Analysis Seminars

Analysis Seminars for this Semester

Tuesday 3rd October, 2017 (place: UCD Science North 125 ) 4.00pm

Speaker: R. Smith
Title: The continuity of betweenness

Abstract Given a set $X$, we can use a suitable ternary relation $[\cdot,\cdot,\cdot] \subseteq X^3$ to express the notion of `betweenness' on $X$: $x$ is between $a$ and $b$ if and only if $[a,x,b]$ holds. We assume that this relation is "basic": $[a,a,b]$ and $[a,b,b]$ always hold, $[a,x,b]$ implies $[b,x,a]$, and $[a,x,a]$ implies $x=a$. Many natural examples of betweenness arise when $X$ is endowed with some additional order-theoretic or topological structure. Given $a,b \in X$, we can define the "interval" $[a,b] = \lbrace x \in X\,:\,[a,x,b]\rbrace\;(= [b,a])$. If $X$ has additional topological structure, it is reasonable to ask whether the assignment $\lbrace a,b\rbrace \mapsto [a,b]$ has good continuity properties, given a suitable hyperspace topology. We examine this question in the context of "Menger betweenness" on metric spaces $(X,d)$ ($[a,x,b]$ holds if and only if $d(a,b)=d(a,x)+d(x,b)$), and the "K-interpretation of betweenness" on topological continua ($[a,x,b]$ holds if and only if $x$ is an element of every subcontinuum that includes $a$ and $b$). This is joint work with Paul Bankston (Marquette University, WI) and Aisling McCluskey (NUI Galway).

Tuesday 10th October (place: UCD Science North 125 ) 3:00pm

Speaker: R. Levene
Title: Non-commutative graph parameters and quantum channel capacities

Abstract We generalise some graph parameters to non-commutative graphs (a.k.a. operator systems of matrices) and quantum channels. In particular, we introduce the quantum complexity of a non-commutative graph, generalising the minimum semidefinite rank. These parameters give upper bounds on the Shannon zero-error capacity of a quantum channel which can beat the best general upper bound in the literature, namely the quantum Lovász theta number. This is joint work with Vern Paulsen (Waterloo) and Ivan Todorov (Belfast).

Tuesday 10th October (place: UCD Science North 125 ) 4:15pm

Speaker: M. Whittaker (Glasgow)
Title: Fractal substitution tilings and applications to noncommutative geometry

Abstract Starting with a substitution tiling, such as the Penrose tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles typically have fractal boundary. As an application of fractal tilings, we construct an odd spectral triple on a C*-algebra associated with an aperiodic substitution tiling. Even though spectral triples on substitution tilings have been extremely well studied in the last 25 years, our construction produces the first truly noncommutative spectral triple associated with a tiling. My work on fractal substitution tilings is joint with Natalie Frank and Sam Webster, and my work on spectral triples is joint with Michael Mampusti.

Tuesday 17th October, 2017 (place: UCD Science North 125 ) 4.00pm

Speaker: N. Dobbs
Title: Nearby Birkhoff averages

Abstract Birkhoff averages (of an observable along orbits) are objects of interest when investigating statistical behaviour of a dynamical system. If there is a unique physical measure, the Birkhoff averages will converge, for almost every orbit, to the space average (i.e. the integral) of the observable, so the physical measure captures important statistical properties of the dynamical system. However, in the quadratic family, for example, physical measures don't always exist, and even when they do, they don't necessarily depend continuously on the parameter. In joint work with Alexey Korepanov, we examine what happens for finite time Birkhoff averages for nearby parameters.

Tuesday 24th October, 2017 (place: UCD Science North 125 ) 4.00pm

Speaker: R. Timoney
Title: TROs and Morita equivalence

Abstract It is possible to recast the theory of Morita equivalence in terms of the elementary theory of Ternary Rings of Operators (TROs). In particular the Morita correspondence between primitive ideals follows by extending irreducible representations from the right C*-algebra to the linking C*-algebra. The celebrated Brown-Green-Reiffel theorem characterising Morita equivalence as stable isomorphism in the separable case follows by using a Lemma of Brown to show that separable stable TROs are TRO isomorphic to C*-algebras.

Tuesday 31st October (place: UCD Science North 125 ) 3:00pm

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Tuesday 31st October (place: UCD Science North 125 ) 4:15pm

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Tuesday 7th November (place: UCD Science North 125 ) 3:00pm

Speaker: C. Boyd
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Tuesday 7th November (place: UCD Science North 125 ) 4:15pm

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Tuesday 14th November (place: UCD Science North 125 ) 3:00pm

Speaker: S. Gardiner
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Tuesday 14th November (place: UCD Science North 125 ) 4:15pm

Speaker: H. Render
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Tuesday 21st November (place: UCD Science North 125 ) 3:00pm

Speaker: B. Lemmens (Kent)
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Tuesday 21st November (place: UCD Science North 125 ) 4:15pm

Speaker: S. Gardiner
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Tuesday 28th November (place: UCD Science North 125 ) 3:00pm

Speaker: M. Ghergu
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Tuesday 28th November (place: UCD Science North 125 ) 4:15pm

Speaker: G. Singh
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Tuesday 5th December (place: UCD Science North 125 ) 3:00pm

Speaker: M. Golitsyna
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Tuesday 5th December (place: UCD Science North 125 ) 4:15pm

Speaker: M. Mackey
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C. Boyd , R. Timoney.

Trinity College Dublin

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