Life and Mathematics
Nalini Joshi
@monsoon0
Life
Work
Reflections
“mathematician” by Trixie Barretto
vimeo.com/33615260
Life
’78-’81
B.Sc. (Hons)
Sydney
’82-’86
PhD
Princeto
• Married 1984
n
’87-’90
PostDoc
ANU
• First child 1988
’90-’96
Lecturer, Senior
Lecturer
UNSW
• Second child 1993
• Gave up tenure
1996
’97-’02
Senior Lecturer,
Associate
Professor
ARC Senior Research
Adelaide
Fellowship
’02-now Professor
Sydney
...
Where I am now
2012 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
๏ Polynomials
• Rational numbers ๏ Rational functions
• Algebraic
๏ Algebraic functions
numbers
• Transcendental
numbers
๏ Transcendental
functions
The first Painlevé Eqn
•P:
• In system form
I
• P has a t-dependent Hamiltonian
I
• 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 P are meromorphic, with
I
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.
L9
Exceptional Lines
S9(z)
L8
L6(3)
(1)
L7(2)
L5
(4)
L3(6)
L4(5)
L1
(8)
L0
L2
(7)
(9)
Exceptional Lie
Algebra
L0(9)
2
4
3
6
5
4
3
2
1
L1(8) L2(7) L3(6) L4(5) L5(4) L6(3) L7(2) L8(1)
Affine extended E8
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.
How many poles?
• 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.
Discrete Equations
• Sakai CMP 2001 classified all possible
•
•
second-order equations whose initial value
space is regularized by a 9-point blow-up of
2
CP .
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
•
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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 researchoriented career
• Maintaining research momentum;
• ....
Georgina Sweet
Fellowship
• 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.
• 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.