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We congratulate PI Yaiza Canzani and her co-author Jeff Galkowski on the publication of their new book: Geodesic beams in eigenfunction analysis.  Yaiza and Jeff have pioneered many uses of geodesic beams to better understand the behavior of eigenfunctions of the Laplacian.  It is exciting to see a recap of their achievements in text form.

The RTG is excited to welcome the Girls Talk Math program back to campus.  This summer camp was started by two alumni, including Katrina Morgan who is beginning a new faculty position at Temple University and has returned for this year’s camp.  Each year a number of high school students are hosted and dig into challenging problems.  See https://girlstalkmath.com for more info and to follow them on social media.

We congratulate Michael Taylor on his retirement after 36 years at UNC.  Michael is a 1970 PhD from Berkeley (with Cordes) and came to UNC in 1987.  Amongst his many accolades, he is an ICM speaker and an elected member of the American Academy of Arts & Sciences.  He has made many notable contributions with the fields of analysis and PDE.  He is the author of more than a dozen books including some absolute classics in microlocal analysis and an elegant and famous three volume set on Partial Differential Equations in addition to his classroom texts on analysis, linear algebra, complex analysis, ODE, and measure theory.

The members of the RTG are deeply grateful to Michael for his mentorship, generous sharing of ideas, and enthusiasm for all things in mathematics.

Congratulations to Dr. Derrick Nowak!  Derrick successfully defended his dissertation “Local smoothing of the Schrödinger equation on a multi-warped product with degenerate trapping on Thursday, May 11.  Congratulations also goes to Hans Christianson, who was Derrick’s thesis advisor.

The most recent Community Spotlight from MSRI / the Simons Lauer Mathematical Sciences Institute has featured RTG faculty member Arunima Bhattacharya.  You can read the article and learn more about Arunima here.  Congratulations to Arunima!

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

• • Presentation: Geometry and Topology Seminar, Duke University (March 20, 2023)
  • Presentation: Continuum Mechanics Seminar, University of Nebraska-Lincoln (March 30, 2023)
  • Upcoming invited talk: Geometric Analysis Seminar, University of Oregon (May 16, 2023)
  • Upcoming invited talk: Differential Geometry/PDE Seminar, University of Washington (May 17, 2023)
  • Upcoming invited talk: Birthday Conference for Mikhail Safonov, Seoul, South Korea (June 20, 2023)
  • Newly accepted: A. Bhattacharya, J. Chen, and M. Warren: Regularity of Hamiltonian stationary equations in symplectic manifolds, Adv. Math., to appear.
  • Newly accepted: A. Bhattacharya, The Dirichlet problem for the Lagrangian mean curvature equation, Analysis & PDE, to appear.

Benjamin Bechtold, Graduate Student

• • Successfully completed the Departmental Oral Exam (February 2023)

Madelyne Brown, Graduate Student

• • Newly accepted: M. Brown: On the growth of generalized Fourier coefficients of restricted eigenfunctions.  Comm. PDE, to appear.
  • Elected President of UNC’s Graduate Math Associate (GMA) for the 2023-24 academic year.

Yaiza Canzani, Associate Professor

• • Newly accepted: Y. Canzani and J. Toth:  Lower bounds for eigenfunction restrictions in lacunary regions.  Communications in Mathematical Physics, to appear.

Hans Christianson, Professor

• • Newly posted: H. Christianson, J. Wang, and R. P. T. Wang: Control estimates for 0th order pseudodifferential operators.
  • Presentation: Tsinghua-Fudan joint online seminar on microlocal analysis (April 28, 2023)

Casey Rodriguez, Assistant Professor

• • Newly posted: C. Rodriguez, Elastic solids with strain-gradient elastic boundary surfaces.
  • Newly posted: K. R. Rajagopal and C. Rodriguez, Special Cosserat rods with rate-dependent evolving natural configurations.
  • Upcoming invited talk: 13th AIMS Conference on Dynamical Systems, Differential Equations and Applications, UNC-Wilmington (May 31-June 4, 2023).

Jian Wang, Postdoctoral Fellow

• • Newly posted: H. Christianson, J. Wang, and R. P. T. Wang: Control estimates for 0th order pseudodifferential operators.
  • Presentation: Analysis and PDE Seminar, University of California at Berkeley (April 10, 2023)
  • Upcoming invited talk: Analysis Seminar, Northwestern University (May 15, 2023).

With graduation approaching, we want to congratulate our graduating class of undergraduate researchers.

• Ameer Qaquish has been working with Hans Christianson and will return to UNC in the biostatistics graduate program.
• David Snider studied with Mark Williams and recently defended his honors thesis “Nonlinear geometric optics for pulses with nonlinear phases”.  He will enrolled in the graduate program in statistics at Columbia University in the fall.
• Xiao-Ming Porter and Yizhou Gu have worked with Jason Metcalfe.  They presented their research at the 2023 UNC Celebration of Undergraduate Research.
  • Xiao-Ming will graduate with honors with a thesis titled “Local energy estimates for damped waves”.  Xiao-Ming will enroll in the graduate program in mathematics at the National University of Singapore.
  • Yizhou will enroll at the University of Michigan in statistics.  She was one of this year’s recipients of the Archibald Henderson Mathematics Medal and received recognition at the 2023 Chancellor’s Awards Ceremony.

We are extremely proud of these students, provide deep congratulations on their graduations, and wish them much luck in their new endeavors.

The members of the RTG’s first cohort of high school researchers are nearing the end of the senior year.  And we congratulate them on their enrollments at outstanding universities.

Liam Buttita has been studying with RTG faculty member Casey Rodriguez.  Liam worked on obtaining helical equilibrium states for a Cosserat rod model recently introduced by Rodriguez and Rajagopal.  These equilibrium states are obtained via solving for certain relative equilibria of a Hamiltonian system that expresses balances of forces and couples in the rod.  In the fall, Liam will attend the University of Illinois Urbana-Champaign.

Zubin Narayan has been working with RTG faculty member Jason Metcalfe and RTG graduate student Ben Bechtold.  They have examined global existence for wave equations satisfying a null condition with initial data that may be large in the energy space.  Zubin will be departing for Carnegie Mellon University come the fall.

We thank Liam and Zubin for all of their hard work and congratulate them!

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.