Author: metcalfe

RTG group members Yaiza Canzani and Jeremy Marzuola recently returned from their last week-long SQuaRE (Structured Quartet Research Ensembles) at the the American Institute of Mathematics in San Jose, CA.  The two RTG members, along with their collaborators Greg Berkolaiko and Graham Cox began this journey in 2018, when their original collaborative proposal was approved by this NSF funded institute that specializes in bringing small groups together for focused collaboration.  Each team is approved for three annual meetings to solidify the collaboration and allow for deeper discussions over a long period of time.  The team has been working on projects related to properties of eigenfunctions in a variety of settings, as well as on the notion of equipartition for domains and manifolds.  They have found beautiful connections between properties of spectral equipartitions and the corresponding Dirichlet-to-Neumann map for the given partition.

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.

Arunima Bhattacharya, Assistant Professor

• • Newly posted: J. Bernstein and A. Bhattacharya: Colding-Minicozzi Entropies in Cartan-Hadamard Manifolds
  • Upcoming invited talk: Geometric Analysis Seminar, University of Chicago (February 2023)
  • Upcoming invited talk: Gauge Theory, Geometric Analysis, and Low-Dimensional Topology, Special Session of an AMS Sectional Virtual Meeting (April 2023)
  • Upcoming invited talk: Geometric and Analytic Methods in PDE, Special Session of an AMS Sectional Meeting (April 2023).
  • Upcoming invited talk: Differential Geometry/PDE Seminar, University of Washington (May 2023).

Madelyne Brown, Graduate Student

• • Presentation: Analysis Seminar, University of Rochester (September 30, 2022)
  • Presentation: Analysis Seminar, University of Wisconsin (October 18, 2022)
  • Attended the Chern-Weil Symposium at the University of Chicago (October 7-9, 2022)
  • Upcoming invited talk:  Analysis and PDE Seminar, Johns Hopkins University (March 6, 2023)

Yaiza Canzani, Associate Professor

• • Newly posted:  Y. Canzani, J. Galkowski, and B. Keeler (former UNC graduate student): Asymptotics for the spectral function on Zoll manifolds.
  • Newly accepted: Y. Canzani and J. Galkowski: Weyl remainders: an application of geodesic beams.  Invent. Math., to appear.
  • Presentation: Münich-Aahrus-Santiago Seminar in Mathematical Physics (virtual) (December 5, 2022)
  • Upcoming invited talk: Geometria em Lisboa Semina (virtual) (January 17, 2023)
  • Upcoming invited talk: Analysis Seminar, Princeton University (February 20, 2023)
  • Upcoming invited talk: Spectral Geometry in the Clouds Seminar (virtual) (March 6, 2023)

Hans Christianson, Professor

• • Newly posted:  H. Christianson and D. Pezzi (former UNC undergraduate): Energy distribution for Dirichlet eigenfunctions on right triangles.

Jeremy Marzuola, Professor

• • Newly posted: Z. Boyd, N. Fraiman, J. Marzuola, P. Mucha, and B. Osting: Escape times for subgraph detection and graph partitioning.
  • Newly accepted: L. Hillairet and J. Marzuola: Eigenvalue spacing for 1d Schrödinger operators, Asymptotical Analysis, to appear.
  • Upcoming invited talk: Analysis Seminar, Oklahoma University (February 13, 2023)
  • Upcoming invited talk: Seminar, Purdue University (February 20, 2023)

Jason Metcalfe, Professor

• • Newly accepted: J. Metcalfe and T. Rhoads (former UNC graduate student): Long-time existence for systems of quasilinear wave equations. Matematica, to appear.
  • Newly accepted: M. Facci (former UNC undergraduate student and current UNC graduate student), A. McEntarffer (former UNC undergraduate student), and J. Metcalfe: An r^p-weighted local energy approach to global existence for null form semilinear wave equations.  Involve, to appear.
  • Presentation: Analysis Seminar, University of New Mexico (virtual) (October 2022)

Casey Rodriguez, Assistant Professor

• • Newly accepted: K. R. Rajagopal and C. Rodriguez: On evolving natural curvature for an inextensible, unshearable, viscoelastic rod. Journal of Elasticity, to appear.
  • Newly accepted: K. R. Rajagopal and C. Rodriguez: On an elastic strain-limiting special Cosserat rod model. Mathematical Models and Methods in Applied Sciences, to appear.
  • Upcoming invited talk:  Special Session, Spring Central Sectional Meeting of the AMS, University of Cincinnati (April 15-16, 2023)

Michael Taylor, William R. Kenan Distinguished Professor

• • Newly submitted: M. Taylor: Gauss-Green formulas on domains with non-rectifiable boundaries
  • Newly posted: M. Taylor: Introduction to Lie Groups, Open Math Notes, AMS.
  • Newly posted: M. Taylor: Short course on pseudodifferential operators, Open Math Notes, AMS

Jian Wang, Postdoctoral Fellow

• • Newly posted: J. Galkowski, P. Marchand, J. Wang, and M. Zworski: The scattering phase: seen at last.

We are excited to welcome our newest faculty member, Arunima Bhattacharya.  Arunima earned her Ph.D. in 2019 from the University of Oregon where she worked with Micah Warren.  She subsequently held postdoctoral positions at the University of Washington and SLMath.  Her research focuses on geometric analysis and nonlinear PDE.  Her office will be PH304A.  Please join us in welcoming Arunima to the department and to the group.

The first year of the Developmental Training Group, which was designed and led by Yaiza Canzani, just wrapped up.  This led to eight applications by students for the NSF Graduate Research Fellowship Program.  This is more than double the number of applications that is typical.  The students were surveyed and the program received unanimous praise.  We are very proud of the work of Yaiza and the students and are looking forward to the results of the applications.

We are delighted to announce the return of the Brauer lectures within the Department of Mathematics at UNC.  The 2023 lectures will be given by Carlos Kenig, Louis Block Distinguished Service Professor in the Department of Mathematics at the University of Chicago.

Carlos earned his PhD at the University of Chicago in 1978 under the direction of Alberto Calderon.  After holding positions at Princeton and Minnesota, he returned to the University of Chicago in 1985.  Carlos’s recognitions include the Salem Prize (1984), the Bocher Prize (2008), three time ICM speaker (1986, 2002, 2010), elected member of the National Academy of Sciences (2014), and current President of the International Mathematical Union.  Carlos is an expert in harmonic analysis and partial differential equations, and he is an excellent expositor and lecturer for both specialized and general audiences.

The local chapter of the AWM (Association for Women in Mathematics) is holding a number of fascinating events.  We want to encourage all of our group to participate.  See https://math.unc.edu/events/awm-lecture/.  This evening’s event (Monday, October 24) is a panel discussion on jobs for mathematicians outside of academia.  There is an upcoming panel on REU opportunities.  Both of these are outstanding opportunities.

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.