Heegaard Floer homologies and rational cuspidal curves

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Heegaard Floer homologies and rational cuspidal curves

joint with Ch. Livingston

Maciej Borodzik

www.mimuw.edu.pl/~mcboro

Institute of Mathematics, University of Warsaw

Edinburgh, September 2013

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Semigroups of singular points

For a singularityx^p− y^q =0withp, q coprime, the semigroup is generated bypandq.

Ifp = 4,q = 7, the semigroup is

S_4,7:= (0, 4, 7, 8, 11, 12, 14, 15, 16, 18, 19, 20, 21, . . .). Thegap sequenceisG_4,7 = {1, 2, 3, 5, 6, 9, 10, 13, 17}. We have#G_4,7= µ/2andmax{x ∈ G_4,7} = 17 = µ − 1. Thegap functionis defined as

I(m) := #{x ∈ Z, x ≥ m, x 6∈ S4,7}. We have

I_4,7(5) = #{5, 6, 9, 10, 13, 17} = 6.

AlwaysI(0) = µ/2,I(x ) = 0forx ≥ µandI(−n) = n + µ/2 forn > 0.

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Semigroups of singular points

S_4,7:= (0, 4, 7, 8, 11, 12, 14, 15, 16, 18, 19, 20, 21, . . .).

Thegap sequenceisG_4,7 = {1, 2, 3, 5, 6, 9, 10, 13, 17}. We have#G_4,7= µ/2andmax{x ∈ G_4,7} = 17 = µ − 1. Thegap functionis defined as

I(m) := #{x ∈ Z, x ≥ m, x 6∈ S4,7}. We have

I_4,7(5) = #{5, 6, 9, 10, 13, 17} = 6.

Maciej Borodzik Heegaard Floer homologies and rational cuspidal curves

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Semigroups of singular points

S_4,7:= (0, 4, 7, 8, 11, 12, 14, 15, 16, 18, 19, 20, 21, . . .).

Thegap sequenceisG_4,7 = {1, 2, 3, 5, 6, 9, 10, 13, 17}.

We have#G_4,7= µ/2andmax{x ∈ G_4,7} = 17 = µ − 1. Thegap functionis defined as

I(m) := #{x ∈ Z, x ≥ m, x 6∈ S4,7}. We have

I_4,7(5) = #{5, 6, 9, 10, 13, 17} = 6.

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Semigroups of singular points

S_4,7:= (0, 4, 7, 8, 11, 12, 14, 15, 16, 18, 19, 20, 21, . . .).

We have#G_4,7= µ/2andmax{x ∈ G_4,7} = 17 = µ − 1.

Thegap functionis defined as

I(m) := #{x ∈ Z, x ≥ m, x 6∈ S4,7}. We have

I_4,7(5) = #{5, 6, 9, 10, 13, 17} = 6.

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Semigroups of singular points

S_4,7:= (0, 4, 7, 8, 11, 12, 14, 15, 16, 18, 19, 20, 21, . . .).

this is a special property of semigroups of singular points!

I(m) := #{x ∈ Z, x ≥ m, x 6∈ S4,7}. We have

I_4,7(5) = #{5, 6, 9, 10, 13, 17} = 6.

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Semigroups of singular points

S_4,7:= (0, 4, 7, 8, 11, 12, 14, 15, 16, 18, 19, 20, 21, . . .).

I(m) := #{x ∈ Z, x ≥ m, x 6∈ S4,7}.

We have

I_4,7(5) = #{5, 6, 9, 10, 13, 17} = 6.

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Semigroups of singular points

S_4,7:= (0, 4, 7, 8, 11, 12, 14, 15, 16, 18, 19, 20, 21, . . .).

Thegap sequenceisG_4,7 = {1, 2, 3,5, 6, 9, 10, 13, 17}.

I(m) := #{x ∈ Z, x ≥ m, x 6∈ S4,7}.

We have

I (5) = #{5, 6, 9, 10, 13, 17} = 6.

