Life and Mathematics. Nalini

Transcription

1 Life and Mathematics Nalini

2 Life Work Reflections

3 mathematician by Trixie Barretto vimeo.com/

4 Life B.Sc. (Hons) Sydney PhD Princeton Married PostDoc ANU First child Lecturer, Senior Lecturer UNSW Second child 1993 Gave up tenure Senior Lecturer, Associate Professor Adelaide ARC Senior Research Fellowship 02-now Professor Sydney...

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6 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,...

7 Integrable Systems u t +6uu x + u xxx =0 Korteweg-de Vries eqn w 00 =6w 2 t First Painlevé eqn v n+1 v n 1 = 1 v n q n v 2 n Discrete first Painlevé eqn

8 Properties of Solutions Integers Rational numbers Algebraic numbers Transcendental numbers Polynomials Rational functions Algebraic functions Transcendental functions e,,,... e x, (x), 2 F 1 (a, b; c; x),... w(t) =} t t 0 ; g 2,g 3...

9 The first Painlevé Eqn PI : w 00 =6w 2 t In system form d dt w1 w 2 = w2 6 w 2 1 t PI has a t-dependent Hamiltonian H = w w tw 1 Solutions are highly transcendental, meromorphic functions.

10 Elliptic Functions ẇ 2 g 2 2 =2w3 2 w g 3 2 w(t) =} t t 0 ; g 2,g 3 Weierstrass elliptic functions

11 A Geometric View y = ẇ, x = w ) y 2 =4x 3 g 2 x g 3 Instead of studying the differential equation, we can study properties of the level curves of f(x, y) =y 2 4 x 3 + g 2 x Initial values for the differential equation identify a curve and a starting point on it.

12 Geometry x x Level curves of y 2

13 Projective Space The solutions of PI are meromorphic, with movable poles. What if x, y become unbounded? We use projective geometry: [x, y, 1] = [u, v, w] 2 CP 2 The level curves are now x = u w,y = v w F = wv 2 4 u 3 + g 2 uw 2 + g 3 w 3 all intersecting at the base point [0, 1, 0]. How to resolve the flow through this point?

14 Resolution Blow up the singularity or base point: Note that f(x, y) =y 2 x 3 (x, y) =(x 1,x 1 y 1 ) )x 2 1 y1 2 x 3 1 =0,x 2 1 (y1 2 x 1 )=0 x 1 = x, y 1 = y/x

15 PI There are nine blow-ups: b 0 : u 031 =0,u 032 =0 b 1 : u 111 =0,u 112 =0 b 2 : u 211 =0,u 212 =0 b 3 : u 311 =4,u 312 =0 b 4 : u 411 =4,u 412 =0 b 5 : u 511 =0,u 512 =0 b 6 : u 611 =0,u 612 =0 b 7 : u 711 = 32,u 712 =0 b 8 : u 811 = 2 8 (5 z),u 812 =0 Only the last one differs from the elliptic case.

16 L9 Exceptional Lines S9(z) L8 (1) L7 (2) L6 (3) L5 (4) (6) L3 L4 (5) L0 (9) L1 (8) L2 (7)

17 L9 Exceptional Lines Line of poles S9(z) L8 (1) L7 (2) L6 (3) L5 (4) (6) L3 L4 (5) L0 (9) L1 (8) L2 (7)

18 Exceptional Lie Algebra L0 (9) L1 (8) L2 (7) L3 (6) L4 (5) L5 (4) L6 (3) L7 (2) L8 (1) Affine extended E8

19 The Repellor Set Definition: For z C\{0}, let S denote the fibre bundle of the Okamoto surfaces S9(z) and I(z) :=[ 8 i=0l (9 i) i (z) This is the infinity set. Proposition: I(z) is a repellor for the flow.

20 The Limit Set Definition: For every solution U(z) S 9(z)\I(z), let n U = s S 9 ( )\I( ) {z j } s.t. z j, This is the limit set. o U(z j ) s as j Lemma: U is a non-empty, connected and compact subset of Okamoto s space.

21 How many poles? Lemma: Every solution of the first Painlevé equation has infinitely many poles. If U intersects L9 then we get infinitely many poles. If not, then U 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.

22 Discrete Equations Sakai CMP 2001 classified all possible secondorder equations whose initial value space is regularized by a 9-point blow-up of CP 2. 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.

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

24 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.

25 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.

26 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

27 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.

28 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;...

29 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.

30 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.