(See links above for my publication and presentation lists, as well as information about my conference activities.)
My research is currently supported in part by a CAREER grant from the National Science Foundation: DMS-2439897.
My primary research is in finite group theory and the representation theory of finite groups. Group theory can be viewed as the study of symmetries of an object, such as those coming from nature, art, communication networks, or any other place that symmetry might play a role. Representation theory is a tool used to better understand the structure of a group and the symmetries it represents by giving us a way to view, in some sense, an abstract group as a group of matrices whose structure is often easier to understand. In particular, the irreducible representations form the "building blocks" of all representations. (Click here for a short introduction to the area by my colleagues Eugenio Giannelli and Jay Taylor for non-experts. Here is an article written for non-experts by myself and collaborator Noelia Rizo for the Oberwolfach Snapshot series, introducing the reader to some of the main types of problems I work in, through the work of Brauer and the legacy he left. Here is also an article by Gabriel Navarro about the area and remembering Marty Isaacs, one of the leaders of the field, who passed away in Feb 2025.)
Most of my research involves the irreducible representations of finite groups of Lie type, sometimes called finite reductive groups. These groups are analogues of Lie groups over finite fields and form the largest collection of finite simple groups. I have been particularly interested in problems concerning local-global conjectures in character theory and problems involving the action of Galois automorphisms of characters.
Click on one of the links at the top for more information!
Group Theory In the Media!?
Here is a Quanta Magazine article about the McKay Conjecture, recently completed by Marc Cabanes and Britta Späth.
Here and Here are articles about Tiep and some of his recent work, including our (Malle, Navarro, myself, and Tiep) proof of Brauer's Height Zero Conjecture.
Here is an article through University of Denver about me, also featuring the proof of BHZ!
(#31) (with G. Malle, G. Navarro, and P.H. Tiep) Brauer's Height Zero Conjecture. Annals of Mathematics 200:2 (2024), 557-608.
Final proof of Brauer's Height Zero Conjecture from 1955
(#38) (with L. Ruhstorfer) The McKay--Navarro conjecture for the prime 2. Advances in Mathematics. 477 (2025), 110369.
Proof of the McKay--Navarro conjecture for p=2; See also (#23--25) for key lead-up results.
(#35) (with A. Moretó and N. Rizo) Brauer's problem 21 for principal blocks. Transactions of the American Mathematical Society. Ser. B Vol. 12 (2025), 38-64.
Proof of Brauers Problem 21 (from 1963) for principal blocks.
(#19) (with G. Navarro, N. Rizo, and C. Vallejo) Characters and generation of Sylow 2-subgroups. AMS Representation Theory. Vol. 25 (2021), 142–165.
Proof of consequence of the Alperin--McKay--Navarro conjecture on 2-generation of Sylow 2-subgroups; See also (#16, 36) for key lead-up and related results.
(#29) (with N. N. Hung) On Héthelyi--Külshammer's conjecture for principal blocks. Algebra & Number Theory, 17:6 (2023), 1127–1151. https://doi.org/10.2140/ant.2023.17.1127.
Proof of Hethelyi--Kulshammer's conjecture for principal blocks; See also (#28, 30, 32) for extensions
(#10) Action of Galois Automorphisms on Harish-Chandra Series and Navarro's Self-Normalizing Sylow 2-Subgroup Conjecture. Trans. Amer. Math. Soc. 372 no. 1 (2019), pp. 457-483.
Final proof of Navarro's conjecture on self-normalizing Sylow subgroups; See also (#6,8,9) for key lead-up results and extensions.
In keeping what I've learned is a common tradition, below are links to my collaborators' websites:
Gabriel Navarro