Interactions between Algebra, Geometry and Combinatorics Karin Baur University of Graz 12 June 2018
Geometry 1 Pythagoras’ theorem Ptolemy’s Theorem Triangulations 2 Polygons Surfaces Cluster theory 3 Cluster algebras Cluster categories Surfaces and combinatorics 4 Categories and diagonals Dimers, boundary algebras and categories of modules Overview and Outlook 5
Pythagoras’s theorem a b c Theorem (Pythagoras, 570-495 BC) The sides of a right triangle satisfy a 2 + b 2 = c 2 . Karin Baur Interactions between Algebra, Geometry and Combinatorics 1 / 25
Ptolemy’s Theorem (Theorema Secundum) a A cyclic quadrilateral is a quadrilateral whose vertices b b lie on a common circle. a Theorem (Ptolemy, 70 - 168 BC) The lengths of a sides and diagonals of a cyclic quadrilateral satisfy: | AC | · | BD | = | AB | · | CD | + | BC | · | DA | Karin Baur Interactions between Algebra, Geometry and Combinatorics 2 / 25
Ptolemy’s Theorem (Theorema Secundum) D AD a A A cyclic quadrilateral is a BD CD quadrilateral whose vertices b b AB lie on a common circle. AC a C BC B Theorem (Ptolemy, 70 - 168 BC) The lengths of a sides and diagonals of a cyclic quadrilateral satisfy: | AC | · | BD | = | AB | · | CD | + | BC | · | DA | Karin Baur Interactions between Algebra, Geometry and Combinatorics 2 / 25
Ptolemy’s Theorem (Theorema Secundum) D AD A A cyclic quadrilateral is a BD CD quadrilateral whose vertices AB lie on a common circle. AC C BC B Theorem (Ptolemy, 70 - 168 BC) The lengths of a sides and diagonals of a cyclic quadrilateral satisfy: | AC | · | BD | = | AB | · | CD | + | BC | · | DA | Karin Baur Interactions between Algebra, Geometry and Combinatorics 2 / 25
Triangulations A triangulation of an n -gon is a subdivision of the polygon into triangles. Result: n − 2 triangles, using n − 3 diagonals ( invariants of the n -gon). Theorem (Euler’s Conjecture, 1751, Proofs: Catalan et al., 1838-39) Number of ways to triangulate a convex n -gon: � 2 n − 4 � 1 C n := Catalan numbers n − 1 n − 2 3 4 5 6 7 8 n 1 2 5 14 42 132 C n Karin Baur Interactions between Algebra, Geometry and Combinatorics 3 / 25
Triangulations A triangulation of an n -gon is a subdivision of the polygon into triangles. Result: n − 2 triangles, using n − 3 diagonals ( invariants of the n -gon). Theorem (Euler’s Conjecture, 1751, Proofs: Catalan et al., 1838-39) Number of ways to triangulate a convex n -gon: � 2 n − 4 � 1 C n := Catalan numbers n − 1 n − 2 3 4 5 6 7 8 n 1 2 5 14 42 132 C n Karin Baur Interactions between Algebra, Geometry and Combinatorics 3 / 25
Figures in the plane: disk, annulus Disk Dynkin type A n -gon: disk with n marked points on boundary. x x Punctured disk Dynkin type D Marked points on boundary, one marked point in interior. (degenerate triangles) Dynkin type ˜ Annulus A Marked points on both boundaries of the figure. Finiteness (Fomin - Shapiro - D. Thurston 2005) S Riemann surface S is a disk and Karin Baur Interactions between Algebra, Geometry and Combinatorics 4 / 25 with marked points M has ⇐ ⇒ M has at most one
