Positivity results for cluster algebras from surfaces

Transcription

1 Positivity esuts fo custe agebas fom sufaces Gegg Musike (MSRI/MIT) (Joint wok with Raf Schiffe (Univesity of Connecticut) and Lauen Wiiams (Univesity of Caifonia, Bekeey)) AMS 009 Easten Sectiona Octobe, 009 http//math.mit.edu/ musike/custesufaceams.pdf Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

2 Outine Intoduction: the Lauent phenomenon, and the positivity conjectue of Fomin-Zeevinsky. Fomin-Shapio-Thuston s theoy of custe agebas aising fom tianguated sufaces. Gaph theoetic constuction fo sufaces with o without punctues (joint wok with Schiffe and Wiiams). Exampes of this constuction. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

3 Intoduction to Custe Agebas In the ate 990 s: Fomin and Zeevinsky wee studying tota positivity and canonica bases of agebaic goups. They noticed ecuing combinatoia and agebaic stuctues. Led them to define custe agebas, which have now been inked to quive epesentations, Poisson geomety, Teichmüe theoy, topica geomety, Lie goups, and othe topics. Custe agebas ae a cetain cass of commutative ings which have a distinguished set of geneatos that ae gouped into oveapping subsets, caed custes, each having the same cadinaity. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

4 What is a Custe Ageba? Definition (Segey Fomin and Andei Zeevinsky 00) A custe ageba A (of geometic type) is a subageba of k(x,...,x n,x n+,...,x n+m ) constucted custe by custe by cetain exchange eations. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

5 What is a Custe Ageba? Definition (Segey Fomin and Andei Zeevinsky 00) A custe ageba A (of geometic type) is a subageba of k(x,...,x n,x n+,...,x n+m ) constucted custe by custe by cetain exchange eations. Geneatos: Specify an initia finite set of them, a Custe, {x,x,...,x n+m }. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

6 What is a Custe Ageba? Definition (Segey Fomin and Andei Zeevinsky 00) A custe ageba A (of geometic type) is a subageba of k(x,...,x n,x n+,...,x n+m ) constucted custe by custe by cetain exchange eations. Geneatos: Specify an initia finite set of them, a Custe, {x,x,...,x n+m }. Constuct the est via Binomia Exchange Reations: x α x α = x d+ i γ i + x d i γ i. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

7 What is a Custe Ageba? Definition (Segey Fomin and Andei Zeevinsky 00) A custe ageba A (of geometic type) is a subageba of k(x,...,x n,x n+,...,x n+m ) constucted custe by custe by cetain exchange eations. Geneatos: Specify an initia finite set of them, a Custe, {x,x,...,x n+m }. Constuct the est via Binomia Exchange Reations: x α x α = x d+ i γ i + x d i γ i. The set of a such geneatos ae known as Custe Vaiabes, and the initia patten B of exchange eations detemines the Seed. Reations: Induced by the Binomia Exchange Reations. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

8 Finite Mutation Type and Finite Type A pioi, get a tee of exchanges. In pactice, often get identifications among seeds. In exteme cases, get ony a finite numbe of exchange pattens as tee coses up on itsef. Such custe agebas caed finite mutation type. Sometimes ony a finite numbe of custes. Caed finite type. Finite type = Finite mutation type. Theoem. (FZ 00) Finite type custe agebas can be descibed via the Catan-Kiing cassification of Lie agebas. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

9 Custe Expansions and the Lauent Phenomenon Exampe. Let A be the custe ageba defined by the initia custe {x,x,x,y,y,y } and the initia exchange patten x x = y + x, x x = x x y +, x x = y + x Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

10 Custe Expansions and the Lauent Phenomenon Exampe. Let A be the custe ageba defined by the initia custe {x,x,x,y,y,y } and the initia exchange patten x x = y + x, x x = x x y +, x x = y + x A is of finite type, type A and coesponds to a tianguated hexagon. { x,x,x, y + x, x x y +, y + x, x x y y + y + x, x x x x x x x y y + y + x x x, x x y y y + y y + x y + x y + x x x x The y i s ae known as pincipa coefficients. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, / }.

