The Work of David Trotman

Transcription

1 The Work of David Trotman L. Wilson University of Hawai i at Mānoa Geometry and Topology of Singular Spaces (in celebration of David Trotman s 60th birthday Luminy, 31 October 2012

2 David John Angelo Trotman

3 David John Angelo Trotman Born September 27, 1951 in Plymouth, Devon, England

4 David John Angelo Trotman Born September 27, 1951 in Plymouth, Devon, England Grandson of the poet and author Oliver W F Lodge and a great-grandson of the physicist Sir Oliver Lodge

5 David John Angelo Trotman Born September 27, 1951 in Plymouth, Devon, England Grandson of the poet and author Oliver W F Lodge and a great-grandson of the physicist Sir Oliver Lodge Entered St John s College, Cambridge in 1969

6 David John Angelo Trotman Born September 27, 1951 in Plymouth, Devon, England Grandson of the poet and author Oliver W F Lodge and a great-grandson of the physicist Sir Oliver Lodge Entered St John s College, Cambridge in 1969 Doctorate from University of Warwick and French Doctorat: Whitney Stratifications: Faults and Detectors; Advisors: Chris Zeeman; René Thom; Unofficial Advisors: C. T. C. Wall; Bernard Teissier

7 David John Angelo Trotman Born September 27, 1951 in Plymouth, Devon, England Grandson of the poet and author Oliver W F Lodge and a great-grandson of the physicist Sir Oliver Lodge Entered St John s College, Cambridge in 1969 Doctorate from University of Warwick and French Doctorat: Whitney Stratifications: Faults and Detectors; Advisors: Chris Zeeman; René Thom; Unofficial Advisors: C. T. C. Wall; Bernard Teissier Positions at University of Paris-Sud, University of Angers, and since 1988 at University of Provence.

8 The Trotman School of Stratifications

9 The Trotman School of Stratifications First generation: Patrice Orro (1984); Karim Bekka (1988); Stephane Simon (1994); Laurent Noirel (1996);, Claudio Murolo (1997); Georges Comte (1998); Didier D Acunto (2001); Dwi Juniati (2002); Guillaume Valette (2003); Saurabh Trivedi (2013)

10 The Trotman School of Stratifications First generation: Patrice Orro (1984); Karim Bekka (1988); Stephane Simon (1994); Laurent Noirel (1996);, Claudio Murolo (1997); Georges Comte (1998); Didier D Acunto (2001); Dwi Juniati (2002); Guillaume Valette (2003); Saurabh Trivedi (2013) The next generation: via Orro: Mohammad Alcheikh, Abdelhak Berrabah, Si Tiep Dinh, Farah Farah, Sébastien Jacquet, Mayada Slayman; via Bekka: Nicolas Dutertre, Vincent Grandjean; via Comte: Lionel Alberti; via Juniati: Mustamin Anggo, M.J. Dewiyani, Sulis Janu, Jackson Mairing, Theresia Nugrahaningsih, Herry Susanto, Nurdin

11 The Trotman School of Stratifications First generation: Patrice Orro (1984); Karim Bekka (1988); Stephane Simon (1994); Laurent Noirel (1996);, Claudio Murolo (1997); Georges Comte (1998); Didier D Acunto (2001); Dwi Juniati (2002); Guillaume Valette (2003); Saurabh Trivedi (2013) The next generation: via Orro: Mohammad Alcheikh, Abdelhak Berrabah, Si Tiep Dinh, Farah Farah, Sébastien Jacquet, Mayada Slayman; via Bekka: Nicolas Dutertre, Vincent Grandjean; via Comte: Lionel Alberti; via Juniati: Mustamin Anggo, M.J. Dewiyani, Sulis Janu, Jackson Mairing, Theresia Nugrahaningsih, Herry Susanto, Nurdin The coauthors: Kambouchner, Brodersen, Navarro Aznar, Orro, Bekka, Kuo, Li Pei Xin, Kwieciński, Risler, Wilson, Murolo, Noirel, Comte, Milman, Juniati, du Plessis, Gaffney, King, Plénat

12 Some Stratification Theory pre-trotman Definition of Stratification Let Z be a closed subset of a differentiable manifold M of class C k. A C k stratification of Z is a filtration by closed subsets Z = Z d Z d 1 Z 1 Z 0 such that each difference Z i Z i 1 is a differentiable submanifold of M of class C k and dimension i, or is empty. Each connected component of Z i Z i 1 is called a stratum of dimension i. Thus Z is a disjoint union of the strata, denoted {X α } α A.

