Max Reinhold Jahnke

Research Project and Collaborations

Research project

My research project aims to advance the fields of Cauchy-Riemann (CR) geometry, linear partial differential equations (PDEs), and analysis on homogeneous manifolds through three complementary directions, connected by a theme: the relationship between the regularity of PDEs and the geometry of the ambient space.

Invariant involutive structures on compact Lie groups and nilmanifolds

In the context of Lie groups and nilmanifolds, my guiding question is:

Let M be a compact homogeneous manifold for a Lie group G, endowed with a G-invariant involutive structure. We consider the differential complex associated with this involutive structure and its corresponding cohomology spaces. Under what conditions can these cohomologies be computed using only Lie algebraic methods?

The question above is very general. Nonetheless, I have obtained results in the case in which the manifold is a compact Lie group and the involutive structure is CR or elliptic.

Regularity of sum-of-squares operators

In the context of PDEs, my guiding question is:

Let M be a compact real-analytic manifold and P a linear partial differential operator with real-analytic coefficients. Under which conditions on M and P can we guarantee that P is globally analytic-hypoelliptic? In other words, if u is a smooth or distributional solution of Pu = f with f real-analytic on M, then u must be real-analytic.

Of course, answering the question in this level of generality is beyond the reach of the current techniques in PDEs. This question has attracted considerable interest, making it a very active area of research. I do have published papers in this direction and the references within my papers provide a general overview of the field.

Solvability for partial differential equations in the context of CR geometry

I have a growing interest in CR geometry because it combines different techniques ranging from geometry, PDEs, summability of divergent series, and many other exciting techniques. I have papers in the field, for example:

Co-authors

  • Antonio Victor da Silva, Hamad bin Khalifa University – Qatar
  • Gabriel Araújo, University of São Paulo – Brazil
  • Gregorio Chinni, University of Bologna – Italy
  • Gustavo Hoepfner, Federal University of São Carlos – Brazil
  • Howard Jacobowitz, Rutgers University – United States
  • Igor A. Ferra, Federal University of São Carlos – Brazil
  • Konstantin Wehler, Philipps-University Marburg – Germany
  • Luis Fernando Ragognette, Federal University of Minas Gerais – Brazil
  • Nicholas Braun Rodrigues, University of São Paulo – Brazil
  • Paulo Domingos Cordaro, University of São Paulo – Brazil
  • Vinícius Novelli, University of Vienna – Austria