11 The Positivity Conjectue of Fomin and Zeevinsky Theoem. (The Lauent Phenomenon FZ 00) Fo any custe ageba defined by initia seed ({x,x,...,x n+m },B), a custe vaiabes of A(B) ae Lauent poynomias in {x,x,...,x n+m } (with no coefficient x n+,...,x n+m in the denominato). Because of the Lauent Phenomenon, any custe vaiabe x α can be expessed as Pα(x,...,x n+m) whee P α Z[x,...,x n+m ] and the α i s Z. x α xn αn Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

12 The Positivity Conjectue of Fomin and Zeevinsky Theoem. (The Lauent Phenomenon FZ 00) Fo any custe ageba defined by initia seed ({x,x,...,x n+m },B), a custe vaiabes of A(B) ae Lauent poynomias in {x,x,...,x n+m } (with no coefficient x n+,...,x n+m in the denominato). Because of the Lauent Phenomenon, any custe vaiabe x α can be expessed as Pα(x,...,x n+m) whee P α Z[x,...,x n+m ] and the α i s Z. x α xn αn Conjectue. (FZ 00) Fo any custe vaiabe x α and any initia seed (i.e. initia custe {x,...,x n+m } and initia exchange patten B), the poynomia P α (x,...,x n ) has nonnegative intege coefficients. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

13 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 00] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G,n+ ). Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

14 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 00] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G,n+ ). [FZ 00] poved positivity fo finite type with bipatite seed. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

15 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 00] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G,n+ ). [FZ 00] poved positivity fo finite type with bipatite seed. [M-Popp 00, Sheman-Zeevinsky 00] poved positivity fo ank two affine custe agebas. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

16 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 00] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G,n+ ). [FZ 00] poved positivity fo finite type with bipatite seed. [M-Popp 00, Sheman-Zeevinsky 00] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 009]. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

17 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 00] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G,n+ ). [FZ 00] poved positivity fo finite type with bipatite seed. [M-Popp 00, Sheman-Zeevinsky 00] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 009]. Wok towads positivity fo acycic seeds [Cadeo-Reineke 006]. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

18 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 00] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G,n+ ). [FZ 00] poved positivity fo finite type with bipatite seed. [M-Popp 00, Sheman-Zeevinsky 00] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 009]. Wok towads positivity fo acycic seeds [Cadeo-Reineke 006]. Positivity fo custe agebas incuding a bipatite seed (which is necessaiy acycic) by [Nakajima 009]. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

19 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 00] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G,n+ ). [FZ 00] poved positivity fo finite type with bipatite seed. [M-Popp 00, Sheman-Zeevinsky 00] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 009]. Wok towads positivity fo acycic seeds [Cadeo-Reineke 006]. Positivity fo custe agebas incuding a bipatite seed (which is necessaiy acycic) by [Nakajima 009]. Custe agebas aising fom unpunctued sufaces [Schiffe-Thomas 00, Schiffe 008], geneaizing Tais mode of Cao-Pice. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

20 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 00] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G,n+ ). [FZ 00] poved positivity fo finite type with bipatite seed. [M-Popp 00, Sheman-Zeevinsky 00] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 009]. Wok towads positivity fo acycic seeds [Cadeo-Reineke 006]. Positivity fo custe agebas incuding a bipatite seed (which is necessaiy acycic) by [Nakajima 009]. Custe agebas aising fom unpunctued sufaces [Schiffe-Thomas 00, Schiffe 008], geneaizing Tais mode of Cao-Pice. Gaph theoetic intepetation fo unpunctued sufaces [M-Schiffe 008]. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

