The Work of David Trotman - Aix-Marseille Universitymurolo/DT/Talks/Wilson.pdf · Bekka, Kuo, Li Pei Xin, Kwiecinski, Risler, Wilson, Murolo,´ ... L. Wilson The Work of David Trotman

The Work of David Trotman

L. Wilson

University of Hawai‘i at Manoa

Geometry and Topology of Singular Spaces(in celebration of David Trotman’s 60th birthday

Luminy, 31 October 2012

L. Wilson The Work of David Trotman

David John Angelo Trotman

Born September 27, 1951 in Plymouth, Devon, EnglandGrandson of the poet and author Oliver W F Lodge and agreat-grandson of the physicist Sir Oliver LodgeEntered St John’s College, Cambridge in 1969Doctorate from University of Warwick and French Doctorat:Whitney Stratifications: Faults and Detectors; Advisors:Chris Zeeman; René Thom; Unofficial Advisors: C. T. C.Wall; Bernard TeissierPositions at University of Paris-Sud, University of Angers,and since 1988 at University of Provence.



Born September 27, 1951 in Plymouth, Devon, England

Grandson of the poet and author Oliver W F Lodge and agreat-grandson of the physicist Sir Oliver LodgeEntered St John’s College, Cambridge in 1969Doctorate from University of Warwick and French Doctorat:Whitney Stratifications: Faults and Detectors; Advisors:Chris Zeeman; René Thom; Unofficial Advisors: C. T. C.Wall; Bernard TeissierPositions at University of Paris-Sud, University of Angers,and since 1988 at University of Provence.



Born September 27, 1951 in Plymouth, Devon, EnglandGrandson of the poet and author Oliver W F Lodge and agreat-grandson of the physicist Sir Oliver Lodge

Entered St John’s College, Cambridge in 1969Doctorate from University of Warwick and French Doctorat:Whitney Stratifications: Faults and Detectors; Advisors:Chris Zeeman; René Thom; Unofficial Advisors: C. T. C.Wall; Bernard TeissierPositions at University of Paris-Sud, University of Angers,and since 1988 at University of Provence.



Born September 27, 1951 in Plymouth, Devon, EnglandGrandson of the poet and author Oliver W F Lodge and agreat-grandson of the physicist Sir Oliver LodgeEntered 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 TeissierPositions at University of Paris-Sud, University of Angers,and since 1988 at University of Provence.



Born September 27, 1951 in Plymouth, Devon, EnglandGrandson of the poet and author Oliver W F Lodge and agreat-grandson of the physicist Sir Oliver LodgeEntered St John’s College, Cambridge in 1969Doctorate 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.



Born September 27, 1951 in Plymouth, Devon, EnglandGrandson of the poet and author Oliver W F Lodge and agreat-grandson of the physicist Sir Oliver LodgeEntered St John’s College, Cambridge in 1969Doctorate from University of Warwick and French Doctorat:Whitney Stratifications: Faults and Detectors; Advisors:Chris Zeeman; René Thom; Unofficial Advisors: C. T. C.Wall; Bernard TeissierPositions at University of Paris-Sud, University of Angers,and since 1988 at University of Provence.


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); DwiJuniati (2002); Guillaume Valette (2003); Saurabh Trivedi(2013)

The next generation: via Orro: Mohammad Alcheikh, AbdelhakBerrabah, Si Tiep Dinh, Farah Farah, Sébastien Jacquet,Mayada Slayman; via Bekka: Nicolas Dutertre, VincentGrandjean; via Comte: Lionel Alberti; via Juniati: MustaminAnggo, M.J. Dewiyani, Sulis Janu, Jackson Mairing, TheresiaNugrahaningsih, Herry Susanto, Nurdin

The coauthors: Kambouchner, Brodersen, Navarro Aznar, Orro,Bekka, Kuo, Li Pei Xin, Kwiecinski, Risler, Wilson, Murolo,Noirel, Comte, Milman, Juniati, du Plessis, Gaffney, King,Plénat

















Some Stratification Theory pre-Trotman

Definition of StratificationLet Z be a closed subset of a differentiable manifold M of classCk . A Ck stratification of Z is a filtration by closed subsets

Z = Zd ⊇ Zd−1 ⊇ · · · ⊇ Z1 ⊇ Z0

such that each difference Zi − Zi−1 is a differentiablesubmanifold of M of class Ck and dimension i , or is empty.Each connected component of Zi − Zi−1 is called a stratum ofdimension i . Thus Z is a disjoint union of the strata, denoted{Xα}α∈A.

Frontier Condition:Each stratum with nonempty intersection with the closure ofanother stratum is a subset of that closure.


