talk will be to motivate a "p-adic" Langlands correspondence, which is at the forefront of contemporary number theory, but still only well-understood for GL(2) over Q_p. We will discuss, in some depth, the case of semistable elliptic curves, which provide the first non-trivial example. This leads naturally to a result we proved recently, which shows the existence of (many) integral structures in locally algebraic representations of "Steinberg" type, for any reductive group G (such as GL(n), symplectic, and orthogonal groups). As a result, there are a host of ways to p-adically complete the Steinberg representation (tensored with an algebraic representation). The ensuing Banach spaces should play a role in a (yet elusive) higher-dimensional p-adic Langlands correspondence. We hope to at least give some idea of the proof, which goes via automorphic representations and the trace formula. For the most part, the colloquium will be very low-key and widely accessible. |