13 Some Stratification Theory pre-trotman Definition of Stratification Let Z be a closed subset of a differentiable manifold M of class C k. A C k stratification of Z is a filtration by closed subsets Z = Z d Z d 1 Z 1 Z 0 such that each difference Z i Z i 1 is a differentiable submanifold of M of class C k and dimension i, or is empty. Each connected component of Z i Z i 1 is called a stratum of dimension i. Thus Z is a disjoint union of the strata, denoted {X α } α A. Frontier Condition: Each stratum with nonempty intersection with the closure of another stratum is a subset of that closure.

14 Regularity conditions of Whitney and Verdier Whitney (a) and (b) The pair of strata (X, Y ), Y in the closure of X, is (a)-regular at y Y if sequences x i X with limit y, one has that the limit of T xi X (if it exists) contains T Y. The pair (X, Y ) (b)-regular at y if sequences x i X and y i Y with limit y such that the limit of T xi X (if it exists) contains the limit of x i y i (if it exists).

15 Regularity conditions of Whitney and Verdier Whitney (a) and (b) The pair of strata (X, Y ), Y in the closure of X, is (a)-regular at y Y if sequences x i X with limit y, one has that the limit of T xi X (if it exists) contains T Y. The pair (X, Y ) (b)-regular at y if sequences x i X and y i Y with limit y such that the limit of T xi X (if it exists) contains the limit of x i y i (if it exists). Verdier s condition (w) This is like condition (a), but with the distance from T xi X to T yi Y being O( x i y i ) (y i the projection of x i onto Y ).

16 Regularity conditions of Whitney and Verdier Whitney (a) and (b) The pair of strata (X, Y ), Y in the closure of X, is (a)-regular at y Y if sequences x i X with limit y, one has that the limit of T xi X (if it exists) contains T Y. The pair (X, Y ) (b)-regular at y if sequences x i X and y i Y with limit y such that the limit of T xi X (if it exists) contains the limit of x i y i (if it exists). Verdier s condition (w) This is like condition (a), but with the distance from T xi X to T yi Y being O( x i y i ) (y i the projection of x i onto Y ). Whitney s Example: Z = {y 2 = t 2 x 2 x 3 }; T = t-axis, X = Z T. (X, T ) satisfies (a) but not (b) or (w) at 0.

17 Equisingularity theorems of Mather and Verdier Whitney (b) implies vector fields on a stratum lift to controlled" vector fields on the adjacent larger strata.

18 Equisingularity theorems of Mather and Verdier Whitney (b) implies vector fields on a stratum lift to controlled" vector fields on the adjacent larger strata. (w) implies the vector fields lift to rugose" vector fields.

19 Equisingularity theorems of Mather and Verdier Whitney (b) implies vector fields on a stratum lift to controlled" vector fields on the adjacent larger strata. (w) implies the vector fields lift to rugose" vector fields. In both cases integrating the vector fields yield topological trivializations of the stratifications along a stratum.

20 Relation between conditions

21 Relation between conditions (b) is equivalent to (w) in the complex analytic case

22 Relation between conditions (b) is equivalent to (w) in the complex analytic case (w) implies (b) in the real subanalytic case

23 Relation between conditions (b) is equivalent to (w) in the complex analytic case (w) implies (b) in the real subanalytic case The converse is not true: first semialgebraic example in Trotman 1976, first algebraic example y 4 = t 4 x + x 3 due to Brodersen-Trotman 1979

24 Relation between conditions (b) is equivalent to (w) in the complex analytic case (w) implies (b) in the real subanalytic case The converse is not true: first semialgebraic example in Trotman 1976, first algebraic example y 4 = t 4 x + x 3 due to Brodersen-Trotman 1979 In the differentiable case neither condition implies the other (the slow spiral satisfies (w) but not (b)).

25 Relation between conditions (b) is equivalent to (w) in the complex analytic case (w) implies (b) in the real subanalytic case The converse is not true: first semialgebraic example in Trotman 1976, first algebraic example y 4 = t 4 x + x 3 due to Brodersen-Trotman 1979 In the differentiable case neither condition implies the other (the slow spiral satisfies (w) but not (b)). Trotman s Arcata paper 1983 is still a beautiful though no longer complete listing of known relationships between stratification conditions.

26 Condition (a) (a) is weaker then (b), and doesn t imply topological triviality; why is it interesting?

27 Condition (a) (a) is weaker then (b), and doesn t imply topological triviality; why is it interesting? Trotman 1979 A locally finite stratification of a closed subset Z of a C 1 manifold M is (a)-regular iff for every C 1 manifold N, {f C 1 (N, M) f is transverse to the strata of Z } is an open set in the Whitney C 1 topology.