Some Stratification Theory pre-Trotman

Definition of StratificationLet Z be a closed subset of a differentiable manifold M of classCk . A Ck stratification of Z is a filtration by closed subsets

Z = Zd ⊇ Zd−1 ⊇ · · · ⊇ Z1 ⊇ Z0

such that each difference Zi − Zi−1 is a differentiablesubmanifold of M of class Ck and dimension i , or is empty.Each connected component of Zi − Zi−1 is called a stratum ofdimension i . Thus Z is a disjoint union of the strata, denoted{Xα}α∈A.

Frontier Condition:Each stratum with nonempty intersection with the closure ofanother stratum is a subset of that closure.


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 aty ∈ Y if ∀ sequences xi ∈ X with limit y , one has that the limitof Txi X (if it exists) contains TY .The pair (X , Y ) (b)-regular at y if ∀ sequences xi ∈ X andyi ∈ Y with limit y such that the limit of Txi X (if it exists)contains the limit of xiyi (if it exists).

Verdier’s condition (w)

This is like condition (a), but with the distance from Txi X toTyi Y being O(|xi − yi |) (yi the projection of xi onto Y ).

Whitney’s Example: Z = {y2 = t2x2 − x3}; T = t-axis,X = Z − T .(X , T ) satisfies (a) but not (b) or (w) at 0.



Whitney (a) and (b)







Whitney (a) and (b)






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 thevector fields lift to “rugose" vector fields. In both casesintegrating the vector fields yield topological trivializations of thestratifications along a stratum.



Whitney (b) implies vector fields on a stratum lift to “controlled"vector fields on the adjacent larger strata. (w) implies thevector fields lift to “rugose" vector fields.

In both casesintegrating the vector fields yield topological trivializations of thestratifications along a stratum.



Whitney (b) implies vector fields on a stratum lift to “controlled"vector fields on the adjacent larger strata. (w) implies thevector fields lift to “rugose" vector fields. In both casesintegrating the vector fields yield topological trivializations of thestratifications along a stratum.


Relation between conditions

(b) is equivalent to (w) in the complex analytic case(w) implies (b) in the real subanalytic caseThe converse is not true: first semialgebraic example inTrotman 1976, first algebraic example y4 = t4x + x3 due toBrodersen-Trotman 1979In 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 nolonger complete listing of known relationships betweenstratification conditions.


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

(w) implies (b) in the real subanalytic caseThe converse is not true: first semialgebraic example inTrotman 1976, first algebraic example y4 = t4x + x3 due toBrodersen-Trotman 1979In 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 nolonger complete listing of known relationships betweenstratification conditions.


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 inTrotman 1976, first algebraic example y4 = t4x + x3 due toBrodersen-Trotman 1979In 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 nolonger complete listing of known relationships betweenstratification conditions.


Relation between conditions(b) is equivalent to (w) in the complex analytic case(w) implies (b) in the real subanalytic caseThe converse is not true: first semialgebraic example inTrotman 1976, first algebraic example y4 = t4x + x3 due toBrodersen-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 nolonger complete listing of known relationships betweenstratification conditions.


Relation between conditions(b) is equivalent to (w) in the complex analytic case(w) implies (b) in the real subanalytic caseThe converse is not true: first semialgebraic example inTrotman 1976, first algebraic example y4 = t4x + x3 due toBrodersen-Trotman 1979In 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 nolonger complete listing of known relationships betweenstratification conditions.


Relation between conditions(b) is equivalent to (w) in the complex analytic case(w) implies (b) in the real subanalytic caseThe converse is not true: first semialgebraic example inTrotman 1976, first algebraic example y4 = t4x + x3 due toBrodersen-Trotman 1979In 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 nolonger complete listing of known relationships betweenstratification conditions.


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 C1

manifold M is (a)-regular iff for every C1 manifold N,{f ∈ C1(N, M)|f is transverse to the strata of Z } is an open setin the Whitney C1 topology.

Orro-Trotman 2012If (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 includesother conditions we haven’t looked at.


Condition (a)


Trotman 1979






Condition (a)


Trotman 1979






Regularity conditions and generic wings

Definition of (E∗)-regularity

X , Y disjoint C2 submfds of a C2 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, densesubset of the Grassmannian of codimension k subspaces ofTyM containing TyY such that if W is a C2 submfd of M withY ⊂ W near y , and TyW ∈ Uk , then W is transverse to X neary and (X ∩W , Y ) is (E)-regular at y .




X , Y disjoint C2 submfds of a C2 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, densesubset of the Grassmannian of codimension k subspaces ofTyM containing TyY such that if W is a C2 submfd of M withY ⊂ W near y , and TyW ∈ Uk , then W is transverse to X neary and (X ∩W , Y ) is (E)-regular at y .




