The Counting CSP (#CSP), where the goal is to find or approximate the number of solutions of a CSP instance has been thoroughly studied from multiple perspectives. In this talk we study Counting CSPs, in which the goal is to find the number of solutions modulo a prime number p, denoted #_pCSP. We first show that the standard algebraic approach works (somewhat surprisingly) for this kind of problems, and then classify the complexity of modular counting for #_pCSP(H), where H is a graph.
Fredholm modules were introduced by M. Atiyah as an abstraction of elliptic differential operators. G. Kasparov used Fredholm modules to construct K-homology, the dual to K-theory, and his KK-theory. A. Connes defined characters of Fredholm modules into cyclic homology. We will define a generalization of Fredholm modules and we will extend Connes' character theory to this setting.
As an application of our generalization, we will extend a result by J. Avron, R. Seiler, and B. Simon about the trace of (powers of) a difference of two unitarily equivalent projections.
Let K be a number field, f(x) \in K(x) be a rational function of degree at least 2, and a \in K. This thesis concerns arithmetic properties of sequences that are similar to a, f(a) , f^2(a), \ldots. We look into primitive prime divisors problems in sequences (\vphi_{n} \circ \cdots \circ \vphi_{1}(a) - b)_{n \geq 1}, where (\vphi_{i})_{i \geq 1} is a sequence of rational functions, sequences (a_{n}-b)_{n \geq 1}, where (a_{n})_{n \geq 1} lives in the backward orbit of dynamical systems, and sequences involving roots of unity. We also study the rings of integers and class numbers of number fields generated by sequences in backward orbits of dynamical systems.
Motivated by the generalized AGT conjecture in this talk I will construct surjective homomorphisms from the affine super Yangians to the universal enveloping algebras of rectangular W-superalgebras. This result is a super affine analogue of a result of Ragoucy and Sorba, which gave surjective homomorphisms from finite Yangians of type to rectangular finite W-algebras of type A.
