Laplacians and Heat Kernels: Theory and Applications (15w5110)

Arriving in Banff, Alberta Sunday, March 22 and departing Friday March 27, 2015

Organizers

(Ecole Polytechnique)

(Yale University)

(University of California in Davis)

Objectives

Objectives of the proposed workshop are to: examine our current understanding on the above aspects of the Laplacian eigenvalue problems and heat kernels; provide and promote new collaborations among the participants ranging from the leading experts to young scientists in broad fields related to the Laplacian eigenvalue problems and heat kernels; and cross-fertilize the ideas generated in the different fields to make further progress on these problems.In particular we plan to organize our 5-day workshop with the following daily themes.

  • Geometrical structure of eigenfunctions
    • Localization by rough boundaries, acoustic and electromagnetic applications
    • Anderson localization and related phenomena
    • Localization of eigenfunctions on graphs and networks

  • Quantum billiards
    • Scarring and quantum billiards
    • Geometry and topology of nodal lines and domains, critical points
    • Measures arising from the Gaussian free field and applications

  • Inverse and optimization problems for the Laplace operator
    • Inequalities for eigenvalues
    • "Hearing" shapes: eigenvalues as "fingerprints" of the geometry
    • Shape optimization with eigenvalues

  • Applications of Laplacian eigenfunctions and heat kernels
    • Coordinate systems, parameterization of manifolds with eigenfunctions
    • Scientific visualization and computer graphics
    • Searching engines and data mining
    • Function estimation on high-dimensional datasets
    • Image processing and analysis using heat kernels and non-local means
    • Diffusion-weighted imaging and its applications in medicine and material sciences


    About 30 scientists among 50 or so in the list of potential participants already indicated their interest in participating this workshop if it is held. We also would like to mention that the related workshop was held in February 2009 at the Institute for Pure and Applied Mathematics (IPAM), UCLA (see http://www.ipam.ucla.edu/programs/le2009/ for the details) with a great success. It attracted 87 participants (including 25 invited speakers) in different fields, ranging from mathematics to physics, neuroscience, medicine, electrical engineering, and computer science. The interactions among the participants including those between speakers and audience were great. Many participants praised the exciting talks and interdisciplinary nature of the program. The participants also consisted of a good mixture of various groups, ranging from graduate students, postdocs, early career academics, senior academics, to people in industry. Quality of the scientific program was obviously high. 29 participants submitted their evaluation forms, and 22 out of 29 said the workshop was very helpful over all, and 20 out of 29 said that the quality of the talks was high and the opportunity to interact with other participants or speakers was high. The following comments by the participants are the proof of the high quality of this workshop:


    "This was an excellent workshop. Unfortunately, this means that I have no suggestions on how to improve it."

    "Fantastic workshop overall. Video taping the lectures to make them available online would be my only request."

    "IPAM was very helpful to bring interesting people from different domains. It was brilliant!"

    "The workshop is a wonderful environment for learning, discussing the theory and applications related with Laplacian eigenvalues. I appreciate the opportunity in participating in this."

    "Excellent workshop, very interdisciplinary. Great interaction with participants."
    In fact, one of our motivations to propose this workshop comes from enthusiastic requests that we, the organizers, received from individuals who participated in that IPAM workshop. At that time, many of them (about 30) indicated to us that they would love to attend a follow-up workshop if one was actually offered. Our proposed workshop at the BIRS in 2015 will allow us an opportunity to examine the progress each person has made in this field over the last five years and see how our previous interdisciplinary meeting at IPAM had influenced collaborations among the participants. Moreover, we recently spoke with about 10 potential participants and confirmed their interest. Hence, altogether, we can say that at least about 40 people expressed their interest in participating our proposed workshop. We would like to clarify that our proposed workshop is not simply a repetition of that we organized at IPAM in 2009. We will invite a new set of mathematicians and scientists, extend the interdisciplinary community gathered at the IPAM, and emphasize the progress made since 2009. We believe that organizing and holding our workshop is a very important thing to do for this broad community of mathematicians and scientists.