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Semigroups of singular points

S_4,7:= (0, 4, 7, 8, 11, 12, 14, 15, 16, 18, 19, 20, 21, . . .).

I(m) := #{x ∈ Z, x ≥ m, x 6∈ S4,7}.

We have

I_4,7(5) = #{5, 6, 9, 10, 13, 17} = 6.

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The Alexander polynomial

For a semigroupSwith a gap sequenceGwe define

∆_S(t) = 1 + (t − 1)X

j∈G

t^j.

For the semigroupS_4,7, the gap sequence is {1, 2, 3, 5, 6, 9, 10, 13, 17}, so we have

∆_4,7(t) = 1 + (t − 1)

t + t²+t³+t⁵+t⁶+t⁹+t¹⁰+t¹³+t¹⁷ or:

∆_4,7 =1 − t + t⁴− t⁵+t⁷− t⁹+t¹¹− t¹³+t¹⁴− t¹⁷+t¹⁸. This is theAlexander polynomial of the knot of the singularity.

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The Alexander polynomial

∆_S(t) = 1 + (t − 1)X

j∈G

t^j.

∆_4,7(t) = 1 + (t − 1)

t + t²+t³+t⁵+t⁶+t⁹+t¹⁰+t¹³+t¹⁷ or:

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The Alexander polynomial

∆_S(t) = 1 + (t − 1)X

j∈G

t^j.

∆_4,7(t) = 1 + (t − 1)

t + t²+t³+t⁵+t⁶+t⁹+t¹⁰+t¹³+t¹⁷ or:

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The Alexander polynomial

∆_S(t) = 1 + (t − 1)X

j∈G

t^j.

∆_4,7(t) = 1 + (t − 1)

t + t²+t³+t⁵+t⁶+t⁹+t¹⁰+t¹³+t¹⁷

or:

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The Alexander polynomial

∆_S(t) = 1 + (t − 1)X

j∈G

t^j.

∆_4,7(t) = 1 + (t − 1)

t + t²+t³+t⁵+t⁶+t⁹+t¹⁰+t¹³+t¹⁷ or:

∆_4,7 =1 − t + t⁴− t⁵+t⁷− t⁹+t¹¹− t¹³+t¹⁴− t¹⁷+t¹⁸.

This is theAlexander polynomial of the knot of the singularity.

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The Alexander polynomial

∆_S(t) = 1 + (t − 1)X

j∈G

t^j.

∆_4,7(t) = 1 + (t − 1)

t + t²+t³+t⁵+t⁶+t⁹+t¹⁰+t¹³+t¹⁷ or:

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The staircase

∆_4,7=t¹⁸− t¹⁷+t¹⁴− t¹³+t¹¹− t⁹+t⁷− t⁵+t⁴− t + 1.

9 = g(T_4,7) 18 − 17 = 1 17 − 14 = 3 14 − 13 = 1 13 − 11 = 2 . . . and so on Symmetry reflects

symmetry of∆

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The staircase

∆_4,7=t¹⁸− t¹⁷+t¹⁴− t¹³+t¹¹− t⁹+t⁷− t⁵+t⁴− t + 1.

(0,9) 9 = 18/2

18 − 17 = 1 17 − 14 = 3 14 − 13 = 1 13 − 11 = 2 . . . and so on Symmetry reflects

symmetry of∆

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The staircase

∆_4,7=t¹⁸− t¹⁷+t¹⁴− t¹³+t¹¹− t⁹+t⁷− t⁵+t⁴− t + 1.

9 = g(T_4,7)

18 − 17 = 1 17 − 14 = 3 14 − 13 = 1 13 − 11 = 2 . . . and so on Symmetry reflects

symmetry of∆

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The staircase

∆_4,7=t¹⁸−t¹⁷+t¹⁴− t¹³+t¹¹− t⁹+t⁷− t⁵+t⁴− t + 1.