28 Condition (a) (a) is weaker then (b), and doesn t imply topological triviality; why is it interesting? Trotman 1979 A locally finite stratification of a closed subset Z of a C 1 manifold M is (a)-regular iff for every C 1 manifold N, {f C 1 (N, M) f is transverse to the strata of Z } is an open set in the Whitney C 1 topology. Orro-Trotman 2012 If (Z, Σ) and (Z, Σ ) are Whitney (b)-regular (resp. (a)-regular, resp. (w)-regular) and have transverse intersections in M, then (Z z, Σ Σ ) is (b)-regular (resp. (a)-regular, resp. (w)-regular). The (b) case was done earlier, the Orro-Trotman result includes other conditions we haven t looked at.

29 Regularity conditions and generic wings Definition of (E )-regularity X, Y disjoint C 2 submfds of a C 2 mfd M,y Y X.

30 Regularity conditions and generic wings Definition of (E )-regularity X, Y disjoint C 2 submfds of a C 2 mfd M,y Y X. Suppose E is a regularity condition (like (b)).

31 Regularity conditions and generic wings Definition of (E )-regularity X, Y disjoint C 2 submfds of a C 2 mfd M,y Y X. Suppose E is a regularity condition (like (b)). (X, Y ) is (E )-regular if 0 k < cody there is an open, dense subset of the Grassmannian of codimension k subspaces of T y M containing T y Y such that if W is a C 2 submfd of M with Y W near y, and T y W U k, then W is transverse to X near y and (X W, Y ) is (E)-regular at y.

32 Navarro Aznar-Trotman 1981: For subanalytic stratifications, (w) = (w ), and if dim Y = 1, (b) = (b ).

33 Navarro Aznar-Trotman 1981: For subanalytic stratifications, (w) = (w ), and if dim Y = 1, (b) = (b ). This property plays an important role in the work of Goresky and MacPherson on existence of stratified Morse functions, and in Teissier s equisingularity results.

34 Navarro Aznar-Trotman 1981: For subanalytic stratifications, (w) = (w ), and if dim Y = 1, (b) = (b ). This property plays an important role in the work of Goresky and MacPherson on existence of stratified Morse functions, and in Teissier s equisingularity results. Juniata-Trotman-Valette 2003 For subanalytic stratifications, (L) = (L ) (where (L) is the condition of Mostowski guaranteeing Lipschitz equisingularity)

35 Condition (t k ) Whitney s Example: Z = {y 2 = t 2 x 2 + x 3 } Recall this satisfies (a) but not (b). The intersections with planes through 0 transverse to the t-axis have constant topological type.

36 Condition (t k ) Whitney s Example: Z = {y 2 = t 2 x 2 + x 3 } Recall this satisfies (a) but not (b). The intersections with planes through 0 transverse to the t-axis have constant topological type. Theorem Kuo 1978 If (X, Y ) is (a)-regular at y Y then (h ) holds, i.e. the germs at y of intersections S X, where S is a C submfd transverse to Y at y and dim S + dim Y = dim M (S is called a direct transversal) are homeomorphic.

37 Definition: Trotman s refinement of Thom s condition (X, Y ) is (t k )-regular at y Y if every C k submfd S transverse to Y at y is transverse to X nearby.

38 Definition: Trotman s refinement of Thom s condition (X, Y ) is (t k )-regular at y Y if every C k submfd S transverse to Y at y is transverse to X nearby. Theorem Trotman 1976 If Y is semianalytic then (t 1 ) is equivalent to (a).

39 Definition: Trotman s refinement of Thom s condition (X, Y ) is (t k )-regular at y Y if every C k submfd S transverse to Y at y is transverse to X nearby. Theorem Trotman 1976 If Y is semianalytic then (t 1 ) is equivalent to (a). In the above result one needs non-direct transversals. In the results below, we always restrict to direct transversals

40 Definition: Trotman s refinement of Thom s condition (X, Y ) is (t k )-regular at y Y if every C k submfd S transverse to Y at y is transverse to X nearby. Theorem Trotman 1976 If Y is semianalytic then (t 1 ) is equivalent to (a). In the above result one needs non-direct transversals. In the results below, we always restrict to direct transversals Theorem Trotman 1985 (t 1 ) is equivalent to (h 1 ).

