Brauer Lectures

 

The UNC Alfred T. Brauer Lectures

Alfred Theodor Brauer, 1894–1985, had a profound impact on the Mathematics Department at UNC. Born in Germany, he studied at the University of Berlin and earned his degree in 1928 under Issai Schur. He continued lecturing there until 1936 when he was forced to resign by the Nazi government due to his Jewish heritage. In 1937 he declined Hermann Weyl’s invitation to come to the United States and, instead, gave the names of other he felt were in more immediate need of help. From 1936-1938, Brauer continued research while teaching junior-high level mathematics at a private school in Berlin until “Kristallnacht.”  He successfully fled the country in 1939, finally accepting Weyl’s invitation to the Institute for Advanced Study in Princeton. He came to North Carolina in 1942, teaching here until his retirement in 1966.

During this time at UNC, he founded the Mathematics and Physics Library, using his knowledge and expertise to establish a superb collection. In appreciation for this effort the library was named for him in 1976, later being absorbed into the UNC Libraries system. Alfred Brauer was honored by the University with the award of a Kenan professorship in 1959, the Tanner Award for excellence in undergraduate teaching in 1965, and an honorary doctor of legal letters degree in 1972. He has also received honors from outside the University, including the Oak Ridge Science Award and the G.W.F. Hegel Medal from the University of Berlin.

In 1975 an Alfred T. Brauer Instructorship was created at Wake Forest University, where he taught after his retirement from the University of North Carolina. The Alfred Brauer Fund was established by the Department of Mathematics in 1984 on the occasion of his ninetieth birthday. For more on Brauer’s biography, see the article “Alfred T. Brauer: Teacher, Mathematician and Developer of Libraries”.

 

2024 Brauer Lecturer

The 2024 Brauer Lectures will be given by Peter J. Olver, Professor in the Department of Mathematics at the University of Minnesota.

Peter earned his Ph.D. from Harvard University in 1976 under the mentorship of Prof. Garrett Birkhoff. Following postdoctoral roles at the Universities of Chicago and Oxford, he has been a faculty member at the University of Minnesota’s School of Mathematics since 1980. He held the position of Department Head from 2008 to 2020. Peter’s research focuses on the application of symmetry and Lie groups to differential equations. Throughout his career, he has made significant contributions to diverse fields such as partial differential equations, calculus of variations, mathematical physics, fluid mechanics, elasticity, quantum mechanics, Hamiltonian mechanics, geometric numerical methods, differential geometry, classical invariant theory, algebra, computer vision, image processing, anthropology, and beyond. In recognition of his work, Peter was elected a fellow of the Institute of Physics in 2004, the American Mathematical Society in 2013, and the Society for Industrial and Applied Mathematics in 2014.

The lectures will take place February 27-29, 2024.

• Lecture 1: Fractalization, quantization, and revivals in dispersive systems
  • The Talbot effect describes the remarkable evolution, through spatially periodic linear dispersion, of rough initial data, producing fractal, non-differentiable profiles at irrational times and, for asymptotically polynomial dispersion relations, quantized structures and revivals at rational times. Such phenomena have been observed in dispersive waves, optics, and quantum mechanics, and have intriguing connections with number theoretic exponential sums. I will present recent results on the analysis and numerics for linear and nonlinear dispersive wave models, both integrable and non-integrable, as well as integro-differential equations modeling interface dynamics and Fermi-Pasta-Ulam-Tsingou systems of coupled nonlinear oscillators.
• Lecture 2: Analysis and reassembly of broken objects
  • The problem of analyzing and reassembling broken objects appears in a broad range of applications, including jigsaw puzzle assembly, archaeology (broken pots and statues), surgery (broken bones and reassembly of histological sections), paleontology (broken fossils and egg shells), and anthropology (broken bones and lithics). I will discuss recent progress on such problems based on new and classical mathematical tools, including differential geometry, invariant signatures, and machine learning.
• Lecture 3: Two new developments concerning Noether’s Two Theorems
  • The first part impacts Noether’s Second Theorem, which concerns variational problems admitting an infinite-dimensional symmetry group. I first recall the two well-known classes of partial differential equations that admit infinite hierarchies of higher order generalized symmetries: 1) linear and linearizable systems that admit a nontrivial point symmetry group; 2) integrable nonlinear equations such as Korteweg–de Vries, nonlinear Schrödinger, and Burgers’. I will then introduce a new general class: 3) underdetermined systems of partial differential equations that admit an infinite-dimensional symmetry algebra depending on one or more arbitrary functions of the independent variables. An important subclass of the latter are the underdetermined Euler–Lagrange equations arising from a variational principle that admits an infinite-dimensional variational symmetry algebra depending on one or more arbitrary functions of the independent variables. According to Noether’s Second Theorem, the associated Euler–Lagrange equations satisfy Noether dependencies and are hence underdetermined; examples include general relativity, electromagnetism, and parameter-independent variational principles.The second part concerns Noether’s First Theorem, which, as originally formulated, relates strictly invariant variational problems and conservation laws of their Euler–Lagrange equations. The Noether correspondence was extended by her student Bessel-Hagen to divergence invariant variational problems. A key issue is when is a divergence invariant variational problem equivalent to a strictly invariant one. Here, I highlight the role of Lie algebra cohomology in resolving this question. The talk concludes with some provocative remarks on the role of invariant variational problems in the modern formulation of fundamental physics.