9 = g(T_4,7) 18 − 17 = 1

17 − 14 = 3 14 − 13 = 1 13 − 11 = 2 . . . and so on Symmetry reflects

symmetry of∆

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The staircase

∆_4,7=t¹⁸−t¹⁷+t¹⁴− t¹³+t¹¹− t⁹+t⁷− t⁵+t⁴− t + 1.

9 = g(T_4,7) 18 − 17 = 1 17 − 14 = 3

14 − 13 = 1 13 − 11 = 2 . . . and so on Symmetry reflects

symmetry of∆

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The staircase

∆_4,7=t¹⁸− t¹⁷+t¹⁴−t¹³+t¹¹− t⁹+t⁷− t⁵+t⁴− t + 1.

9 = g(T_4,7) 18 − 17 = 1 17 − 14 = 3 14 − 13 = 1

13 − 11 = 2 . . . and so on Symmetry reflects

symmetry of∆

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The staircase

∆_4,7=t¹⁸− t¹⁷+t¹⁴−t¹³+t¹¹− t⁹+t⁷− t⁵+t⁴− t + 1.

9 = g(T_4,7) 18 − 17 = 1 17 − 14 = 3 14 − 13 = 1 13 − 11 = 2

. . . and so on Symmetry reflects

symmetry of∆

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The staircase

∆_4,7=t¹⁸− t¹⁷+t¹⁴− t¹³+t¹¹− t⁹+t⁷− t⁵+t⁴− t + 1.

9 = g(T_4,7) 18 − 17 = 1 17 − 14 = 3 14 − 13 = 1 13 − 11 = 2 . . . and so on

Symmetry reflects

symmetry of∆

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The staircase

∆_4,7=t¹⁸− t¹⁷+t¹⁴− t¹³+t¹¹− t⁹+t⁷− t⁵+t⁴− t + 1.

9 = g(T_4,7) 18 − 17 = 1 17 − 14 = 3 14 − 13 = 1 13 − 11 = 2 . . . and so on Symmetry reflects

symmetry of∆

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The staircase complex

PlaceZ2for each vertex.

Differential is given by lines as

depicted. Type A vertices. Type B vertices. Bifiltrationis given by coordinates. Absolute grading of a type A vertex is0, of type B is1.

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The staircase complex

depicted. Type A vertices. Type B vertices. Bifiltrationis given by coordinates. Absolute grading of a type A vertex is0, of type B is1.

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The staircase complex

depicted.

Type A vertices. Type B vertices. Bifiltrationis given by coordinates. Absolute grading of a type A vertex is0, of type B is1.

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The staircase complex

depicted.

Type A vertices.

Type B vertices. Bifiltrationis given by coordinates. Absolute grading of a type A vertex is0, of type B is1.

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The staircase complex

depicted.

Type A vertices.

Type B vertices.

Bifiltrationis given by coordinates. Absolute grading of a type A vertex is0, of type B is1.

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The staircase complex

depicted.

Type A vertices.

Type B vertices.

Bifiltrationis given by coordinates.

Absolute grading of a type A vertex is0, of type B is1.

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The staircase complex

depicted.

Type A vertices.

Type B vertices.

Bifiltrationis given by coordinates.

Absolute grading of a type A vertex is0, of type B is1.

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Now there comes something really scary

We will tensor the staircase complex byZ2[U, U⁻¹].

Are you ready for the challenge?

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Now there comes something really scary

We will tensor the staircase complex byZ2[U, U⁻¹].

Are you ready for the challenge?

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Tensoring

TensorSt(K )by Z2[U, U⁻¹].

U changes the filtration level by (−1, −1)and the absolute grading by

−2.

The resulting complex is

CFK^∞(K )ifK is an algebraic knot.

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Tensoring

−2.

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Tensoring

(0) (−2) (−4) (−6)

−2.

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Tensoring

−2.

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The function J(m)

J(m) = −2 min

(v₁,v₂)∈Vert_Amax(v₁,v₂− m)

m ∈ Z. Here m = 1.

The subcomplex C(i < 0, j < m). Look at the quotientC+. DefineJ(m)as the minimal absolute grading of an element non-trivial in homology of the quotient.

