MIT 2020 Abstracts and Presentation Slides

August 31, 2020 — Calculus of Variations


'Regularity of the free boundary for the two-phase Bernoulli problem and eigenvalues partitions' with Guido De Philippis (Courant Institute of Mathematical Sciences - NYU)

  • Abstract: De Philippis will briefly review some optimal partitions problem and some of their applications. Eventually I will illustrate a recent result obtained in collaboration with L. Spolaor and B. Velichkov concerning the regularity of the free boundaries in the two phase Bernoulli problems and its applications to partition problems.
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‘A capillarity model for soap films’ with Francesco Maggi (University of Texas)

  • Abstract: Soap films are modeled, rather than as surfaces with zero mean curvature, as regions with small volume satisfying a spanning condition of homotopic type. We discuss qualitative properties of such soap films and their convergence towards minimal surfaces when the volume parameter goes to zero. This talk is based on a series of joint works with Antonello Scardicchio (ICTP), Darren King (UT Austin) and Salvatore Stuvard (UT Austin).
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'Optimal Shapes of Level Sets' with David Jerison (Massachusetts Institute of Technology)

  • Abstract: Level surfaces of eigenfunctions, free boundaries, and isoperimetric surfaces divide space and partition energy in optimal ways. The calculus of variations allows us to see strong parallels among the methods used to understand all of these types of surfaces. We begin by explaining the Hot Spots Conjecture of Jeff Rauch concerning the shape of the first non-constant Neumann eigenfunction in convex domains. One approach to proving it is to establish clean separation of level sets, a kind of Harnack inequality. So far we are only able to prove analogous results for free boundaries and for isoperimetric surfaces, not level surfaces of eigenfunctions. For free boundaries, the key step is understanding how a convex cone divides itself into two cones so as to minimize a suitable energy, an extension of the classical Friedland-Hayman inequality from all of space to convex cones. The work presented is mostly joint with one or another combination of Thomas Beck, Guy David, and Sarah Raynor.
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Discussion Topic 1: 'Formation of Materials' with input from Professor Craig Carter (Massachusetts Institute of Technology)

  • Details TBA

Discussion Topic 2: 'Soap Film Capillary Model' with Francesco Maggi (University of Texas)

  • Details: A motivation for introducing the soap film capillarity model discussed in my talk is introducing a length scale in the description of soap films. Indeed, when modeling soap films as minimal surfaces (surfaces with zero mean curvature), which is the classical approach to soap films, we obtain a soap film model without length scales: in particular, no effect related to physical features of actual soap films like thickness or volume can be described with minimal surfaces. Thickness seem relevant in determining the characteristic existence time of a soap film, and gravity when present should be a major factor in determining how thickness is changing over time, eventually leading to bursting by formations of hole. The analysis of our model seems to indicate that the thickness of a soap film, even in the absence of gravity, should be considerably larger near soap film singularities (like the Y points and T points of Plateau's laws). Is there any experimental evidence of such behavior?
  • Prerequisites: Familiarity with the geometric notions of mean curvature of a surface and with the capillarity theory (the introduction of Robert Finn's book on Capillarity Surfaces could be a good reference).

 

September 1, 2020 — Homogenization


‘Blackhole waves at corners of negative material’ with Anne-Sophie Bonnet-Ben Dhia (Centre national de la recherche scientifique)

  • Abstract: In this talk, we consider electromagnetic waves in presence of materials which have, in a given frequency range, a dielectric permittivity with a very small imaginary part, that will be neglected,  and a negative real part. This occurs for instance for metals (like gold or silver) at optical frequencies and for homogenized models of some metamaterials. For such materials, very unsual singular phenomena take place at corners. In particular, for some configurations, a part of the energy may be trapped by the corners: this is the so-called blackhole effect [1]. In this presentation, we first  give a mathematical analysis of this blackhole phenomenon, based on a detailed description of the corner's singularities, in the 2D case. Then we show that this phenomenon leads to numerical instabilities of finite element simulations. The solution that we have found and validated is to introduce a complex scaling at the corners. Finally, we  compute the plasmonic eigenvalues  of a 2D subwavelength particle with a corner. While a smooth particle has a discrete sequence of eigenvalues, blackhole waves at the corner lead to the presence of an essential spectrum filling an interval. Numerical results show that the complex scaling deforms this essential spectrum, so as to unveil both embedded eigenvalues and complex plasmonic resonances. The later are analogous to well-known complex scattering resonances, with the local behavior at the corner playing the role of the behavior at infinity.
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‘On Einstein’s effective viscosity formula’ with Antoine Gloria (Sorbonne Université)

  • Abstract: In one of his most cited papers, Einstein argued in 1905 that a Stokes fluid with a random suspensions of colloidal particles behaves at large scales like a homogeneous Stokes fluid with some effective viscosity, and he gave a first order expansion of the latter in the low density regime. Seventy years later, Batchelor and Green extended this expansion to second order. The aim of this talk is to rigorously justify such low density regime expansions at arbitrary order under minimal assumptions. The starting point is the definition of the effective viscosity by homogenization, which we analyse in the low density regime via a cluster expansion, combined with combinatorial arguments, PDE analysis, probability, and variational analysis. This is joint work with Mitia Duerinckx (CNRS, Université d’Orsay).

