Jitendra Prakash, Department of Pure Mathematics, University of Waterloo
"Review of Reproducing Kernel Hilbert Spaces"
In this talk we shall review some of the basic results in the theory of reproducing kernel Hilbert spaces and their multiplier algebras.
MC 5403
Jitendra Prakash, Department of Pure Mathematics, University of Waterloo
"Nevanlinna-Pick Interpolation Problems"
Nevanlinna-Pick interpolation problem seeks a holomorphic function on the unit disk of norm at most one which interpolates two sets of points. This problem can be viewed as a question about the multiplier algebra of the Hardy space. We shall outline this operator-theoretic approach.
MC 5417
Jitendra Prakash, Department of Pure Mathematics, University of Waterloo
"Factorisation problems in RKHS"
We introduce the problem of factorisation of a positive operator P as P=UU* where U is upper triangular, and relate it to RKHS. We shall also discuss some generalisations of the classical Fejer-Riesz theorem.
MC 5403
Michael Deveau, Department of Pure Mathematics, University of Waterloo
"Games, Determinacy and Martin's Theorem"
We first define the notion of a game and then discuss the Axiom of Determinacy (AD), which is inconsistent with ZFC. We work instead in ZF + AD, and explore the consequences of AD as it relates to Turing degrees. In particular, we prove Martin's Theorem: assuming AD, every set of Turing degrees contains or omits a cone of degrees.
MC 5403
Janez Bernik, University of Ljubljana
"Quasi-filiform Lie algebras of maximum length revisited"
Ross Willard, Department of Pure Mathematics, University of Waterloo
"Uncountable SIs in residually small varieties in a countable signature, part III"
Continuing the proof of "the theorem" of McKenzie and Shelah, I will prove a technical theorem about the automorphism groups of maximal special subdirectly irreducible members in the relevant varieties. Then I will assemble the pieces to show that the types realized in a maximal special subdirectly irreducible algebra form a compact second countable topological space.
MC 5479
Mohammad Mahmoud, Department of Pure Mathematics, University of Waterloo
"Degree invariant functions and Martin's Conjectures"
We will go through a theorem which proves a special case of one of Martin's conjectures about degree (Turing) invariant functions. The theorem says: Assuming the Axiom of Determinacy, a uniformly degree invariant function that is not increasing on a cone must be constant on a cone. The theorem is about a uniformly degree invariant function while the conjecture is about any degree invariant function, not necessarily uniformly.
Ross Willard, Department of Pure Mathematics, University of Waterloo
"Uncountable SIs in residually small varieties in a countable signature, part IV"
Continuing the proof of "the theorem" of McKenzie and Shelah, I will define how the types realized over a countable set in a maximal special subdirectly irreducible algebra form a compact T1 second countable topological space.
MC 5479
Mohammad Mahmoud, Department of Pure Mathematics, University of Waterloo
"Degree invariant functions and Martin's Conjectures"
We will continue the proof we started for the special case of Martin's conjecture. That special case result says: Assuming the Axiom of Determinacy, a uniformly degree invariant function that is not increasing on a cone must be constant on a cone.
MC 5403
Jacob Campbell, Department of Pure Mathematics, University of Waterloo
We will study a class of compact quantum groups whose structure and representation theory depend on the combinatorics of set partitions, and we will also study some "noncommutative spaces" on which these quantum groups act. The emphasis of the presentations will be placed on the illustrative examples provided by the quantum permutation group and by the free orthogonal quantum group.