J(m) =

−2I(m + g).

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The function J(m)

J(m) = −2 min

m ∈ Z. Here m = 1.

The subcomplex C(i < 0, j < m).

Look at the quotientC+.

DefineJ(m)as the minimal absolute grading of an element non-trivial in homology of the quotient.

J(m) =

−2I(m + g).

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The function J(m)

J(m) = −2 min

m ∈ Z. Here m = 1.

Look at the quotientC+. DefineJ(m)as the minimal absolute grading of an element non-trivial in homology of the quotient.

J(m) =

−2I(m + g).

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The function J(m)

(−6)

J(m) = −2 min

m ∈ Z. Here m = 1.

J(m) =

−2I(m + g).

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The function J(m)

J(m) = −2 min

m ∈ Z. Here m = 1.

J(m) =

−2I(m + g).

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The function J(m)

J(m) = −2 min

m ∈ Z. Here m = 1.

J(m) =

−2I(m + g).

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The function J(m)

m ∈ Z. Here m = 1.

J(m) =

−2I(m + g).

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The function J(m)

J(m) = −2 min

m ∈ Z. Here m = 1.

J(m) =

−2I(m + g).

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Question

Now you may start wondering:

Oh where, oh where has the true

mathematics gone?

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Question

Now you may start wondering:

Oh where, oh where has the true mathematics gone?

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d -invariants of Ozsváth and Szabó

Proposition

LetK be an L–space knot.

Letq > 2g(K )and m ∈ [−q/2, q/2]. Then

d (S_q³(K ), s_m) = (q − 2m)²− q

4q − 2I(m + g).

Theorem

IfM³bounds asmooth negative definitemanifoldW⁴andsis a spin^c structure onM³, that is a restriction of aspin^c structuret onW, then

d (M, s) ≥ c²₁(t) −2χ(W ) − 3σ(W )

4 .

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d -invariants of Ozsváth and Szabó

Proposition

LetK be analgebraic knot.

Letq > 2g(K )and m ∈ [−q/2, q/2]. Then

d (S_q³(K ), s_m) = (q − 2m)²− q

4q − 2I(m + g).

Theorem

d (M, s) ≥ c²₁(t) −2χ(W ) − 3σ(W )

4 .

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d -invariants of Ozsváth and Szabó

Proposition

LetK be an L–space knot. Letq > 2g(K )and m ∈ [−q/2, q/2]. Then

d (S³_q(K ), s_m) = (q − 2m)²− q

4q +J(m).

Theorem

d (M, s) ≥ c²₁(t) −2χ(W ) − 3σ(W )

4 .

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d -invariants of Ozsváth and Szabó

Proposition

LetK be an L–space knot. Letq > 2g(K )and m ∈ [−q/2, q/2]. Then

d (S_q³(K ), s_m) = (q − 2m)²− q

4q − 2I(m + g).

Theorem

d (M, s) ≥ c₁²(t) −2χ(W ) − 3σ(W )

4 .

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Applications. The FLMN conjecture.

C degreed rational cuspidal curve inCP².

N neighbourhood ofC,M = ∂N,W = CP²\ N.W is rational homology ball.

ThenM = S_d³2(K ),K is connected sum of links of singularities.

Suppose thatCis unicuspidal.

Thespin^c structures that extend overW are those with m = kd ford odd orkd /2ford even,k ∈ Z.

By Ozsváth and Szabó,d–invariant must vanish fors_m. Theorem (—,Livingston, 2013)

IfIis the gap function associated with the single singular point, j = 0, . . . , d − 3. Then

I(jd + 1) = (d − j − 1)(d − j − 2)

2 .

Generalizations apply for many singular points.

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Applications. The FLMN conjecture.

I(jd + 1) = (d − j − 1)(d − j − 2)

2 .

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Applications. The FLMN conjecture.

I(jd + 1) = (d − j − 1)(d − j − 2)

2 .

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Applications. The FLMN conjecture.

