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Life and Mathematics

Life and Mathematics. Nalini Joshi @monsoon0. Life Work Reflections. “mathematician” by Trixie Barretto. vimeo.com/33615260. Life. Where I am now. Integrable Systems. Properties of Solutions. Integers Rational numbers Algebraic numbers Transcendental numbers. Polynomials

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Life and Mathematics

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  1. Life and Mathematics • Nalini Joshi • @monsoon0

  2. Life • Work • Reflections

  3. “mathematician” by Trixie Barretto vimeo.com/33615260

  4. Life

  5. Where I am now

  6. Integrable Systems

  7. Properties of Solutions • Integers • Rational numbers • Algebraic numbers • Transcendental numbers • Polynomials • Rational functions • Algebraic functions • Transcendental functions

  8. The first Painlevé Eqn • PI : • In system form • PI has a t-dependent Hamiltonian • Solutions are highly transcendental, meromorphic functions.

  9. Elliptic Functions • Weierstrass elliptic functions

  10. 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.

  11. Geometry Level curves of

  12. 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?

  13. Resolution • “Blow up” the singularity or base point: • Note that

  14. PI • There are nine blow-ups: • Only the last one differs from the elliptic case.

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

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

  17. The Repellor Set • Definition: For z ∈ ℂ\{0}, let S denote the fibre bundle of the Okamoto surfacesS9(z) and This is the infinity set. • Proposition: I(z) is a repellor for the flow.

  18. Lemma: is a non-empty, connected and compact subset of Okamoto’s space. The Limit Set • Definition: For every solution U(z) ∈ S9(z)\I(z), let This is the limit set.

  19. How many poles? • Lemma: Every solution of the first Painlevé equation has infinitely many poles. • If intersects L9then 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.

  20. 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.

  21. 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.”

  22. 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.

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

  24. 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

  25. 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.

  26. 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; • ....

  27. 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.

  28. 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.

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