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Paul Pollack, University of Georgia

How nonunique is your factorization?

Number theorists learn early on not to take unique factorization for granted. In 1980, R.J. Valenza introduced the "elasticity” of an integral domain as a means of measuring how far away the domain is from possessing unique factorization. I will survey what is known about elasticity of rings of number theoretic interest. Particular attention will be paid to recent work of the speaker on elasticity of orders in quadratic fields. Much of this is joint with Steve Fan (UGA) and Enrique Trevino (Lake Forest).

MC 5417

Jeremy Champagne, University of Waterloo 

On the generalisation of a theorem of Watson

This talk is a continuation of the one I gave in March. In essence, we are discussing the set of real valued functions f(n) such that gcd(n,[f(n)])=1 happens with probability 1/zeta(2) (in the sense of natural densities), and related problems. I will give a general gameplan to establishing such results, and I will prove that gcd(n, [alpha_1n], [alpha_2n^2],...,[alpha_kn^k])=1 happens with probability 1/zeta(k+1) foralpha_1,...,alpha_k irrational.

MC 5403

Larissa Kroell, University of Waterloo

Partial C*-Dynamical Systems: Injective Envelopes and the Ideal Intersection Property

Partial C*-dynamical systems are a generalization of ordinary C*-dynamical systems and were first introduced by Ruy Exel (1994) to express certain C*-algebras as crossed products by a single partial automorphism. In this talk, we will show the existence of an equivariant injective envelope for partial C*-dynamical systems — a concept which has been a driving force in many recent results regarding the ideal structure of ordinary C*-dynamical systems. We will motivate our construction by highlighting its connection to ordinary C*-dynamical systems via enveloping actions. Furthermore, we provide a characterization of the ideal intersection property for partial C*-dynamical systems as an application of equivariant injective envelopes in this setting. This is based on joint work with Matthew Kennedy and Camila Sehnem.

This seminar is held jointly with the Canadian Operator Symposium.

M3 1006

Dashen Yan, Stony Brook University

Non-degenerate Z_2 harmonic 1-forms with shrinking branching sets

In this talk, we will explain the technical aspects of the gluing construction presented in Thursday’s talk. Specifically, we adapt Donaldson’s framework for deforming multivalued harmonic functions to our gluing setting and establish a weighted version of the Hamilton–Nash–Moser–Zehnder implicit function theorem to prove the gluing result.

MC 5417

Dashen Yan, Stony Brook University

Non-degenerate Z_2 harmonic 1-forms on R^n and their geometric applications

The Z_2 harmonic 1-form arises in various compactification problems in gauge theory, including those involving PSL(2,C) connections and Fueter sections. In this talk, we will describe a recent construction of non-degenerate Z_2 harmonic 1-forms on R^n for n >(=) 3 , and explore their relation to Lawlor’s necks—a family of special Lagrangian submanifolds in C^n.

We will also discuss a gluing construction in which these examples are glued to a regular zero of a harmonic 1-form on a compact manifold. This yields a sequence of non-degenerate Z_2 harmonic 1-forms whose branching sets shrink to points. As a result, we obtain many new examples of non-degenerate Z_2 harmonic 1-forms on compact manifolds.

STC 0010

Jashan Bal, University of Waterloo

More on Ramsey degrees

We continue discussing various dynamical reformulations of having finite Ramsey degree.

MC 5417

Kain Dineen, University of Waterloo

Linear maps preserving powers of the symplectic form

Let Ω denote the standard symplectic form on ℝ^{2m} For k = 1,..., m, we will describe the subgroup of GL(2m, ℝ) which fixes Ω^k.

STC 0010

Cynthia Dai, University of Waterloo

Categories as Complex Numbers

Let T be the set of all binary trees. It is well-known that T = 1+T^$. This implies that T^2-T=-1, and solving for T over the complex numbers, we can conclude it must be a 6th primitive root of unity. Hence we have an isomorphism T = T^7. Come to this talk and learn why this nonsense works.

MC 5403

Andy Zucker, University of Waterloo

On Ramsey Degrees

We discuss some dynamical reformulations of the notion of Ramsey degree. This meeting also serves as an organizational meeting to plan the rest of the learning seminar.

MC 5417

AJ Fong, University of Waterloo

The Markov equation and birational geometry

We will briefly talk about the basics on the Markov equation and its solutions, and producing Hirzebruch--Jung continued fractions from their weights. We will also describe some connections to certain degenerations of the complex projective plane. This talk is based on work of Urzúa and Zúñiga.

MC 5417