Author: metcalfe

We are now accepting applications for postdoctoral positions within the RTG.  Please see our ad on mathjobs for application instructions.  We anticipate hiring two postdocs.  Recommendations from colleagues are very welcome, and we would be glad to have our ad shared widely.

We want to proudly highlight some of the summer work of Maddie Brown, one of our graduate students.  In addition to speaking in Bonn at the Young Women in Geometric Analysis conference and the CMS Summer Meeting in Newfoundland, she attended the Harmonic Analysis and Waves conference in Seattle and participated in the Harmonic Analysis on Manifolds Summer School in Madison.  Moreover, she volunteered as a mentor at the University of Michigan – Dearborn REU site where she led projects with Yunus Zeytuncu on Spectral Theory and CR Geometry.

We congratulate Maddie on spectacularly productive and engaged summer.

Jeremy Marzuola will offer a Graduate Topics Course (Math 891) in Spring 2023.  It will be titled: Stability of Nonlinear States in Evolution Equations.  We encourage you to enroll / watch for the course (after its completion) on the upcoming Online Topics Course Collaborative.

Description: We cover existence theory for nonlinear states in a variety of nonlinear evolution equations: 1. Dispersive models such as Korteweg-de Vries; nonlinear Schrödinger or Gross-Pitaevski, Dirac and Klein-Gordon equations, as well as other models like Density Functional Theory in quantum mechanics, Dispersion Managed models from optics, Wave and Schrödinger Maps, Schrödinger-Coulomb models, models from weak turbulence, etc.; and 2. Parabolic models such as Ginzburg-Landau equations, thin-films, and models from material science.  While these may seem disparate and unrelated, we will see that in many cases finding stationary states for such models follows from similar tools.  We will consider the existence of nonlinear states from the point of view of ode theory, the calculus of variations, and by using bifurcation theory in function spaces applied to a number of examples.  We will also introduce some numerical schemes for locating these states and discuss their convergence properties. Then, we will discuss aspects of spectral stability and address this question in a variety of examples.  Since the linearized operators around these states are non-self-adjoint, the analysis of these operators can be rather challenging, but we will consider several representative examples to identify key ideas.  In particular, we will address issues surrounding existence of stable and unstable eigenvalues, quantifying the null-space, and existence/non-existence of embedded eigenvalues.  Then, given time, we will then discuss the notions of orbital and asymptotic stability and dynamics, and how they relate to dynamical properties of the corresponding evolution equations.  Throughout we will present open problems that might be of interest for young researchers and focus on insightful computations.

We will regularly announce newly posted articles and newly accepted articles within the group.  We are proud to announce the following updates from the past few months.

Yaiza Canzani (faculty)

• • • Newly posted: Y. Canzani and J. Toth: Lower bounds for eigenfunction restrictions in lacunary regions.
    • Newly accepted: G. Berkolaiko, Y. Canzani, G. Cox, and J. Marzuola: Stability of spectral partitions and the Dirichlet-to-Neumann map.  Calculus of Variations and Partial Differential Equations, to appear.
    • Newly accepted: T. Beck, Y. Canzani, and J. Marzuola: Quantitative bounds on impedance-to-impedance operators with applications to fast direct solvers for PDEs.  Pure and Applied Analysis, to appear.

Jeremy Marzuola (faculty)

• • • New submitted: K. Datchev, J. Marzuola, and J. Wunsch: Newton polygons and resonances of multiple delta-potentials.
    • New submitted: D. Baskin, J. Gell-Redman, and J. Marzuola: Price’s law on Minkowski space in the presence of an inverse square potential
    • Newly accepted: G. Berkolaiko, Y. Canzani, G. Cox, and J. Marzuola: Stability of spectral partitions and the Dirichlet-to-Neumann map.  Calculus of Variations and Partial Differential Equations, to appear.
    • Newly accepted: T. Beck, Y. Canzani, and J. Marzuola: Quantitative bounds on impedance-to-impedance operators with applications to fast direct solvers for PDEs.  Pure and Applied Analysis, to appear.
    • Newly accepted: D. Baskin and J. Marzuola: The radiation field on product cones.  Advances in Mathematics, to appear.
    • Newly accepted: M. Holst, H. Hu, J. Lu, J. Marzuola, D. Song, and J. Weare: Symmetry breaking in density functional theory due to Dirac exchange for a hydrogen molecule.  Journal of Nonlinear Science, to appear.