41 Definition: Trotman s refinement of Thom s condition (X, Y ) is (t k )-regular at y Y if every C k submfd S transverse to Y at y is transverse to X nearby. Theorem Trotman 1976 If Y is semianalytic then (t 1 ) is equivalent to (a). In the above result one needs non-direct transversals. In the results below, we always restrict to direct transversals Theorem Trotman 1985 (t 1 ) is equivalent to (h 1 ). Trotman-Wilson 1999 For subanalytic strata (t k ) is equivalent to the finiteness of the number of topological types of germs at y of S X for S a C k transversal to Y (1 k ).

42 The proofs use the Grassmann blowup": like the regular blowup, but with lines through y replaced with all linear subspaces through y of dimension equal to the codimension of Y.

43 The proofs use the Grassmann blowup": like the regular blowup, but with lines through y replaced with all linear subspaces through y of dimension equal to the codimension of Y. Kuo-Trotman 1988, Trotman-Wilson 1999 (X, Y ) is (t k )-regular at 0 Y iff its Grassmann blowup ( X, Ỹ ) is (t k 1 )-regular at every point of Ỹ (k 1).

44 The proofs use the Grassmann blowup": like the regular blowup, but with lines through y replaced with all linear subspaces through y of dimension equal to the codimension of Y. Kuo-Trotman 1988, Trotman-Wilson 1999 (X, Y ) is (t k )-regular at 0 Y iff its Grassmann blowup ( X, Ỹ ) is (t k 1 )-regular at every point of Ỹ (k 1). A definition of (t k ) is given so that (t 0 ) is equivalent to (w)

45 The proofs use the Grassmann blowup": like the regular blowup, but with lines through y replaced with all linear subspaces through y of dimension equal to the codimension of Y. Kuo-Trotman 1988, Trotman-Wilson 1999 (X, Y ) is (t k )-regular at 0 Y iff its Grassmann blowup ( X, Ỹ ) is (t k 1 )-regular at every point of Ỹ (k 1). A definition of (t k ) is given so that (t 0 ) is equivalent to (w) So (t 1 ) = (h 1 ) follows from blowup and then applying the Verdier Isotopy Theorem.

46 Koike-Kucharz Example Let Z = {x 3 3xy 5 + ty 6 = 0}, with Y the t-axis and X = Z Y. (X, Y ) is (t 2 ), not (t 1 ). There are two topological types of germs at 0 of intersections S X where S is a C 2 submfd transverse to Y at 0. However the number of topological types of such germs for S of class C 1 is uncountable.

47 Koike-Kucharz Example Let Z = {x 3 3xy 5 + ty 6 = 0}, with Y the t-axis and X = Z Y. (X, Y ) is (t 2 ), not (t 1 ). There are two topological types of germs at 0 of intersections S X where S is a C 2 submfd transverse to Y at 0. However the number of topological types of such germs for S of class C 1 is uncountable. Trotman-Wilson 1999

48 Koike-Kucharz Example Let Z = {x 3 3xy 5 + ty 6 = 0}, with Y the t-axis and X = Z Y. (X, Y ) is (t 2 ), not (t 1 ). There are two topological types of germs at 0 of intersections S X where S is a C 2 submfd transverse to Y at 0. However the number of topological types of such germs for S of class C 1 is uncountable. Trotman-Wilson 1999 Also there is a theory of (t k ) such that (t 1 ) is essentially (a) holding for all sequences going to 0 not tangent to Y.

49 Koike-Kucharz Example Let Z = {x 3 3xy 5 + ty 6 = 0}, with Y the t-axis and X = Z Y. (X, Y ) is (t 2 ), not (t 1 ). There are two topological types of germs at 0 of intersections S X where S is a C 2 submfd transverse to Y at 0. However the number of topological types of such germs for S of class C 1 is uncountable. Trotman-Wilson 1999 Also there is a theory of (t k ) such that (t 1 ) is essentially (a) holding for all sequences going to 0 not tangent to Y. The (t k ) and (t k )-conditions were formulated for jets of transversals

50 Koike-Kucharz Example Let Z = {x 3 3xy 5 + ty 6 = 0}, with Y the t-axis and X = Z Y. (X, Y ) is (t 2 ), not (t 1 ). There are two topological types of germs at 0 of intersections S X where S is a C 2 submfd transverse to Y at 0. However the number of topological types of such germs for S of class C 1 is uncountable. Trotman-Wilson 1999 Also there is a theory of (t k ) such that (t 1 ) is essentially (a) holding for all sequences going to 0 not tangent to Y. The (t k ) and (t k )-conditions were formulated for jets of transversals The (t k ) and (t k )-conditions were then used to characterize sufficiency of jets of functions, generalizing theorems of Bochnak, Kuo, Lu and others.

