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

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.

SymplecticBasis_202208

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).

General tips for studying mathematics

I don’t know that I would have anything to say that’s not a platitude, but here are some thoughts.

It’s a hard topic. Recognise that it’s hard. Don’t expect to learn everything in one go. Don’t expect to solve every problem on the first attempt. Take the time to let ideas sink in.

Don’t feel intimidated by others who do things faster, nor superior to those who do things slower. Don’t let anyone put anyone else down for finding the subject difficult, comprehending things more slowly, etc. Try to include people who feel excluded. It’s difficult for everybody, and everybody has different difficulties. We don’t need to make them worse.

Ask questions of your teachers. Chances are if you’re thinking something in a lecture was unclear, many other people are too, but not all of them are brave enough to raise their hand. And ask outside of class too if you want. For most lecturers, most of the questions they have to answer are about homework extensions, repeating for the hundredth time some administrative detail, etc. They love answering questions about their actual subject. Ask them.

If you can, find other students to work with. People vary, but for many people, working together productively with others on problems makes the whole experience much more pleasant, and it’s better for solving problems and bouncing around ideas too. Learn from others, and they can learn from you.

Be curious. Ask how and why things work. Think about the big picture and how everything fits together. Ask why we study the things we study.

Enjoy the subject. Mathematics is full of patterns, curiosities, profound truths, breathtaking theorems, hard-fought triumphs, ancient mysteries. Its roots in culture stretch back as far as humans have walked the earth. Play with the patterns, play with as much of it as you can get your hands on, it’s much better if you an understand it that way. Understand it your way, and tell stories about it.

Wonder at the profound beauty and elegance of deep theories. Take the time to properly understand and appreciate what theorems say. They surprise, they flabbergast, they connect oceans of thought. They reduce the incomprehensible to the world of the living. Their proofs are often even more instructive than their statements. These things are among the greatest achievements of humanity. They are the eternal harmonies of this tragic universe.

Also, savour the enjoyment when you solve a difficult problem. It’s a hard subject and we need to take all the wins we can get.

Summarise your maths research in one slide, Dan

I’ve never much been one for concision. To say that the contemporary attention economy, not to mention professional communication, not to mention mainstream politics, trades on absurdly short, necessarily incomplete, if not wrong, attention-grabbing quips and barbs and slogans and pitches, effectively giving most social discourse a lobotomy, is a cliche. That doesn’t make it less true or less important. Equally, to insinuate that you have much more to say, and have such nuanced and complex ideas, that you’re above that sort of thing, that you’re so sophisticated as to be impossible to summarise or condense to that form, is not only irritating and self-important, it’s probably false most of the time too.

Anyway, as part of an upcoming workshop participants were asked to introduce themselves with a one-page slide. I took it as an extreme form of concision: summarise your maths research in one slide, Dan. In practice it turned into a word cloud with a bunch of pictures taken from my papers.

Anyway, I thought the pictures were, if not pretty to look at, a bit fun. Well, of course I would say that, I do this stuff for a living. (Or, at least, I’m supposed to. What it seems I actually do for a living is drown in email and admin and committees and so on.)

So why not post the one slide here too?

slide summarising my maths research