X , Y disjoint C2 submfds of a C2 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, densesubset of the Grassmannian of codimension k subspaces ofTyM containing TyY such that if W is a C2 submfd of M withY ⊂ W near y , and TyW ∈ Uk , then W is transverse to X neary and (X ∩W , Y ) is (E)-regular at y .


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 Goreskyand MacPherson on existence of stratified Morse functions, andin Teissier’s equisingularity results.

Juniata-Trotman-Valette 2003For subanalytic stratifications, (L) =⇒ (L∗) (where (L) is thecondition of Mostowski guaranteeing Lipschitz equisingularity)










Condition (tk)

Whitney’s Example: Z = {y2 = t2x2 + x3}Recall this satisfies (a) but not (b). The intersections withplanes through 0 transverse to the t-axis have constanttopological type.

Theorem Kuo 1978If (X , Y ) is (a)-regular at y ∈ Y then (h∞) holds, i.e. the germsat y of intersections S ∩ X , where S is a C∞ submfd transverseto Y at y and dim S + dim Y = dim M (S is called a directtransversal) are homeomorphic.


Condition (tk)

Whitney’s Example: Z = {y2 = t2x2 + x3}Recall this satisfies (a) but not (b). The intersections withplanes through 0 transverse to the t-axis have constanttopological type.

Theorem Kuo 1978If (X , Y ) is (a)-regular at y ∈ Y then (h∞) holds, i.e. the germsat y of intersections S ∩ X , where S is a C∞ submfd transverseto Y at y and dim S + dim Y = dim M (S is called a directtransversal) are homeomorphic.


Definition: Trotman’s refinement of Thom’s condition

(X , Y ) is (tk )-regular at y ∈ Y if every Ck submfd S transverseto Y at y is transverse to X nearby.

Theorem Trotman 1976

If Y is semianalytic then (t1) is equivalent to (a).

In the above result one needs non-direct transversals. In theresults below, we always restrict to direct transversals


(t1) is equivalent to (h1).

Trotman-Wilson 1999

For subanalytic strata (tk ) is equivalent to the finiteness of thenumber of topological types of germs at y of S ∩ X for S a Ck

transversal to Y (1 ≤ k ≤ ∞).









Trotman-Wilson 1999











Trotman-Wilson 1999











Trotman-Wilson 1999











Trotman-Wilson 1999




The proofs use the“Grassmann blowup": like the regularblowup, but with lines through y replaced with all linearsubspaces through y of dimension equal to the codimension ofY .

Kuo-Trotman 1988, Trotman-Wilson 1999

(X , Y ) is (tk )-regular at 0 ∈ Y iff its Grassmann blowup (X , Y )is (tk−1)-regular at every point of Y (k ≥ 1).

A definition of (tk ) is given so that (t0) is equivalent to (w)So (t1) =⇒ (h1) follows from blowup and then applying theVerdier Isotopy Theorem.










A definition of (tk ) is given so that (t0) is equivalent to (w)

So (t1) =⇒ (h1) follows from blowup and then applying theVerdier Isotopy Theorem.







Koike-Kucharz Example

Let Z = {x3 − 3xy5 + ty6 = 0}, with Y the t-axis andX = Z − Y . (X , Y ) is (t2), not (t1). There are two topologicaltypes of germs at 0 of intersections S ∩ X where S is a C2

submfd transverse to Y at 0. However the number oftopological types of such germs for S of class C1 isuncountable.

Trotman-Wilson 1999

Also there is a theory of (tk−) such that (t1−) is essentially(a) holding for all sequences going to 0 not tangent to Y .The (tk ) and (tk−)-conditions were formulated for jets oftransversalsThe (tk ) and (tk−)-conditions were then used tocharacterize sufficiency of jets of functions, generalizingtheorems of Bochnak, Kuo, Lu and others.





Trotman-Wilson 1999






Trotman-Wilson 1999

Also there is a theory of (tk−) such that (t1−) is essentially(a) holding for all sequences going to 0 not tangent to Y .

The (tk ) and (tk−)-conditions were formulated for jets oftransversalsThe (tk ) and (tk−)-conditions were then used tocharacterize sufficiency of jets of functions, generalizingtheorems of Bochnak, Kuo, Lu and others.





Trotman-Wilson 1999

Also there is a theory of (tk−) such that (t1−) is essentially(a) holding for all sequences going to 0 not tangent to Y .The (tk ) and (tk−)-conditions were formulated for jets oftransversals

The (tk ) and (tk−)-conditions were then used tocharacterize sufficiency of jets of functions, generalizingtheorems of Bochnak, Kuo, Lu and others.