51 Gaffney-Trotman-Wilson 2009 We expressed (t k ) in terms of integral closure of modules, giving more algebraic techniques for computations. In the complex analytic case, (t k ) is characterized by the genericity of the multiplicity of a certain submodule.

52 Normal cone C Y Z of a set Z to a submfd Y ; p the projection Theorem (Hironaka in analytic (b) case, Trotman-Orro 2002 generalize to smooth (a) + (r e ))

53 Normal cone C Y Z of a set Z to a submfd Y ; p the projection Theorem (Hironaka in analytic (b) case, Trotman-Orro 2002 generalize to smooth (a) + (r e )) A stratification of Z satisfying the above regularity conditions is (npf ) normally pseudo-flat, i.e. p is an open map, and

54 Normal cone C Y Z of a set Z to a submfd Y ; p the projection Theorem (Hironaka in analytic (b) case, Trotman-Orro 2002 generalize to smooth (a) + (r e )) A stratification of Z satisfying the above regularity conditions is (npf ) normally pseudo-flat, i.e. p is an open map, and (n) the fibre of the normal cone is the tangent cone of the fibre

55 Normal cone C Y Z of a set Z to a submfd Y ; p the projection Theorem (Hironaka in analytic (b) case, Trotman-Orro 2002 generalize to smooth (a) + (r e )) A stratification of Z satisfying the above regularity conditions is (npf ) normally pseudo-flat, i.e. p is an open map, and (n) the fibre of the normal cone is the tangent cone of the fibre Orro-Trotman 2002 show the Theorem fails for (a)-regularity.

56 Normal cone C Y Z of a set Z to a submfd Y ; p the projection Theorem (Hironaka in analytic (b) case, Trotman-Orro 2002 generalize to smooth (a) + (r e )) A stratification of Z satisfying the above regularity conditions is (npf ) normally pseudo-flat, i.e. p is an open map, and (n) the fibre of the normal cone is the tangent cone of the fibre Orro-Trotman 2002 show the Theorem fails for (a)-regularity. Trotman-Wilson 2006 show that it also fails in the non-polynomial bounded o-minimal category; our example is z = f (x, y) = x x ln(y + x 2 + y 2 )/ ln(x).

57 Nash fiber = limits of tangent planes at a singular point

58 Nash fiber = limits of tangent planes at a singular point Kwieciński-Trotman 1995 Every continuum can be realized as the Nash fiber of a Whitney stratified set.

59 David Trotman s other research

60 David Trotman s other research Reducing the Arbitrariness of Our Description of Balinese Dance Or Balinese Dance as a Descriptive Text in Neonatal Motricity Author: David Trotman, 1982, 36 pages

61 Trotman Style Powerful geometric intuition

62 Trotman Style Powerful geometric intuition Creation of insightful examples

63 Trotman Style Powerful geometric intuition Creation of insightful examples Impressive knowledge of what has been done and is being done in many areas, allowing him to make contributions to the real differentiable, subanalytic, o-minimal and complex analytic versions of singularity theory, but also to such fields as microlocal analysis and robotics.

64 Trotman Style Powerful geometric intuition Creation of insightful examples Impressive knowledge of what has been done and is being done in many areas, allowing him to make contributions to the real differentiable, subanalytic, o-minimal and complex analytic versions of singularity theory, but also to such fields as microlocal analysis and robotics. Writer of interesting surveys

65 Trotman Style Powerful geometric intuition Creation of insightful examples Impressive knowledge of what has been done and is being done in many areas, allowing him to make contributions to the real differentiable, subanalytic, o-minimal and complex analytic versions of singularity theory, but also to such fields as microlocal analysis and robotics. Writer of interesting surveys Great friend and collaborator

66 Trotman Style Powerful geometric intuition Creation of insightful examples Impressive knowledge of what has been done and is being done in many areas, allowing him to make contributions to the real differentiable, subanalytic, o-minimal and complex analytic versions of singularity theory, but also to such fields as microlocal analysis and robotics. Writer of interesting surveys Great friend and collaborator Producer of many active and appreciative students

67 Trotman Style Powerful geometric intuition Creation of insightful examples Impressive knowledge of what has been done and is being done in many areas, allowing him to make contributions to the real differentiable, subanalytic, o-minimal and complex analytic versions of singularity theory, but also to such fields as microlocal analysis and robotics. Writer of interesting surveys Great friend and collaborator Producer of many active and appreciative students (which is why we are here)

68 David Trotman Congratulations!