Sample2 Proposal for a Joint Summer Research Conference 1. Proposal submitted by: Peter Kuchment, Professor Mathematics and Statistics Department Wichita State University Wichita, KS 67260-0033 Voice (316)978-3939, FAX (316)978-3748 e-mail kuchment@math.twsu.edu Web page http://www.math.twsu.edu/faculty/kuchment/ Starting September 2001 at Math.Dept., Texas A&M University 2. Title of proposed conference: Waves in periodic and random media 3. Organizing committee: David Dobson, Mathematics Department, Texas A&M University Alex Figotin, Mathematics Department, University of California, Irvine Peter Kuchment (co-Chair), Mathematics Department, Wichita State University        —(Starting September 2001 at Math.Dept., Texas A&M University) Stephanos Venakides (co-Chair), Mathematics Department, Duke University 4. Program information: 4.1. Program summary The topic of wave propagation in periodic and random linear and non-linear media arises in physics and engineering and stems in particular rom quantum solid state physics, material science, electromagnetics, optics, and acoustics. The mathematical problems that appear in this area are of practical importance and pose high challenges to pure and applied mathematicians.They have attracted a lot of attention of researchers lately.The methods involved come from a wide range of areas of pure and applied mathematics ranging rom spectral theory to PDE, to complex analysis, to numerical methods.It is expected that the topics of the conference will be addressed rom analytic, numerical, and physics perspectives. In order to make the discussions more focused, it is planned to specifically target the interrelated areas that are described in more detail below. 4.1.1 Photonic crystal The idea of photonic crystals (photonic band-gap materials)was coined in 1987 ([40 ] , [67 ] ). Since then it has created a huge wave of physics, engineering (e.g., [12 ] , [35 ] , [39 ] , [48 ] , [52 ] , [53 ] , [56 ] , [57 ] , [62 ] , [66 ] ), and recently mathematicsresearch (e.g., [3 ] , [5 ] , [7 ] , [10 ] , [13 ]–[16 ] , [19 ] , [21 ] –[31 ] , [44 ] , [45 ] , [65 ] ). Photonic crystals are artificial optic materials with periodic structure that behave with respect to electromagnetic waves in a manner similar to the behavior of solid semiconductors with respect to electronic waves. One of the main properties of interest is existence of ranges of frequencies (called stop bands or band gaps) or which electromagnetic waves cannot propagate through the material. Mathematically this corresponds to gaps in the spectra of the pertinent differential operators.The first samples of PBG materials were created or the microwave frequency range in 1991 and or the visual light range in 1998.The world wide high activity around photonic crystals is explained by the enormous range of already achieved and predicted applications, which include, but are not limited to, high efficiency light sources, low threshold lasers, efficient optic waveguides, efficient antennas or airplanes and cellular phones, information transmission, and quantum computers, where PBG microcavities are suggested to be used or creating a qubit.Band-gap materials have recently started being considered also or molding the sound, elastic, and heat wave propagation with many obvious applications in mind.Many problems related to creation and studying properties of these materials are of mathematical nature.One can mention for instance studying spectral properties of the corresponding periodic differential operators (e.g., the Maxwell operator) in relation with the geometric and physical parameters of the materials. Effects of impurities, random disorder, nonlinearities, and sample surfaces are also of high interest. Research in this young area of mathematics is becoming more and more active lately.Several advances have been achieved, but the theory has not reached its adolescence yet. It is planned to invite main active researchers in it to a ruitful discussion during the conference. 4.1.2 Anderson localization Anderson electron localization in random media has been one of the hot topics of solid state physics or quite some time. P. Anderson was awarded Nobel prize in physics or his theoretical prediction of this effect.The mathematical theory of electron localization, although signi .cantly advanced during the past decade (e.g., [1 ] , [2 ] , [60 ] , although any complete list would be too extensive), is still far from being complete.This is even more true or localization of classical waves (acoustic and electromagnetic), which has attracted increasing attention of researchers in the past decade in particular due to the study of photonic crystals (e.g., [4 ] , [14 ] , [25 ] , [26 ] , [41 ] ). Localization of classical waves, besides being an important topic on its own, is expected to be much more observable in experiments in comparison with its electron analog and predecessor, which is obscured by additional effects like electron-electron interaction.It is expected that discussion of the problems of the localization theory during the conference would lead to further advances in this important area of mathematical physics. 