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Brauer Lectures

 

The UNC Alfred T. Brauer Lectures

Alfred T. Brauer

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”.

 

2025 Brauer Lecturer

The 2025 Brauer Lectures will be given by Ingrid Daubechies, James B. Distinguished Professor Emeritus of Mathematics and Electrical and Computer Engineering at Duke University.

Ingrid Daubechies majored in physics at the Free University of Brussels where she went on to earn her doctorate in theoretical physics. She joined the Mathematics Research Center at AT&T Bell Laboratories in 1987 and, that same year, published the paper that launched her global reputation in the field of wavelets. These mathematical tools allow efficient information packaging, especially when signals undergo sudden dramatic changes.

Undaunted by the entrenched belief that putting mathematics to work compromises its underlying beauty, Dr. Daubechies aimed to create practical tools that retained the wavelets’ theoretical integrity. The wavelets that she developed transformed signal processing in many digital settings. In images, for instance, they facilitate the ability to minimize bandwidth while maximizing quality. This work underlies many applications, including the JPEG2000 image compression and coding system. Dr. Daubechies has exploited wavelets and numerous other mathematical tools to solve problems in a tremendous range of fields, including brain imaging, geology, evolution, and even art restoration.

Dr. Daubechies joined the mathematics faculty at Rutgers University in 1991 and, three years later, Princeton University recruited her. In 2011, she moved to Duke University, where she is currently the James B. Duke Distinguished professor of Mathematics and Electrical and Computer Engineering.

Dr. Daubechies was named a MacArthur Fellow, and her many honors include the National Academy of Sciences Award in Mathematics, the L’Oréal-UNESCO International Award for Women in Science, the 2023 Wolf Prize in Mathematics, and the 2025 National Medal of Science. She has held numerous leadership positions, including president of the International Mathematical Union. In this role, she promoted equal opportunities in science and math education, especially in developing countries. She also co-founded Duke University’s Summer Workshop in Mathematics for rising female high school seniors, and she serves on the board of directors of the EDGE Foundation, which aims to increase gender and racial diversity and equity in the mathematics community.

Away from academia, Dr. Daubechies teamed up with fiber artist Dominique Ehrmann to create an exhibit called Mathemalchemy. This traveling museum piece aims to convey the beauty and fun of mathematics by celebrating the relationship between math and art.

The lectures will take place February 3-5, 2025.

  • Lecture 1: Mathematicians helping art historians and art conservators
    • In recent years, mathematical algorithms have helped art historians and art conservators putting together the thousands of fragments into which an unfortunate WWII bombing destroyed world famous frescos by Mantegna, decide that certain paintings by masters were “roll mates” (their canvases were cut from the same bolt), virtually remove artifacts in preparation for a restoration campaign, get more insight into paintings hidden underneath a visible one. The presentation reviews these applications, and gives a glimpse into the mathematical aspects that make this possible.
  • Lecture 2: Discovering low-dimensional manifolds in high-dimensional data
    • This talk reviews diffusion methods to identify low-dimensional manifolds underlying high-dimensional datasets, and illustrates that by pinpointing additional mathematical structure, improved results can be obtained. Much of the talk draws on a case study from a collaboration with biological morphologists, who compare different phenotypical structures to study relationships of living or extinct animals with their surroundings and each other. This is typically done from carefully defined anatomical correspondence points (landmarks) on e.g. bones; such landmarking draws on highly specialized knowledge. To make possible more extensive use of large (and growing) databases, algorithms are required for automatic morphological correspondence maps, without any preliminary marking of special features or landmarks by the user.
  • Lecture 3: The wavelet synthesis and beyond 
    • Wavelets emerged in the 1970s and 80s as the synthesis of ideas stemming from many different directions; the list of ingredients includes items as diverse as the study of singular integral kernel operators in harmonic analysis, square integrable group representations in quantum mechanics, the role of scaling in computer vision, efficient algorithms in computer graphics, the power on nonlinear expansion in approximation theory, and deep theoretical insights from statistics. All these fields not only contributed to the wavelet synthesis — they also all benefitted from it. With hindsight, wavelet expansions were a first example of sparse approximation — a forerunner for what has become known as compressed or compressive sensing, a development that has had its own tremendous impact in many fields of application. Compressive sensing is possible when we know the signal of interest can be represented sparsely in an appropriate dictionary of “elementary building blocks”. There are some indications that deep neural networks, the success of which is at present very little understood, may similarly use smaller “elementary” building blocks in their inner workings. The presentation will give a high level overview of these different facets of modern signal learning and processing.

