Life and Mathematics

Nalini Joshi@monsoon0

Life

Work

Reflections

“mathematician” by Trixie Barrettovimeo.com/33615260

Life’78-’81 B.Sc. (Hons) Sydney

’82-’86 PhDPrinceton •Married 1984

’87-’90 PostDoc ANU •First child 1988

’90-’96Lecturer, Senior Lecturer

UNSW

•Second child 1993

•Gave up tenure 1996

’97-’02Senior Lecturer, Associate Professor

Adelaide

ARC Senior Research Fellowship

’02-now

Professor Sydney ...

Where I am now2012 Georgina Sweet Australian Laureate Fellow (Australian Research Council)Along the way:•Head of School•President of Australian Mathematical Society•Chair of the National Committee for Mathematical Sciences, Member of Council of the Australian Academy of Science, ...

Integrable Systems

Korteweg-de Vries eqn

First Painlevé eqn

Discrete first Painlevé eqn

Properties of Solutions•Integers

•Rational numbers

•Algebraic numbers

•Transcendental numbers

๏ Polynomials๏Rational

functions๏Algebraic

functions๏ Transcendental

functions

The first Painlevé Eqn

•In system form

•PI has a t-dependent Hamiltonian

•PI :

•Solutions are highly transcendental, meromorphic functions.

Elliptic Functions

Weierstrass elliptic functions

A Geometric View

•Instead of studying the differential equation, we can study properties of the level curves of

•Initial values for the differential equation identify a curve and a starting point on it.

Geometry

Level curves of

Projective Space•The solutions of PI are meromorphic, with

movable poles. What if x, y become unbounded?

•We use projective geometry:

•The level curves are now

all intersecting at the base point [0, 1, 0].

•How to resolve the flow through this point?

Resolution•“Blow up” the singularity or base point:

•Note that

PI

•There are nine blow-ups:

•Only the last one differs from the elliptic case.

Exceptional LinesLine of polesL9

L8(1)

L7(2)

L6(3)

L5(4)

L4(5)

L3(6)

L0(9)

L1(8)

L2(7)

S9(z)

Exceptional Lie Algebra

Affine extended E8

L1(8)

2

L2(7)

4

L0(9)

3

L3(6)

6

L4(5)

5

L5(4)

4

L6(3)

3

L7(2)

2

L8(1)

1

The Repellor Set

•Definition: For z ∈ ℂ\{0}, let S denote the fibre bundle of the Okamoto surfaces S9(z) and

This is the infinity set.•Proposition: I(z) is a repellor for the

flow.

The Limit Set•Definition: For every solution U(z) ∈

S9(z)\I(z), let

This is the limit set.•Lemma: is a non-empty,

connected and compact subset of Okamoto’s space.

•Lemma: Every solution of the first Painlevé equation has infinitely many poles.

If intersects L9 then we get infinitely many poles. If not, then must be a compact subset of S9\{S9,∞ U L9}. Since holomorphic, the limit set must equal one point. But the autonomous system has two points ⇒ contradiction.

How many poles?

Discrete Equations•Sakai CMP 2001 classified all possible

second-order equations whose initial value space is regularized by a 9-point blow-up of CP2.

•He found all the known Painlevé equations, their recurrence relations and many new difference equations.

•How do we describe their solutions? My plan: use geometry.

Reflections • PhD: “Come and read my poster, it’s much better than

hers.”

• PostDoc: “Babies need mothers.”

• Tenure-track: “We note that all of her papers are with XXX.”

• Tenured: “Your area of research is very narrow.”

•Mid-career: “‘Asymptotic’ does not appear in list of keywords in the NSF database.”

•Mid-career: “We have to thank Nalini for reminding us of what Boutroux did in 1913.”

• Senior Researcher: “She may be well known in Australia, but is not known overseas.”

Even Nobel-Prize Winners ...

•Elizabeth Blackburn (Nobel Prize for Medicine, 2009) New York Times 09 April 2013: She enjoys being free to explore territory where she would not have ventured before. “I would have been a little afraid to do things, because my male colleagues wouldn’t have taken me seriously as a molecular biologist,”she said.

Microaggression n. Brief and commonplace daily verbal, behavioural, and environmental indignities, whether intentional or unintentional, that communicate hostile, derogatory, or negative racial, gender, sexual orientation.

How I survived•More than 20 grants, totalling over $5M

•Two 5-year research fellowships, one of which saved my career

•Papers with 40 collaborators•More than 20 postdocs,10 PhD students

What saves everything, for me, is that mathematics is

๏ Creative play at a deep level.

๏ Creating with friends.

๏ Inventing new ways of seeing.

๏ Contributing to understanding the world.

Collective Wisdom

•The “impostor syndrome”

•Dual careers or the two-body problem: options, examples and solutions

•Work–family balance in a research-oriented career

•Maintaining research momentum;

•....

•To support the promotion of women in research in Australia and the mentoring of early career researchers, particularly women.

•Events at annual meetings of the Australian Mathematical Society and Australian Academy of Science, highlighting the life and careers of female speakers and spreading knowledge.

Georgina Sweet Fellowship

Why do I do Mathematics?

๏ The adventure of exploring the unknown.

๏ The dream that I could understand the structures of the Universe.

๏ The fact that Mathematics has no boundaries or borders.