'Quantitative Stochastic Homogenization and Large-Scale Regularity' with Tuomo Kuusi (University of Helsinki)

  • Abstract: One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In our recent book, jointly with S. Armstrong and J.-C. Mourrat, we have addressed this problem from a new perspective. Essentially, we use regularity theory for stochastic homogenization to accelerate the weak convergence of the energy density, flux and gradient of the solutions as we pass to larger and larger length scales, until it saturates at the CLT scaling. I will discuss our approach and give, at the same time, an introduction to the mathematical theory of stochastic homogenization.

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Discussion Topic 1: TBA

  • Details TBA

Discussion Topic 2 with Antoine Gloria

  1. Can suspensions in a fluid generate localization?
  2. What drives long range order (like hyperuniformity) in physical disordered systems? Can it have an influence on wave propagation?


 

September 2, 2020 — Disordered Environments


‘Quantum Brownian motion as a scaling limit’ with László Erdös (Institute of Science and Technology, Austria)

  • Abstract: We review the mathematically rigorous derivations of the Boltzmann equation and the quantum Brownian motion (or quantum diffusion) starting from first principle Schrodinger dynamics with weak random potential. The main method is Feynman diagrammatic expansion with rigorous control on the error terms. The talk is based upon older joint works with H.T. Yau and M. Salmhofer.
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‘Bogoliubov theory for trapped Bose-Einstein condensates’ with Benjamin Schlein (University of Zürich)

  • Abstract: We consider systems of N trapped particles with bosonic statistics interacting through a repulsive potential with scattering length of the order 1/N (Gross-Pitaevskii regime). We prove the emergence of complete Bose-Einstein condensation and we determine the form of the low-energy spectrum, in the limit of large N. Our results confirm the predictions of Bogoliubov theory. This talk is based on joint works with C. Boccato, C. Brennecke and S. Cenatiempo.
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‘Delocalization of random band matrices’ with H.T. Yau (Harvard University)

  • Random band matrix ensembles have been used as toy models for random Schrodinger equations. In this talk, we will explain the recent work regarding  the delocalization of   eigenvectors of  random band matrices   in high dimensions. This talk is a joint work with  Fan Yang and Jun Yin.

Discussion Topic 1: 'Excitons' with Moungi Bawendi (Massachusetts Institute of Technology)

  • The heat equation and the Boltzmann equation can be derived from the Schr\"{o}dinger evolution in suitable limits when the potential is small in $d \geq 3$. In general, when the potential is $O(1)$, Quantum Brownian motion (QBM) conjecture says: the location of the particle is governed by a diffusive equation that on large scales mimics the Schr\"{o}dinger evolution. Moreover, the joint distribution of the quantum densities at different times converges to the corresponding finite-dimensional marginals of the Wiener process in suitable limits.

    This QBM conjecture extends the extended state conjecture, which states that in $d \geq 3$, with small potential, the spectrum of the Schrödinger Hamiltonian separates into absolutely continuous and pure points spectrums at the mobility edge. The QBM conjecture is stronger in the sense that it states explicitly how states will localize/extend.


Discussion Topic 2 with Benjamin Schlein (University of Zürich)

  • Can one prove the emergence of complete Bose-Einstein condensation and determine the form of the low-energy spectrum in the thermodynamics limit?

September 3, 2020 — Large Group Collaboration Discussion

  • Collaboration Working Meeting — List of possible topics:
    • Regularity of free boundary
      • How smooth is the boundary separating two phases in a free boundary problem (e.g. ice melting)?
    • Formation and growth of nitrite in semiconductor materials
      • What are suitable formation and growth mechanisms for GaN and InGaN in semiconductors? In each model, how can one do simulation and numerical analysis?
    • Stochastic homogenization
      • A heterogeneous media exhibiting random behavior on a microscopic scale can have a deterministic effective behavior on a macroscopic scale after the randomness is averaged out. How can one isolate the effective behaviors in various disordered/random microscopic systems?
    • Justification for the mesoscopic Poisson-Schrodinger equations
      • In which regime can one rigorously justify the use of band off-set $\delta E_c$ and $\delta E_v$ in the Poisson-Schrodinger equation, instead of solving the full equation before homogenizing the background crystal structure?
    • Band structure in non-periodic semiconductors
      • What are suitable notions of band structure in non-periodic semiconductors? When are they valid? (e.g. what if the disorder in the semiconductor is on the scale of the lattice constant?)
    • Phonon coupling in semiconductor and their effective 1-body theory
      • How to include phonon coupling in semiconductor equations? Phonons can be coupled to the electrons Hamiltonian on the level of 2nd quantization. It might be hopeful to use a self-consistent method to derive an effective 1-body equation with phonon coupling. One way to do this is through quasi-free reduction. What are suitable quasi-free/Gaussian states for the phonon coupled Hamiltonian?
    • Evolution of BEC under disorder
      • In a disordered median, what is the behavior of an initially trapped BEC after the trap is removed? To what extent does localization persist?
    • Validity of smoothed out point like dopant charge
      • In a semiconductor, one smooths out point-like doped charge distributions to have some scale length $\sigma$. How should this choice of $\sigma$ be made?
    • Many-body landscape to 1-body landscape
      • Starting from an N-body Hamiltonian H (with 2 body interaction terms), we can define the landscape function in the usual way: Hu = 1. In this case $u \in R^{3N}$ is extremely difficult to study. Is there an effective 1-body theory for u under a suitable regime?