# 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
LovĂˇsz 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

*Speaker:*

*Title:*

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

*Speaker:*

*Title:*

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

*Speaker:* C. Boyd

*Title:*

**Tuesday 7th November **
(place: UCD
Science North 125 )
4:15pm

*Speaker:*

*Title:*

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

*Speaker:* S. Gardiner

*Title:*

**Tuesday 14th November **
(place: UCD
Science North 125 )
4:15pm

*Speaker:* H. Render

*Title:*

**Tuesday 21st November **
(place: UCD
Science North 125 )
3:00pm

*Speaker:* B. Lemmens (Kent)

*Title:*

**Tuesday 21st November **
(place: UCD
Science North 125 )
4:15pm

*Speaker:* S. Gardiner

*Title:*

**Tuesday 28th November **
(place: UCD
Science North 125 )
3:00pm

*Speaker:* M. Ghergu

*Title:*

**Tuesday 28th November **
(place: UCD
Science North 125 )
4:15pm

*Speaker:* G. Singh

*Title:*

**Tuesday 5th December **
(place: UCD
Science North 125 )
3:00pm

*Speaker:* M. Golitsyna

*Title:*

**Tuesday 5th December **
(place: UCD
Science North 125 )
4:15pm

*Speaker:* M. Mackey

*Title:*

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