21 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 00] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G,n+ ). [FZ 00] poved positivity fo finite type with bipatite seed. [M-Popp 00, Sheman-Zeevinsky 00] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 009]. Wok towads positivity fo acycic seeds [Cadeo-Reineke 006]. Positivity fo custe agebas incuding a bipatite seed (which is necessaiy acycic) by [Nakajima 009]. Custe agebas aising fom unpunctued sufaces [Schiffe-Thomas 00, Schiffe 008], geneaizing Tais mode of Cao-Pice. Gaph theoetic intepetation fo unpunctued sufaces [M-Schiffe 008]. Positivity fo abitay sufaces [M-Schiffe-Wiiams 009]. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

22 Main Theoem Theoem. (Positivity fo custe agebas fom sufaces MSW 009) Let A be any custe ageba aising fom a suface (with o without punctues), whee the coefficient system is of geometic type, and et Σ be any initia seed. Then the Lauent expansion of evey custe vaiabe with espect to the seed Σ has non-negative coefficients. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

23 Main Theoem Theoem. (Positivity fo custe agebas fom sufaces MSW 009) Let A be any custe ageba aising fom a suface (with o without punctues), whee the coefficient system is of geometic type, and et Σ be any initia seed. Then the Lauent expansion of evey custe vaiabe with espect to the seed Σ has non-negative coefficients. We pove this theoem by exhibiting a gaph theoetic intepetation fo the Lauent expansions coesponding to custe vaiabes. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

24 Main Theoem Theoem. (Positivity fo custe agebas fom sufaces MSW 009) Let A be any custe ageba aising fom a suface (with o without punctues), whee the coefficient system is of geometic type, and et Σ be any initia seed. Then the Lauent expansion of evey custe vaiabe with espect to the seed Σ has non-negative coefficients. We pove this theoem by exhibiting a gaph theoetic intepetation fo the Lauent expansions coesponding to custe vaiabes. Due to wok of Feikson-Shapio-Tumakin, we get Cooay. Positivity fo any seed, fo a but skew-symmetic custe agebas of finite mutation type. (Rank two skew-symmetic cases by Cadeo-Reineke) Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

25 Custe Agebas of Tianguated Sufaces We foow (Fomin-Shapio-Thuston), based on eaie wok of Fock-Gonchaov and Gekhtman-Shapio-Vainshtein. We have a suface S with a set of maked points M. (If P M is in the inteio of S, i.e. S \ δs, then P is known an a punctue). Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

26 Custe Agebas of Tianguated Sufaces We foow (Fomin-Shapio-Thuston), based on eaie wok of Fock-Gonchaov and Gekhtman-Shapio-Vainshtein. We have a suface S with a set of maked points M. (If P M is in the inteio of S, i.e. S \ δs, then P is known an a punctue). An ac γ satisfies (we cae about acs up to isotopy) The endpoints of γ ae in M. γ does not coss itsef. except fo the endpoints, γ is disjoint fom M and the bounday of S. γ does not cut out an unpunctued monogon o unpunctued bigon. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

27 Custe Agebas of Tianguated Sufaces We foow (Fomin-Shapio-Thuston), based on eaie wok of Fock-Gonchaov and Gekhtman-Shapio-Vainshtein. We have a suface S with a set of maked points M. (If P M is in the inteio of S, i.e. S \ δs, then P is known an a punctue). An ac γ satisfies (we cae about acs up to isotopy) The endpoints of γ ae in M. γ does not coss itsef. except fo the endpoints, γ is disjoint fom M and the bounday of S. γ does not cut out an unpunctued monogon o unpunctued bigon. Seed Tianguation T = {,,..., n } Custe Vaiabe Ac γ (x i i T) Custe Mutation Ptoemy Exchanges (Fipping Diagonas). Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

28 Exampe of Hexagon Conside the tianguated hexagon (S,M) with tianguation T H γ 6 6 x x = y (x x 9 ) + x (x 8 ) Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