2024 Brauer Lecturer

The 2024 Brauer Lectures were 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 took 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.

2023 Brauer Lecturer

The 2023 Brauer Lectures were given by Carlos Kenig, Louis Block Distinguished Service Professor in the Department of Mathematics at the University of Chicago.

Carlos earned his PhD at the University of Chicago in 1978 under the direction of Alberto Calderon.  After holding positions at Princeton and Minnesota, he returned to the University of Chicago in 1985.  Carlos’s recognitions include the Salem Prize (1984), the Bocher Prize (2008), three time ICM speaker (1986, 2002, 2010), elected member of the National Academy of Sciences (2014), and former President of the International Mathematical Union.  Carlos is an expert in harmonic analysis and partial differential equations, and he is an excellent expositor and lecturer for both specialized and general audiences.

The lectures took place February 6-8, 2023.

  • Lecture 1: Asymptotic simplification for solutions of nonlinear wave equations
    • In this lecture I will describe the progress made in the last 14 years, in our understanding of the long-time behavior of large solutions to the energy critical focusing nonlinear wave equation. In the last part of the talk I will concentrate on progress (with Duyckaerts and Merle) on the asymptotic simplification for large time, into sums of modulated static solutions plus a linear term, in all odd dimensions, in the radial case.
  • Lecture 2: Wave maps into the sphere
    • In this talk we will introduce wave maps, an important geometric flow, and discuss, for the case when the target is the sphere, the asymptotic behavior near the ground state (without symmetry) and recent results in the general case (under co-rotational symmetry) in joint work with Duyckaerts, Martel and Merle.
  • Lecture 3: New channels of energy for wave equations, new non-radiative solutions and soliton resolution
    • We will discuss the role of non-radiative solutions to nonlinear wave equations, in connection with soliton resolution. Using new channels of energy estimates we characterize all radial non-radiative solutions for a general class of nonlinear wave equations. This is joint work with C.Collot, T. Duyckaerts and F. Merle.

Pre-Pandemic Brauer Lecturers

To honor the memory of Alfred Brauer and to recognize his many contributions to the Mathematics Department at UNC, the Alfred Brauer Lectures were begun in 1985.

  • The 2019 Brauer Lectures were given by Dr. Peter Constantin of Princeton University.
  • The 2018 Brauer Lectures were given by Dr. Mina Aganagic on April 10-12, 2018.
  • The 2017 Brauer Lectures were given by Mikhail Khovanov, on March 29-31, 2017.
  • The 2016 Brauer Lectures were given by Stan Osher, on April 20-22, 2016.
  • The 2015 Brauer Lectures were given by Michael Hopkins, on March 23-25, 2015.
  • The 2014 Brauer Lectures were given by Simon Donaldson, on March 24-26, 2014.
  • The 2013 Brauer Lectures were given by Vaughan Jones, University of California at Berkeley and Vanderbilt University, on March 4-6, 2013.
  • The 2012 Brauer Lectures were given by Alex Lubotzky, Hebrew University, on April 16-18, 2012.
  • The 2011 Brauer Lectures were given by Gerard Laumon of CNRS and Paris-Sud, Orsay, on March 28-30, 2011.
  • The 2010 Brauer Lectures were given by Alex Eskin of the University of Chicago.
  • Brauer Lectures prior to 2010