4.1.3 Nonlinear effect Nonlinear issues of the theory of wave propagation in periodic media have been only briefly touched in very ew mathematical studies (among those closely related to the topics of the conference one could mention [6 ], [30 ] , and [31 ] ).On the other hand, its importance is well recognized by physicists and engineers ([49 ] , [59 ] , [63 ] ).Nonlinearities, on one hand, can distort the effects predicted by the linear theory, and on the other hand offer immense new opportunities not available in the linear setting.Such effects are proposed to be used in many ways (in ormation transmission and optical diodes are among the examples). One can mention or instance the necessity of development of a theory of gap solitons ([49 ] , [63 ] ). The main idea of a gap soliton is that due to nonlinearity a wave with a requency in a orbidden gap (and hence orbidden in the linear regime) could modulate the dielectric permittivity of the medium in such a way as to tune itself out of the gap and hence propagate through the medium. Although there are analogies between gap solitons and well known solitons arising in fiber optics, there are also some distinctions that make gap solitons a favorite object. The study of gap solitons, rather complete in 1 D , is still barely touched in the most attractive two-and three-dimensional cases. It is believed that this and other issues of nonlinear optics of periodic media could be pushed forward by presentations and discussions during the conference. 4.1.4 Waves in mesoscopic media and surface waves Another hot young field of physics and material science is mesoscopic physics ([17 ] , [18 ] , [20 ] , [36 ] , [33 ] , [34 ] , [37 ] , [38 ] , [54 ] , [55 ] , [58 ] ).Contemporary technology enables one to manufacture semiconductor (or super-conducting) objects whose sizes in one, two, or all three dimensions are reduced to a few nanometers. These objects look like surfaces (quantum walls), segments (quantum wires)or dots (quantum dots).A host of important applications of such systems has been achieved or predicted. In many cases the wave propagation in such media can be modeled by quasi-one-dimensional differential (or pseudo-differential) operators and equations on graphs. Such models are known to describe quantum wires, atomic and molecular wires, high temperature granular superconductors, and free-electronics, optics, and other areas require development and study of wave propagation in periodic circuits of quantum wires and in analogous structures. This problem has been addressed mathematically only very recently ([5 ] , [10 ] , [11 ] , [22 ] , [27 ] , [28 ] , [32 ] , [45 ] , [47 ] , [51 ] , [54 ] ), and its consideration at the conference would stimulate further developments. Another kind of a related problem that deserves attention of experts invited to the conference is the one on surface waves. The role of such waves has been well recognized by physicists, while their mathematical theory just starts to develop. Several very recent works addressed electron localization due to surface potentials. Another direction that needs to be addressed is the study of surface waves arising on the interface of two periodic materials.Very little is known rigorously about this. 4.2 Conference description Due to the character of the topic of the conference, not only mathematicians actively working in this and adjacent areas are invited, but also several physicists who could communicate with mathematicians on these issues.We envision that the conference will stimulate the very much needed progress in the directions described above. It is planned to set up a Web site on the conference and to publish proceedings. About 30 invited one hour and 45 minute talks will be delivered, which would leave sufficient time for fruitful discussions. It is also planned to have poster presentations. If the schedule permits to do so without overloading it, a very limited number of short contributed talks might be allowed. Involvement of young researchers, graduate students, minorities, and women will be strongly encouraged. This will be emphasized in the announcements of the conference and on the Web site. The announcement will be e-mailed to a list of researchers who might be interested in attending. Invited speakers will be encouraged to involve their graduate students and/or young faculty. 4.3 List of principal speakers We expect the total number of participants to be between sixty and seventy five. Here is the list of speakers who have already agreed to present a lecture (with affiliation and area indicated): Note: This list, while included in the original proposal, has been deleted from this sample for reasons of confidentiality. It is tentatively expected that the time will be allocated between different topics and approaches as follows: Physics (4 speakers), Numerics and optimization (6 speakers), Nonlinearity (5 speakers), Localization and surface effects (7 speakers), Mesoscopic and other aspects of wave propagation in periodic and random media (8-9 speakers). Note: The name of the speakers in each group, while included in the original proposal, have been deleted from this sample for reasons of confidentiality. It is tentatively planned to invite among other participants: Note: This list, while included in the original proposal, has been deleted from this sample for reasons of confidentiality. 4.4 Some recent conferences on related topics Mathematical Results in Quantum Mechanics, Prague, 1998, Summer Research Conference on Waves in Complex Media, Boulder, CO, 1999, Workshop on Spectral Theory, Matrei, Austria, 1999, International workshop on Quantum Spectra and Dynamics, Haifa and Jerusalem, 2000, International Conference Waves 2000, Spain, 2000, NATO Summer Institute on Photonic Crystals, Crete, Greece, 2000. The proposed conference, however, has a unique character, since it would be the first conference in the mathematics community concentrating solely on emerging field of photonic crystals, classical wave localization, and mesoscopic systems. 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