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6th Annual LAA Lecturer

Professor William Fulton of the Department of Mathematics of the University of Michigan gave the Sixth Annual LAA Lecture on April 27, 2001. The title of his lecture was ``Eigenvalues of sums of Hermitian Matrices.'' Professor Fulton discussed the history of the problem of characterizing the possible eigenvalues of two Hermitian matrices and their sum, and the recent solution by A. A. Klyachko with A. Knutson and T. Tao. He also discussed several related problems in algebra, geometry, and representation theory. Dr.  Fulton's article ``Eigenvalues, Invariant Factors, Highest Weights, and Schubert Calculus'' appeared in the Bulletin of the American Mathematical Society 37 (2000), 209-249.
        William Fulton

6th Wolfgang Wasow Memorial Lecturer

Peter Lax
On November 7, 2001 Professor Peter D. Lax of the Courant Institute of Mathematical Sciences (NYU) gave the Sixth Wolfgang Wasow Memorial Lecture on ``Multiple Eigenvalues.'' (This lecture was originally scheduled to be given in September but was postponed due to the tragic events that occurred on September 11, 2001.) This lecture was concerned with the discriminant of a real symmetric matrix of order n (i.e. of its eigenvalues). The discriminant is a homogeneous nonnegative polynomial of degree n(n-1) in the entries of the matrix. As first shown by Hilbert, nonnegative polynomials of degree greater than 2 and in more than two variables, cannot always be written as a sum of squares of polynomials. In the lecture it was demonstrated that the discriminant of real symmetric matrices can always be so written. In fact, N. Ilyusheckin gave an explicit representation as a sum of n! squares. Introducing Professor Lax, Shi Jin said: ``Dr. Lax is one of the greatest applied and theoretical mathematicians of the 20th century who has made significant contributions to mathematics and scientific computing, including partial differential equations, integral systems, computational fluid dynamics, and hyperbolic conservation laws. His name is connected with many major mathematical results and numerical methods, such as the Lax-Milgram Lemma, the Lax Equivalence Theorem, the Lax-Wendruff Scheme, the Lax-Friedrichs Scheme, the Lax Entropy Condition, the Lax Pair, the Lax-Levermore Theory, etc. His scientific contributions have been recognized by many awards: the National Medal of Science, the Wolf Prize, the Wiener Prize, election to the National Academy of Science, .... His expository skills have led to the Lester R. Ford Award and the Chauvenet Prize.''

Professor Lax is also a distinguished educator who has mentored numerous graduate students and postdocs. He gave a lecture in our 9th floor conference room on ``The Paradox of Education.''

Distinguished Lecturer

Cumrun Vafa
Dr. Cumrun Vafa, Professor of Physics of Harvard University, was our Distinguished Lecturer in the spring of 2001. He visited UW-Madison during the week of May 7, 2001 and gave two lectures. One was a joint Mathematics and Physics Colloquium on ``Geometric Physics.'' The other was a Graduate Student String Theory Seminar given in our 9th floor conference room. In the seminar he discussed the relationships between the Homfly polynomial of a knot and Chern-Simon gauge theory.

Dr. Vafa's research interests lie in theoretical high energy physics with emphasis on such fundamental questions as the nature of quantum gravity and the relation between geometry and quantum field theories. String theory is a strong candidate for a unified description of gravity and other forces. With A. Strominger he discovered that the Bekenstein-Hawking entropy of a black hole can be accounted for by solitonic states of string theory. One of Professor Vafa's current interests is application of string theory to some unsolved questions of particle physics such as the hierarchy problem and the cosmological constant problem.

The Special Lectures Committee has chosen the 2001-02 LAA and Wasow Memorial Lecturers. They are Thomas Kailath of Stanford University, who will give his LAA Lecture in April of this year, and John Erik Fornaess of the University of Michigan, who will give his Wasow Memorial Lecture in October.

Distinguished Lecturers for 2001-02 have also been announced. George Papanicolaou, Stanford University, will be Distinguished Lecturer visiting the department on April 24-26, 2002. Armand Borel, Institute for Advanced Study, will also be a Distinguished Lecturer, visiting the department on May 7-8, 2002.