ThenM = S_d³2(K ),K is connected sum of links of singularities. Suppose thatCis unicuspidal.

I(jd + 1) = (d − j − 1)(d − j − 2)

2 .

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Applications. The FLMN conjecture.

I(jd + 1) = (d − j − 1)(d − j − 2)

2 .

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Applications. The FLMN conjecture.

By Ozsváth and Szabó,d–invariant must vanish fors_m.

Theorem (—,Livingston, 2013)

I(jd + 1) = (d − j − 1)(d − j − 2)

2 .

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Applications. The FLMN conjecture.

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Applications. The FLMN conjecture.

I(jd + 1) = (d − j − 1)(d − j − 2)

2 .

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Applications. Semigroup semicontinuity.

K₁,K₂two knots. Suppose there is a PSI cobordism from K₁toK₂withk double points.

Wqis a cobordism betweenS_q³(K₁)andS_q+4k³ (K₂). Theorem (—,Livingston 2013)

IfK₁andK₂are two L–space knots,g₁andg₂their genera,I₁ andI₂gap functions, then for anym ∈ Z:

I₂(m + g₂+k ) ≤ I₁(m + g₁). Example

SetK₁unknot,K₂=T_p,q,m = 0,k = g₂− 1 = ^{(p−1)(q−1)}₂ − 1. ThenI₂(2g₂− 1) = 1 6≤ I1(g₁) =0.

This detects the unknotting number of torus knots.

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Applications. Semigroup semicontinuity.

Wqis a cobordism betweenS_q³(K₁)andS_q+4k³ (K₂).

Theorem (—,Livingston 2013)

I₂(m + g₂+k ) ≤ I₁(m + g₁). Example

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Applications. Semigroup semicontinuity.

I₂(m + g₂+k ) ≤ I₁(m + g₁).

Example

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Applications. Semigroup semicontinuity.

I₂(m + g₂+k ) ≤ I₁(m + g₁).

Example

SetK₁unknot,K₂=T_p,q,m = 0,k = g₂− 1 = ^{(p−1)(q−1)}₂ − 1.

ThenI₂(2g₂− 1) = 1 6≤ I₁(g₁) =0.

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Applications. Semigroup semicontinuity.

I₂(m + g₂+k ) ≤ I₁(m + g₁).

Example

(p−1)(q−1)

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Applications. Semigroup semicontinuity.

In the case ofδ–constantdeformation,k = g₂− g1.

We get#S₂∩ [0, m) ≤ #S₁∩ [0, m). Semigroup semicontinuity property.

First obtained by Gorsky–Némethi in January 2013 (withoutδ–constant assumption).

Another proof by Javier Fernandez de Bobadilla. In general weak, but it usessmoothstructure, unlike semicontinuity of spectrum.

(6; 7)cannot be perturbed to(4; 9), even though the spectrum allows it.

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Applications. Semigroup semicontinuity.

In the case ofδ–constantdeformation,k = g₂− g1. We get#S₂∩ [0, m) ≤ #S₁∩ [0, m). Semigroup semicontinuity property.

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Applications. Semigroup semicontinuity.

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Applications. Semigroup semicontinuity.

Another proof by Javier Fernandez de Bobadilla.

In general weak, but it usessmoothstructure, unlike semicontinuity of spectrum.

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Applications. Semigroup semicontinuity.

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Applications. Semigroup semicontinuity.

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. . . you must have been waiting long time for this slide.

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. . . you must have been waiting long time for this slide.

Thank you!

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. . . you must have been waiting long time for this slide.

Thank you!

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. . . you must have been waiting long time for this slide.

Thank you!

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. . . you must have been waiting long time for this slide.

Thank you!

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. . . you must have been waiting long time for this slide.

Thank you!

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Perspectives

Generalize for curves with higher genus (joint project with Ch. Livingston).

Generalize for curves in Hirzebruch surfaces (joint project with K. Moe).

Relate staircases to lattice homology by András Némethi.

Can one classify all the rational unicuspidal curves inCP²? For many cusps other tools are more useful.