29 Exampe of Hexagon Conside the tianguated hexagon (S,M) with tianguation T H x x = y (x x 9 ) + x (x 8 ) x x = y y x (x 9 ) + x (x ) Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

30 Exampe of Hexagon Conside the tianguated hexagon (S,M) with tianguation T H γ = 6 6 x x = y (x x 9 ) + x (x 8 ) x x = y y x (x 9 ) + x (x ) x x = y x (x 6 ) + x (x ) Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

31 Exampe of Hexagon (continued) 9 8 γ By using the Ptoemy eations on,, then, we obtain x = x γ = ( x x x x (x x 8 ) + y x (x x x 9 ) + y x (x x 6 x 8 ) ) + y y (x x 6 x x 9 ) + y y y x x (x 6 x 9 ). 6 Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

32 Exampe of Hexagon (continued) Conside the gaph G TH,γ = G TH,γ has five pefect matchings (x,x,...,x 9 = ): (x 9 )x x (x 6 ), (x 9 x x x 6 ), x (x 8 )(x x 6 ), (x 9 x )x (x ), x (x 8 )x (x ). A pefect matching M E is a set of distinguished edges so that evey vetex of V is coveed exacty once. The weight of a matching M is the poduct of the weights of the constituent edges, i.e. x(m) = e M x(e). Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

33 Exampe of Hexagon (continued) Conside the gaph G TH,γ = G TH,γ has five pefect matchings (x,x,...,x 9 = ): (x 9 )x x (x 6 ), (x 9 x x x 6 ), x (x 8 )(x x 6 ), (x 9 x )x (x ), x x y y y +y y +x y +x y +x x (x 8 )x (x ). x x x These five monomias exacty match those appeaing in the numeato of the expansion of x γ. The denominato of x x x coesponds to the abes of the thee ties. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

34 A Gaph Theoetic Appoach Fo evey tianguation T (in a suface with o without punctues) and an odinay ac γ though odinay tianges, we constuct a snake gaph G T,γ such that pefect matching M of G x γ = T,γ x(m)y(m) x e (T,γ) x e (T,γ) xn en(t,γ). x γ is custe vaiabe (coesp. to γ w..t. seed given by T) with pincipa coefficients. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

35 A Gaph Theoetic Appoach Fo evey tianguation T (in a suface with o without punctues) and an odinay ac γ though odinay tianges, we constuct a snake gaph G T,γ such that pefect matching M of G x γ = T,γ x(m)y(m) x e (T,γ) x e (T,γ) xn en(t,γ). x γ is custe vaiabe (coesp. to γ w..t. seed given by T) with pincipa coefficients. e i (T,γ) is the cossing numbe of i and γ (min. int. numbe), x(m) is the weight of M, y(m) is the height of M (to be defined ate), Simia fomua wi hod fo non-odinay acs (o though sef-foded tianges). Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

36 Exampes of G T,γ Exampe. Using the above constuction fo 9 8 γ 6 : Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

37 Exampes of G T,γ Exampe. Using the above constuction fo 9 8 γ 6 : Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

38 Exampes of G T,γ Exampe. Using the above constuction fo 8 γ 6 9 : , 9 8 Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

39 Exampes of G T,γ Exampe. Using the above constuction fo 9 8 γ 6 : , , 8. Thus G TH,γ = Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

40 Height Functions (of Pefect Matchings of Snake Gaphs) We now wish to give fomua fo y(m) s, i.e. the tems in the F-poynomias. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

41 Height Functions (of Pefect Matchings of Snake Gaphs) We now wish to give fomua fo y(m) s, i.e. the tems in the F-poynomias. We use height functions which ae due to Wiiam Thuston, and Conway-Lagaias. Invoves measuing contast between a given pefect matching M and a fixed minima matching M. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

42 Height Function Exampes Reca that G TH,γ has thee faces, abeed, and. G TH,γ has five pefect matchings (x,x,...,x 9 = ): y y y, y y, y, y, This matching is M. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

43 Height Function Exampes Reca that G TH,γ has thee faces, abeed, and. G TH,γ has five pefect matchings (x,x,...,x 9 = ): y y y, y y, y, y, This matching is M. Fo exampe, we get heights y y y, y y, and y because of supepositions:,, and Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

44 Height Function Exampes (continued) Fo G TA,γ =, M is. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

45 Height Function Exampes (continued) Fo G TA,γ =, M is. One of the matchings, M, is, Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

46 Height Function Exampes (continued) Fo G TA,γ =, M is. One of the matchings, M, is, so M M =, Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

47 Height Function Exampes (continued) Fo G TA,γ =, M is. One of the matchings, M, is, so M M =, which has height y y. So one of the tems in the custe expansion of x x γ is (using FST convention) (x 6 x 8 )x (x )x (x 8 ) x x x (y y x ). Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

48 Summay Theoem. (M-Schiffe-Wiiams 009) Fo evey tianguation T of a suface (with o without punctues) and an odinay ac γ, we constuct a snake gaph G γ,t such that pefect matching M of G γ,t x(m)y(m) x γ = x e (T,γ) x e (T,γ) xn en(t,γ) Hee e i (T,γ) is the cossing numbe of i and γ, x(m) is the edge-weight of pefect matching M, and y(m) is the height of pefect matching M. (x γ is custe vaiabe with pincipa coefficients.). Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

49 Summay Theoem. (M-Schiffe-Wiiams 009) Fo evey tianguation T of a suface (with o without punctues) and an odinay ac γ, we constuct a snake gaph G γ,t such that pefect matching M of G γ,t x(m)y(m) x γ = x e (T,γ) x e (T,γ) xn en(t,γ) Hee e i (T,γ) is the cossing numbe of i and γ, x(m) is the edge-weight of pefect matching M, and y(m) is the height of pefect matching M. (x γ is custe vaiabe with pincipa coefficients.) Theoem. (M-Schiffe-Wiiams 009) An anaogous expansion fomua hods fo acs with notches (ony aise in the case of a punctued suface). Cooay. The F-poynomia equas M y(m), is positive, and has constant tem. The g-vecto satisfies x g = x(m ). Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /.

50 Exampe (Odinay Ac though Sef-foded Tiange) (S, M) γ p Figue: Idea Tianguation T of (S, M) and coesponding Snake Gaph G T,γ. Note the thee consecutive ties of ou snake gaph with abes, and, as γ taveses the oop twice and the encosed adius. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, /

51 Exampe (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

52 Exampe (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

53 Exampe (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

54 Exampe (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

55 Exampe (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

56 Exampe (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

57 Exampe (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

58 Exampe (Notched Ac in Punctued Suface) (S, M) 0 00 γ p Figue: Idea Tianguation T of (S, M) and coesponding Snake Gaph G T,γ. We obtain the Lauent expansion fo x γ by summing ove so caed γ-symmetic matchings of G T,γ, those that agee on the two bod ends. Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /

59 Thank You Fo Listening Positivity fo Custe Agebas fom Sufaces (with Raf Schiffe and Lauen Wiiams), axiv:math.co/ Custe Expansion Fomuas and Pefect Matchings (with Raf Schiffe), axiv:math.co/ A Gaph Theoetic Expansion Fomua fo Custe Agebas of Cassica Type, musike/finite.pdf (To appea in the Annas of Combinatoics) Fomin, Shapio, and Thuston. Custe Agebas and Tianguated Sufaces I: Custe Compexes, Acta Math. 0 (008), no., 8 6. Fomin and Zeevinsky. Custe Agebas IV: Coefficients, Compos. Math. (00), no., 6. Sides Avaiabe at http//math.mit.edu/ musike/custesufaceams.pdf Musike (MSRI/MIT) Positivity esuts fo custe agebas fom sufaces Octobe, 009 /