Jason Metcalfe (faculty)

• • • Newly accepted:  J. Metcalfe and A. Stewart: On a system of weakly null semilinear wave equations.  Analysis and Mathematical Physics, to appear.
    • Newly accepted:  M. Facci and J. Metcalfe: Global existence for quasilinear wave equations satisfying the null condition.. Houston Journal of Mathematics, to appear.
    • Newly accepted:  K. Hepditch and J. Metcalfe: A local energy estimate for 2-dimensional Dirichlet wave equations.  Involve, to appear.

Casey Rodriguez (faculty)

• • • Newly submitted: K. Rajagopal and C. Rodriguez: On an elastic strain-limiting special Cosserat rod model.
    • Newly appeared: M. Bulíček, J. Málek, and C. Rodriguez: Global well-posedness for two-dimensional flows of viscoelastic rate-type fluids with stress diffusion.  Journal of Mathematical Fluid Mechanics 24 (2022)

Michael Taylor (faculty)

• • • Newly submitted: X. Huang, C. Sogge, and M. Taylor: Product manifolds with improved spectral cluster and Weyl remainder estimates.

Mark Williams (faculty)

• • • Newly submitted: M. Williams: Reflection of conormal pulse solutions to large variable-coefficient semilinear hyperbolic systems.

We proudly alert you to a number of upcoming presentations by RTG members.

• Maddie Brown (graduate student)
  • • Analysis Seminar at the University of New Mexico on Friday, September 9.
• Yaiza Canzani (faculty)
  • • QMATH 15 at the University of California – Davis, September 12-16
    • Colloquium at Northwestern University, October 5
    • Chern-Weil Symposium at the University of Chicago, October 7-9
    • Panelist for “Exploring a research landscape” at the GROW conference at Duke University, October 22
    • Colloquium at Dartmouth University, October 29
• Hans Christianson (faculty)
  • • SEARCDE Conference at North Carolina State University, November 12-13
• Jeremy Marzuola (faculty)
  • • Attending “At the interface between semiclassical analysis and numerical analysis of wave scattering problems” at Mathematisches Forschungsinstitut Oberwolfach (Germany)
    • Applied math seminar at the University of North Carolina – Greensboro, October 3
    • SEARCDE Conference at North Carolina State University, November 12-13
• Casey Rodriguez (faculty)
  • • Trends in Soliton Dynamics and Singularity Formation for Nonlinear Dispersive PDEs, October 21-23, 2022, Texas A&M University.
    • Workshop on Continuum Thermodynamics Analysis, Modeling, and Numerics, Prague, December 1-4.

We are happy to announce that two of the upcoming colloquium speakers have research that is aligned with the RTG.  While we encourage you to attend all of the colloquia, we particularly alert you to the upcoming visits by:

• Yannick Sire, Johns Hopkins University, October 27 (3:30pm talk)
• Daniel Tataru, University of California – Berkeley, November 10 (3:30pm talk)

We are very happy to announce that the 2022-23 UNC PDE Mini-school will feature a series of lectures by Peter Hintz from ETH Zürich.  We will have additional talks by Lili He, Katrina Morgan, and Ethan Sussman. The mini-school will take place March 24-26, 2023.  Watch the RTG webpage for forthcoming details on registration, a schedule, etc.

We wanted to alert you to the upcoming Conference in honor of Steve Zelditch, which will take place virtually September 8-11, 2022.  Steve has had a profound impact on the careers of several of the RTG faculty.  Please follow the link to register.

We will advertise other conference opportunities through Twitter.  Follow our account so you don’t miss any announcements!

We want to highlight the outreach work that co-PI Hans Christianson has been doing through the Correctional Education Program.  This past summer, Hans spent two nights a week teaching Math 118 at the Dan River Work Farm in Yanceyville.  Hans reports the work to be quite rewarding, and we are proud of his efforts.