Charles Waldo Rezk (born 26 January 1969) is an American mathematician, specializing in algebraic topology, category theory, and spectral algebraic geometry.
Rezk matriculated at the University of Pennsylvania in 1987 and graduated there in 1991 with B.A. and M.A. in mathematics.In 1996 he received his PhD from MIT with thesis Spaces of Algebra Structures and Cohomology of Operads and advisor Michael J. Hopkins. At Northwestern University Rezk was a faculty member from 1996 to 2001. At the University of Illinois he was an assistant professor from 2001 to 2006 and an associate professor from 2006 to 2014 and is a full professor since 2014.
He was at the Institute for Advanced Study in the fall of 1999, the spring of 2000, and the spring of 2001.He held visiting positions at MIT in 2006 and at Berkeley's MSRI in 2014. Since 2015 he has been a member of the editorial board of Compositio Mathematica .
Rezk was an invited speaker at the International Congress of Mathematicians in Seoul in 2014.He was elected a Fellow of the American Mathematical Society in the class of 2015 (announced in late 2014).
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
Vladimir Alexandrovich Voevodsky was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002. He is also known for the proof of the Milnor conjecture and motivic Bloch–Kato conjectures and for the univalent foundations of mathematics and homotopy type theory.
Daniel Gray "Dan" Quillen was an American mathematician.
In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor.
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.
Jeffrey Henderson Smith is a former professor of mathematics at Purdue University in Lafayette, Indiana. He received his Ph.D. from the Massachusetts Institute of Technology in 1981, under the supervision of Daniel Kan, and was promoted to full professor at Purdue in 1999. His primary research interest is algebraic topology; his best-cited work consists of two papers in the Annals of Mathematics on "nilpotence and stable homotopy".
In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams (1958). Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.
Alexander A. Beilinson is the David and Mary Winton Green University Professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1999 Beilinson was awarded the Ostrowski Prize with Helmut Hofer. In 2017 he was elected to the National Academy of Sciences.
In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer n there is a topological space , and these spaces are equipped with certain maps between them, so that for any topological space X, one obtains an abelian group structure on the set of homotopy classes of continuous maps from X to . One feature that distinguishes tmf is the fact that its coefficient ring, (point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring.
Michael Jerome Hopkins is an American mathematician known for work in algebraic topology.
Steve Shnider is a retired professor of mathematics at Bar Ilan University. He received a PhD in Mathematics from Harvard University in 1972, under Shlomo Sternberg. His main interests are in the differential geometry of fiber bundles; algebraic methods in the theory of deformation of geometric structures; symplectic geometry; supersymmetry; operads; and Hopf algebras. He retired in 2014.
In mathematics, a highly structured ring spectrum or -ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an -ring is called an -ring. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory.
Mark Edward Mahowald was an American mathematician known for work in algebraic topology.
Douglas Conner Ravenel is an American mathematician known for work in algebraic topology.
In mathematics, the first Blakers–Massey theorem, named after Albert Blakers and William S. Massey, gave vanishing conditions for certain triad homotopy groups of spaces.
Charles Alexander Weibel is an American mathematician working on algebraic K-theory, algebraic geometry and homological algebra.
In topology, a discipline within mathematics, the Brown–Gitler spectrum is a spectrum whose cohomology is a certain cyclic module over the Steenrod algebra.
Kari Kaleva Vilonen is a Finnish mathematician, specializing in geometric representation theory. He is currently a professor at the University of Melbourne.
William Gerard Dwyer is an American mathematician specializing in algebraic topology and group theory. For many years he was a professor at the University of Notre Dame, where he is the William J. Hank Family Professor Emeritus.
Jean E. Lannes is a French mathematician, specializing in algebraic topology and homotopy theory.