Tsinghua topology seminar: A symplectic approach to 3-manifold triangulations and hyperbolic structures

On 20 September 2022 I gave a zoom talk at the Yau Mathematical Sciences Center at Tsinghua University in Beijing. The talk was in the YMSC Topology Seminar.

Title: A symplectic approach to 3-manifold triangulations and hyperbolic structures

Abstract: In the 1980s, Neumann and Zagier introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold, such as a knot complement. We give an interpretation for this symplectic structure in terms of the topology of the 3-manifold, via intersections of certain curves on a Heegaard surface. We also give an algorithm to construct curves forming a symplectic basis for this vector space. This approach gives a description of hyperbolic structures on a knot complement via Ptolemy equations, which can be used to calculate the A-polynomial. This talk involves joint work with Jessica Purcell and Joshua Howie.

Video: The talk was recorded and a video of the talk is available through the YMSC website here.

Slides: from the talk are below.

22-09-20_Tsinghua_talk_slides

Monash topology talk on Symplectic approach to 3-manifold Triangulations, September 2022

On 14 September 2022 I gave a talk (in person!) in the Monash Topology seminar.

Title: A symplectic approach to 3-manifold triangulations and hyperbolic structures

Abstract: In the 1980s, Neumann and Zagier introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold, such as a knot complement. We give an interpretation for this symplectic structure in terms of the topology of the 3-manifold, via intersections of certain curves on a Heegaard surface. We also give an algorithm to construct curves forming a symplectic basis for this vector space. This approach gives a description of hyperbolic structures on a knot complement via Ptolemy equations, which can be used to calculate the A-polynomial. This is joint work with Jessica Purcell and Joshua Howie.

Slides from the talk are below

22-09-14_monash_talk

“There is always room for a new theory in mathematics”: Maryna Viazovska and her mathematics

On 30 August 2022 I gave a talk about recent Fields medallist Prof Maryna Viazovska and some of her mathematical work. This was a Monash LunchMaths seminar.

Title: “There is always room for a new theory in mathematics”: Maryna Viazovska and her mathematics

Abstract: In July of this year, Maryna Viazovska became the second woman and second Ukrainian to be awarded a Fields medal. Among other amazing achievements, she gave a wonderful proof of the most efficient way to pack spheres in 8 dimensions, bringing together ideas from all over mathematics. In this talk I’ll give a brief biography of Prof Viazovska and attempt to explain some of her mathematics.

The slides from my talk are available here (15mb pptx and 5mb pdf).

viazovska_talk

A symplectic basis for 3-manifold triangulations

Joint with Jessica Purcell – (45 pages) – on the arxiv – accepted for publication in Communications in Analysis and Geometry

Abstract: In the 1980s, Neumann and Zagier introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold, such as a knot complement. We give a geometric interpretation for this symplectic structure in terms of the topology of the 3-manifold, via intersections of certain curves on a Heegaard surface. We also give an algorithm to construct curves forming a symplectic basis.

Symplectic_basis_published_version

Invitation to Quantum Topology, MATRIX GT3 July 2022

On Wednesday 27 July 2022 I gave a talk at MATRIX, as part of the PhD Student Symposium: Graduate Talks in Geometry and Topology Get-Together, also known as (GT)^3.

Title:

Invitation to quantum topology

Abstract:

Over the last few decades there has been an explosion of developments in topology that, for lack of a better word, are “quantum” in nature. The relationship of this mathematics with physics is complex and wonderful. Ideas in this field include knots and links, topological quantum field theory, Khovanov homology, quantum groups, categorification, and many others. I’ll try to give an introduction to some of the ideas of this extraordinary subject.

Talk at Knots in Washington 49.75

On 22 April 2022 I gave a (virtual) talk at the 49.75’th (!) Knots in Washington conference, an international conference on knot theory held regularly since 1995.

Title: A symplectic basis for 3-manifold triangulations

Abstract: Neumann and Zagier in the 1980s introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold, such as a knot complement. We will discuss an interpretation for this symplectic structure in terms of the topology of the 3-manifold. This joint work with Jessica Purcell involves train tracks, Heegaard splittings, and is related to Ptolemy varieties, geometric quantisation, and the A-polynomial.

Slides from the talk are below.

washington_talk

A Symplectic Basis for 3-manifold Triangulations, AustMS 2021

On 8 December 2021 I gave a (virtual) talk in the Topology session of the 2021 meeting of the Australian Mathematical Society.

Title: A symplectic basis for 3-manifold triangulations

Abstract: Walter Neumann and Don Zagier in the 1980s introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold. We will discuss an interpretation for this symplectic structure in terms of the topology of the 3-manifold. This work, joint with Jessica Purcell, involves train tracks, Heegaard splittings, and is related to Ptolemy varieties, geometric quantisation, and the
A-polynomial.

Slides from the talk are below.

austms_talk_v2

Five-minute surrealist antiwar exposition of topological data analysis

On Remembrance Day 2021 (11 November) I have a talk at a session of Lightning Talks at a session on “Mathematics for Data Analysis, AI & Machine Learning” organised by the Monash Data Futures Institute.

This was a “Lightning Talk” — 5 minutes only. In which I attempted to explain what topological data analysis is and how it works.

It had to be impressionistic, but it turns out surrealism is better for this kind of thing. For what is topological data analysis — or more explicitly, one of its main tools, persistent homology — if not the Persistence of Memory of Topological Contortion?

Being Remembrance Day, a day for ending war, and Topological Data Analysis having been funded for military applications, no better time to mention the campaign against lethal autonomous weapons systems.

Five minutes.

The slides from my talk are available below (2mb pptx).

Slides from my MDFI talk 11/11/21Download

An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes

In this article, by my Monash colleague Jerome Droniou, and Daniel A Di Pietro, I am acknowledged for providing an explicit basis for the space \( \mathcal{G}^{c,l} (T) \), which is used in a software implementation.

Di Pietro, Daniele A, Droniou, Jérôme, An arbitrary-order discrete de Rham complex on polyhedral meshes: exactness, Poincaré inequalities, and consistency. Found. Comput. Math. 23 (2023), no.1, 85–164.

The article has open access online and is available here at the journal website.