Trotman-Wilson 1999



Gaffney-Trotman-Wilson 2009

We expressed (tk ) in terms of integral closure of modules,giving more algebraic techniques for computations. In thecomplex analytic case, (tk ) is characterized by the genericity ofthe multiplicity of a certain submodule.


Normal cone CY Z of a set Z to a submfd Y ; p theprojection

Theorem (Hironaka in analytic (b) case, Trotman-Orro 2002generalize to smooth (a) + (re))

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 isz = f (x , y) = x − x ln(y +

√x2 + y2)/ ln(x).




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



√x2 + y2)/ ln(x).







√x2 + y2)/ ln(x).





Orro-Trotman 2002 show the Theorem fails for (a)-regularity.

Trotman-Wilson 2006 show that it also fails in thenon-polynomial bounded o-minimal category; our example isz = f (x , y) = x − x ln(y +

√x2 + y2)/ ln(x).







√x2 + y2)/ ln(x).


Nash fiber = limits of tangent planes at a singular point

Kwiecinski-Trotman 1995Every continuum can be realized as the Nash fiber of a Whitneystratified set.


Nash fiber = limits of tangent planes at a singular point

Kwiecinski-Trotman 1995Every continuum can be realized as the Nash fiber of a Whitneystratified set.


David Trotman’s other research

Reducing the Arbitrariness of Our Description of BalineseDance Or Balinese Dance as a Descriptive Text in NeonatalMotricity Author: David Trotman, 1982, 36 pages


David Trotman’s other research

Reducing the Arbitrariness of Our Description of BalineseDance Or Balinese Dance as a Descriptive Text in NeonatalMotricity Author: David Trotman, 1982, 36 pages


Trotman Style

Powerful geometric intuition

Creation of insightful examplesImpressive knowledge of what has been done and is beingdone in many areas, allowing him to make contributions tothe real differentiable, subanalytic, o-minimal and complexanalytic versions of singularity theory, but also to suchfields as microlocal analysis and robotics.Writer of interesting surveysGreat friend and collaboratorProducer of many active and appreciative students (whichis why we are here)


Trotman Style

Powerful geometric intuitionCreation of insightful examples

Impressive knowledge of what has been done and is beingdone in many areas, allowing him to make contributions tothe real differentiable, subanalytic, o-minimal and complexanalytic versions of singularity theory, but also to suchfields as microlocal analysis and robotics.Writer of interesting surveysGreat friend and collaboratorProducer of many active and appreciative students (whichis why we are here)


Trotman Style

Powerful geometric intuitionCreation of insightful examplesImpressive knowledge of what has been done and is beingdone in many areas, allowing him to make contributions tothe real differentiable, subanalytic, o-minimal and complexanalytic versions of singularity theory, but also to suchfields as microlocal analysis and robotics.

Writer of interesting surveysGreat friend and collaboratorProducer of many active and appreciative students (whichis why we are here)


Trotman Style

Powerful geometric intuitionCreation of insightful examplesImpressive knowledge of what has been done and is beingdone in many areas, allowing him to make contributions tothe real differentiable, subanalytic, o-minimal and complexanalytic versions of singularity theory, but also to suchfields as microlocal analysis and robotics.Writer of interesting surveys

Great friend and collaboratorProducer of many active and appreciative students (whichis why we are here)


Trotman Style

Powerful geometric intuitionCreation of insightful examplesImpressive knowledge of what has been done and is beingdone in many areas, allowing him to make contributions tothe real differentiable, subanalytic, o-minimal and complexanalytic versions of singularity theory, but also to suchfields as microlocal analysis and robotics.Writer of interesting surveysGreat friend and collaborator

Producer of many active and appreciative students (whichis why we are here)


Trotman Style

Powerful geometric intuitionCreation of insightful examplesImpressive knowledge of what has been done and is beingdone in many areas, allowing him to make contributions tothe real differentiable, subanalytic, o-minimal and complexanalytic versions of singularity theory, but also to suchfields as microlocal analysis and robotics.Writer of interesting surveysGreat friend and collaboratorProducer of many active and appreciative students

(whichis why we are here)


Trotman Style

Powerful geometric intuitionCreation of insightful examplesImpressive knowledge of what has been done and is beingdone in many areas, allowing him to make contributions tothe real differentiable, subanalytic, o-minimal and complexanalytic versions of singularity theory, but also to suchfields as microlocal analysis and robotics.Writer of interesting surveysGreat friend and collaboratorProducer of many active and appreciative students (whichis why we are here)


David Trotman—Congratulations!


Documents

The Work of David Trotman - Aix-Marseille Universitymurolo/DT/Talks/Wilson.pdf · Bekka, Kuo, Li Pei Xin, Kwiecinski, Risler, Wilson, Murolo,´ ... L. Wilson The Work of David Trotman