Figures in the plane: disk, annulus Disk Dynkin type A n -gon: disk with n marked points on boundary. Punctured disk Dynkin type D Marked points on boundary, one marked point in interior. (degenerate triangles) x x x Dynkin type ˜ Annulus A Marked points on both boundaries of the figure. Finiteness (Fomin - Shapiro - D. Thurston 2005) S Riemann surface S is a disk and with marked points M has ⇐ ⇒ M has at most one Karin Baur Interactions between Algebra, Geometry and Combinatorics 4 / 25
Figures in the plane: disk, annulus Disk Dynkin type A n -gon: disk with n marked points on boundary. Punctured disk Dynkin type D Marked points on boundary, one marked point in interior. (degenerate triangles) Dynkin type ˜ Annulus A Marked points on both boundaries of the figure. x x Finiteness (Fomin - Shapiro - D. Thurston 2005) S Riemann surface S is a disk and with marked points M has ⇐ ⇒ M has at most one Karin Baur Interactions between Algebra, Geometry and Combinatorics 4 / 25
Figures in the plane: disk, annulus Disk Dynkin type A n -gon: disk with n marked points on boundary. Punctured disk Dynkin type D Marked points on boundary, one marked point in interior. (degenerate triangles) Dynkin type ˜ Annulus A Marked points on both boundaries of the figure. Finiteness (Fomin - Shapiro - D. Thurston 2005) S Riemann surface S is a disk and with marked points M has ⇐ ⇒ M has at most one finitely many triangulations point in S \ ∂ S Karin Baur Interactions between Algebra, Geometry and Combinatorics 4 / 25
Triangles, diagonals ‘Triangles’ Figures with three or less edges: degenerate triangles. ‘Diagonals’ (FST 2005) The number of diagonals is constant. It is the rank of the surface: where: p marked points, q punctures, p + 3 q − 3(2 − b ) + 6 g , b boundary components, g genus. today: q ∈ { 0 , 1 } , b ∈ { 1 , 2 } , g = 0. Karin Baur Interactions between Algebra, Geometry and Combinatorics 5 / 25
Cluster algebras \In an attempt to create an algebraic framework for dual canonical bases and total positivity in semi simple groups, we initiate the study of a new class of commutative algebras." Fomin-Zelevinsky 2001 recursively defined algebras ⊆ Q ( x 1 , . . . , x n ) grouped in overlapping sets of generators many relations between the generators Karin Baur Interactions between Algebra, Geometry and Combinatorics 6 / 25
Cluster algebras \In an attempt to create an algebraic framework for dual canonical bases and total positivity in semi simple groups, we initiate the study of a new class of commutative algebras." Fomin-Zelevinsky 2001 recursively defined algebras ⊆ Q ( x 1 , . . . , x n ) grouped in overlapping sets of generators many relations between the generators Karin Baur Interactions between Algebra, Geometry and Combinatorics 6 / 25
Pentagon-recurrence (Spence, Abel, Hill) f m +1 = f m +1 Sequence ( f i ) i ⊆ Q ( x 1 , x 2 ): with f 1 := x 1 , f 2 := x 2 . f m − 1 f 3 = x 2 +1 x 1 , f 4 = x 1 + x 2 +1 , f 5 = x 1 +1 x 2 , f 6 = f 1 , f 7 = f 2 , etc. x 1 x 2 1 1 1 1 1 5 5 5 1 1 f 4 = x 1 + x 2 +1 x 1 x 2 f 4 1 1 1 2 2 2 f 5 f 3 = x 2 +1 1 1 x 1 4 4 4 1 1 3 3 3 Cluster algebra A := � ( f i ) i � = � f 1 , f 2 , . . . , f 5 � ⊆ Q ( x 1 , x 2 ). Relations: f 1 f 3 = f 2 + 1, f 2 f 4 = f 3 + 1, etc. Karin Baur Interactions between Algebra, Geometry and Combinatorics 7 / 25
Pentagon-recurrence (Spence, Abel, Hill) f m +1 = f m +1 Sequence ( f i ) i ⊆ Q ( x 1 , x 2 ): with f 1 := x 1 , f 2 := x 2 . f m − 1 f 3 = x 2 +1 x 1 , f 4 = x 1 + x 2 +1 , f 5 = x 1 +1 x 2 , f 6 = f 1 , f 7 = f 2 , etc. x 1 x 2 Cluster algebra A := � ( f i ) i � = � f 1 , f 2 , . . . , f 5 � ⊆ Q ( x 1 , x 2 ). Relations: f 1 f 3 = f 2 + 1, f 2 f 4 = f 3 + 1, etc. Karin Baur Interactions between Algebra, Geometry and Combinatorics 7 / 25
Cluster algebras Start with { x 1 , . . . , x n } cluster , B = ( b ij ) n × n a sign-skew symmetric matrix over Z . The pair ( x , B ) is a seed. Relations through mutation at k ( B mutates similarly): � � x b ik x − b ik x k · x ′ k = + i i b ik > 0 b ik < 0 → x ′ ( x , B ) � ( { x 1 , . . . , x ′ k , . . . , x n } , B ′ ). Mutation at k : x k �− k , and so Cluster variables all the x i , the x ′ i , etc. Cluster algebra A = A ( x , B ) ⊂ Q ( x 1 , . . . , x n ) generated by all cluster variables. Karin Baur Interactions between Algebra, Geometry and Combinatorics 8 / 25
Cluster algebras A = A ( x , B ) ⊂ Q ( x 1 , . . . , x n ). Properties Laurent phenomenon: A ⊂ Z [ x ± 1 , x ± 2 , . . . , x ± n ]: Fomin-Zelevinsky. Finite type follows Dynkin type: Fomin-Zelevinsky. Positivity: coefficients in Z > 0 : Musiker-Schiffler-Williams 2011, Lee-Schiffler 2015, Gross-Hacking-Keel-Kontsevich 2018. Examples C [SL 2 ], C [Gr(2 , n )] (Fomin-Zelevinsky), C [Gr( k , n )] (Scott). Karin Baur Interactions between Algebra, Geometry and Combinatorics 9 / 25
Overview Poisson geometry integrable systems Total positivity Gekhtman-Shapiro-Vainshtein, 2003 canonical basis of U q ( g ) Teichm ¨ uller theory Fock-Goncharov, 2006 Karin Baur Interactions between Algebra, Geometry and Combinatorics 10 / 25
In focus Gratz, 2016 Parsons, 2015 Parsons, 2013 Grabowski-Gratz, 2014 B-Marsh 2012 B-Marsh w Thomas, 2009 Lamberti, 2011, 2012 B-Dupont, 2014 Gratz, 2015 Tschabold, arXiv 2015 B-Bogdanic, 2016 B-Marsh 2007, 2008, 2012 B-Parsons-Tschabold, 2016 B-Bogdanic-Garc ´ ı a Elsener ∗ B-Buan-Marsh, 2014 Vogel, 2016 B-Coelho Simoes-Pauksztello ∗ B-King-Marsh, 2016 Aichholzer-Andritsch-B-Vogtenhuber, 2017 B-Bogdanic-Pressland ∗ Lamberti, 2014 Gunawan-Musiker-Vogel, 2018 B-Bogdanic-Garc ´ ı a Elsener-Martsinkovsky ∗ B-Torkildsen, arXiv 2015 B-Schi ffl er ∗ B-Fellner-Parsons-Tschabold, 2018 B-Gratz, 2018 B-Nasr Isfahani ∗ B-Martin, 2018 McMahon, arXiv 2016, 2017 B-Gekhtman ∗ Coelho Simoes-Parsons 2017 B-Martin, arXiv 2017 B-Faber-Gratz-Serhiyenko-Todorov, 2017 B-Coelho Simoes, arXiv 2018 Andritsch, 2018 B-Marsh ∗ B-Faber-Gratz-Serhiyenko-Todorov ∗ B-Schroll ∗ B-Laking ∗ B-Beil ∗ ∗ ongoing projects Karin Baur Interactions between Algebra, Geometry and Combinatorics 11 / 25
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