The "Goodwillie" Calculus of Functors allows us to decompose any homotopy functor (eg. the identity functor on topological spaces) into a "Taylor Tower" of degree n functors. These are analoguous to the degree n Taylor approximations of a continuous function on the Reals p_n f(x) = f(0) + f'(0) x + ... + f^n (0)/n! x^n. However, in functor calculus our coefficients are spectra (or equivalently (co)homology theories) rather than real numbers. We will give an overview of Goodwillie Calculus focusing on the equivalence between linear functors, spectra and (co)homology theories.
We explain consequences of recent work of Frank Quinn for computations of Nil groups in algebraic K-theory, in particular the Nil groups occurring in the K-theory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group.
Let G be a compact Lie group. Then the complexification of G has the structure of a reductive algebraic group over the field C of complex numbers. This algebraic group is canonically defined over the ring of integers Z. In this talk, I will discuss the problem of defining this group "over the sphere spectrum" (based on joint work with Dennis Gaitsgory).
Newton introduced divided differences to define interpolation polynomials fitting data points. More recently, Demazure and Bernstein-Gelfand-Gelfand have used divided difference operators when studying the cohomology of flag varieties. These operators may be defined more generally on Borel's equivariant cohomology. I shall begin with a brief introduction to equivariant cohomology, discuss the construction of the divided difference operators, and show how they can be used computationally. This talk is based on joint work with Reyer Sjamaar.
The v_1-periodic stable homotopy groups of spheres at an odd prime are generated by the alpha family, and the orders of these groups admit a global description in terms of denominators of Bernoulli numbers. I will describe a similarly global description of the v_2-periodic beta family in terms of explicit congruences of modular forms, for primes greater than 3.
University of Chicago Algebraic Topology Seminar
University of Chicago Algebraic Topology Seminar