Spring 2023 Topics Course

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.

Research Round-up

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)

Jeremy Marzuola (faculty)

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)

Michael Taylor (faculty)

Mark Williams (faculty)

Upcoming talks by RTG members

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

  • Maddie Brown (graduate student)
  • 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)
  • 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)

Girls Talk Math

It was a pleasure to welcome the Girls Talk Math program back to campus this past summer.  The RTG is proud to support this free educational summer program that strives to make advanced mathematics accessible to students of underrepresented genders in STEM.  Its founders, Katrina Morgan and Francesca Bernardi, are former UNC graduate students.  GTM has expanded to sites at the University of Maryland and at Worcester Polytechnic Institute in addition to the original location at UNC.  The Association for Women in Mathematics recently highlighted this story regarding the work at WPI.

Conference Opportunities

As we welcome back graduate student Maddie Brown from giving talks at

Please let me alert you to a couple of upcoming conference opportunities, which feature RTG members: