In double breakthrough, mathematician helps solve two long-standing problems with paper and pencil

Good news! What progress will we make in mathematics once AI is being utilized?

Caveat: I am almost clueless as to what these achievements in the queen of all sciences (Carl Friedrich Gauss) are all about!

"A Rutgers University-New Brunswick professor who has devoted his career to resolving the mysteries of higher mathematics has solved two separate, fundamental problems that have perplexed mathematicians for decades. ..."

"... completed a proof of the 1955 Height Zero Conjecture posed by Richard Brauer, a leading German-American mathematician who died in 1977. Proof of the conjecture – commonly viewed as one of the most outstanding challenges in a field of math known as the representation theory of finite groups ...

In a sense, Tiep and his colleagues have been following a blueprint of challenges Brauer laid out for them in a series of mathematical conjectures posed and published in the 1950-60s.

“Some mathematicians have this rare intellect,” Tiep said of Brauer. “It’s as though they came from another planet or from another world. They are capable of seeing hidden phenomena that others can’t.” ...

In the second advance, Tiep solved a difficult problem in what is known as the Deligne-Lusztig theory, part of the foundational machinery of representation theory. The breakthrough touches on traces, an important feature of a rectangular array known as a matrix. The trace of a matrix is the sum of its diagonal elements. The work is detailed in two papers ...

Both breakthroughs are advances in the field of representation theory of finite groups, a subset of algebra. Representation theory is an important tool in many areas of math, including number theory and algebraic geometry as well as in the physical sciences, including particle physics. Through mathematical objects known as groups, representation theory also has been used to study symmetry in molecules, encrypt messages and produce error-correcting codes.

Following the principles of representation theory, mathematicians take abstract shapes that exist in Euclidean geometry – some of them extremely complex – and transform them into arrays of numbers. This can be achieved by identifying certain points that exist in each three- or higher-dimensional shape and converting them to numbers placed in rows and columns. ..."

From the abstract (an unusually short one, but the proof is 110 pages long):

"We complete the proof of Brauer's Height Zero Conjecture from 1955 by establishing the open implication for all odd primes."

In double breakthrough, mathematician helps solve two long-standing problems

In Double Breakthrough, Mathematician Solves Two Long-Standing Problems (original news release) "Rutgers professor’s proofs will provide deeper understanding of math mysteries and may lead to advances in science and technology"

Brauer's Height Zero Conjecture (pre-print, open access, first published September 2022, 110 pages)

Character levels and character bounds for finite classical groups (open access)

Uniform Character Bounds for Finite Classical Groups (pre-print, open access)