色色研究所

A BRIEF HISTORY OF TRIANGULATIONS

ABOUT THIS EVENT 

鈥淎 Brief History of Triangulations鈥�

A fundamental problem of mathematics 鈥� and of science in general 鈥� is to decompose a complex object into simpler pieces (e.g., integers into products of primes, or functions into Fourier waveforms). One of the oldest such problems is to cut a general polygon into triangles, the simplest type of polygon.

In this lecture, mathematician Christopher Bishop will survey about 200 years of progress on triangulations, starting with the Wallace-Bolyai-Gerwien theorem, which states that any two equal area polygons can be dissected into identical sets of triangles. The 3D analog of this 鈥� cutting equal volume polyhedra into identical sets of tetrahedra 鈥� is known as Hilbert's third problem and is somewhat more involved. Around 1960, it was proven that any polygon can be triangulated using only acute triangles (i.e., triangles whose angles are all less than 90 degrees). Such triangulations are important in applications to numerical analysis and computer graphics, and practical applications lead to consideration of computational complexity. For simple planar polygons, it was proven in the 1990s that the work needed is proportional to the number of sides of the polygon. However, for general polygonal regions, a rigorous polynomial time bound was only proven about ten years ago, and the sharp polynomial power remains unknown. If time permits, Professor Bishop will also discuss even more recent developments involving optimal angle bounds, 3D analogs and connections to other parts of mathematics.

ABOUT THE SPEAKER

professor christopher bishop, phD
STONY BROOK UNIVERSITY

Christopher Bishop received his bachelor's degree in mathematics at Michigan State University and attended the University of Cambridge as a Churchill Scholar, where he did Part III of the Mathematical Tripos. He earned his PhD from the University of Chicago under the direction of Peter W. Jones, spending two years at Chicago and two years at Yale. As an NSF postdoctoral fellow, Bishop spent a year at MSRI in Berkeley and three years as a Hedrick Instructor at UCLA before moving to Stony Brook University as an associate professor. Since then, he has been a Sloan Fellow, a Simons Fellow, a Fellow of the American Mathematical Society and an invited speaker to the 2018 International Congress of Mathematicians. In 2020, Bishop became a SUNY Distinguished Professor and, in 2024, was awarded the Senior Berwick Prize by the London Mathematical Society.

Bishop's work centers around complex function theory, exploring its connections to many diverse areas such as Brownian motion, geometric measure theory, Riemann surfaces, hyperbolic 3-manifolds, minimal surfaces, numerical analysis, optimal meshing algorithms, rational approximation and holomorphic dynamics. Among his noteworthy results are (with P.W. Jones) the resolution of a problem of Thurston on the fractal dimension of Kleinian limit sets, a linear time algorithm for computing conformal mappings onto polygons, the first polynomial time algorithm for triangulating general polygonal regions with acute triangles, the solution of several long standing problems in transcendental dynamics, the extension (with A. Eremenko and K. Lazebnik) of Hilbert's polynomial lemniscate theorem to rational leminscates, the proof (with L. Rempe) that every Riemann surface is a holomorphic cover of the sphere with finitely many branch points, and several geometric descriptions of Weil-Petersson curves, a class of closed planar loops arising from string theory. With Y. Peres, he is co-author of the book "Fractals in Probability and Analysis." However, he is most proud of having married far out of his league and of having both his children graduate college as valedictorians. 

PARKING

Guest parking is available in Lots 21 and 37.   
 
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DIRECTIONS

 
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