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(This is a reading seminar for graduate students.)
Algebraic K-theory originated with the Grothendieck-Riemann-Roch theorem, a generalization of Riemann-Roch theorem to higher dimensional varieties. For this, we shall discuss the definitions of $K_0$-theory of a variety, its connection with intersection theory, $lambda$-operation, $gamma$-filtration, Chern classes and Adams operations.
I report on work with M. Gubinelli and T. Oh on the
renormalized nonlinear wave equation
in 2d with monomial nonlinearity and in 3d with quadratic nonlinearity.
Martin Hairer has developed an efficient machinery to handle elliptic
and parabolic problems with additive white noise, and many local
existence questions are by now well understood. In contrast not much is
known for hyperbolic equations. We study the simplest nontrivial
examples and prove local existence and weak universality, i.e. the
nonlinear wave equations with additive white noise occur as scaling
limits of wave equations with more regular noise.
A result due to Gyárfás, Hubenko, and Solymosi, answering a question of Erdős, asserts that if a graph does not contain as an induced subgraph yet has at least edges, then has a complete subgraph on at least vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced , which allows us to extend their result to -uniform hypergraphs. Our result also has interesting consequences in topological combinatorics and abstract convexity, where it can be used to answer questions by Bukh, Kalai, and several others.