How is the hypersimplex like the amplituhedron? Lauren K. Williams, Harvard Slides at http://people.math.harvard.edu/~williams/Williams.pdf Based on: \The positive tropical Grassmannian, the hypersimplex, and the m = 2 amplituhedron," with Tomasz Lukowski and Matteo Parisi, arXiv:2002.06164 \The positive Dressian equals the positive tropical Grassmannian," with David Speyer, arXiv:2003.10231 Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 1 / 36 Overview of the talk I. Hypersimplex and moment map '87 II. Amplituhedron '13 Gelfand-Goresky-MacPherson-Serganova Arkani-Hamed{Trnka matroids, torus orbits on Grk;n N = 4 SYM III. Positive tropical Grassmannian '05 Speyer{W. associahedron, cluster algebras Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 2 / 36 Overview of the talk I. Hypersimplex and moment map '87 II. Amplituhedron '13 Gelfand-Goresky-MacPherson-Serganova Arkani-Hamed{Trnka matroids, torus orbits on Grk;n N = 4 SYM III. Positive tropical Grassmannian '05 Speyer{W. associahedron, cluster algebras Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 2 / 36 Overview of the talk I. Hypersimplex and moment map '87 II. Amplituhedron '13 Gelfand-Goresky-MacPherson-Serganova Arkani-Hamed{Trnka matroids, torus orbits on Grk;n N = 4 SYM III. Positive tropical Grassmannian '05 Speyer{W. associahedron, cluster algebras Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 2 / 36 Program Geometry of Grassmannian and the moment map. GGMS '87. Hypersimplex, matroid stratification, matroid polytopes. Add positivity to the previous picture. Postnikov '06. Positroid stratification of (Grkn)≥0, positroid polytopes, plabic graphs. Simultaneous generalization of (Grk;n)≥0 and polygons: amplituhedron. Arkani-Hamed and Trnka '13. Triangulations of the amplituhedron and triangulations of the hypersimplex. Connection with positive tropical Grassmannian Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 3 / 36 Program Geometry of Grassmannian and the moment map. GGMS '87. Hypersimplex, matroid stratification, matroid polytopes. Add positivity to the previous picture. Postnikov '06. Positroid stratification of (Grkn)≥0, positroid polytopes, plabic graphs. Simultaneous generalization of (Grk;n)≥0 and polygons: amplituhedron. Arkani-Hamed and Trnka '13. Triangulations of the amplituhedron and triangulations of the hypersimplex. Connection with positive tropical Grassmannian Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 3 / 36 Program Geometry of Grassmannian and the moment map. GGMS '87. Hypersimplex, matroid stratification, matroid polytopes. Add positivity to the previous picture. Postnikov '06. Positroid stratification of (Grkn)≥0, positroid polytopes, plabic graphs. Simultaneous generalization of (Grk;n)≥0 and polygons: amplituhedron. Arkani-Hamed and Trnka '13. Triangulations of the amplituhedron and triangulations of the hypersimplex. Connection with positive tropical Grassmannian Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 3 / 36 Program Geometry of Grassmannian and the moment map. GGMS '87. Hypersimplex, matroid stratification, matroid polytopes. Add positivity to the previous picture. Postnikov '06. Positroid stratification of (Grkn)≥0, positroid polytopes, plabic graphs. Simultaneous generalization of (Grk;n)≥0 and polygons: amplituhedron. Arkani-Hamed and Trnka '13. Triangulations of the amplituhedron and triangulations of the hypersimplex. Connection with positive tropical Grassmannian Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 3 / 36 Program Geometry of Grassmannian and the moment map. GGMS '87. Hypersimplex, matroid stratification, matroid polytopes. Add positivity to the previous picture. Postnikov '06. Positroid stratification of (Grkn)≥0, positroid polytopes, plabic graphs. Simultaneous generalization of (Grk;n)≥0 and polygons: amplituhedron. Arkani-Hamed and Trnka '13. Triangulations of the amplituhedron and triangulations of the hypersimplex. Connection with positive tropical Grassmannian Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 3 / 36 Program Geometry of Grassmannian and the moment map. GGMS '87. Hypersimplex, matroid stratification, matroid polytopes. Add positivity to the previous picture. Postnikov '06. Positroid stratification of (Grkn)≥0, positroid polytopes, plabic graphs. Simultaneous generalization of (Grk;n)≥0 and polygons: amplituhedron. Arkani-Hamed and Trnka '13. Triangulations of the amplituhedron and triangulations of the hypersimplex. Connection with positive tropical Grassmannian Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 3 / 36 The Grassmannian and the moment map n The Grassmannian Grk;n(C) := fV j V ⊂ C ; dim V = kg Represent an element of Grk;n by a full-rank k × n matrix A. 1 0 0 −3 0 1 2 1 [n] Given I 2 k , the Pl¨ucker coordinate pI (A) is the minor of the k × k submatrix of A in column set I . ∗ n Have torus action of H = (C ) on Grk;n which rescales columns of each A 2 Grk;n. Gives moment map ... n P Let fe1;:::; eng be basis of R ; for I ⊂ [n], let eI := i2I ei : n The moment map µ : Grk;n ! R is defined by P 2 [n] jpI (A)j eI I 2( k ) n µ(A) = P 2 ⊂ R : [n] jpI (A)j I 2( k ) Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 4 / 36 The Grassmannian and the moment map n The Grassmannian Grk;n(C) := fV j V ⊂ C ; dim V = kg Represent an element of Grk;n by a full-rank k × n matrix A. 1 0 0 −3 0 1 2 1 [n] Given I 2 k , the Pl¨ucker coordinate pI (A) is the minor of the k × k submatrix of A in column set I . ∗ n Have torus action of H = (C ) on Grk;n which rescales columns of each A 2 Grk;n. Gives moment map ... n P Let fe1;:::; eng be basis of R ; for I ⊂ [n], let eI := i2I ei : n The moment map µ : Grk;n ! R is defined by P 2 [n] jpI (A)j eI I 2( k ) n µ(A) = P 2 ⊂ R : [n] jpI (A)j I 2( k ) Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 4 / 36 The Grassmannian and the moment map n The Grassmannian Grk;n(C) := fV j V ⊂ C ; dim V = kg Represent an element of Grk;n by a full-rank k × n matrix A. 1 0 0 −3 0 1 2 1 [n] Given I 2 k , the Pl¨ucker coordinate pI (A) is the minor of the k × k submatrix of A in column set I . ∗ n Have torus action of H = (C ) on Grk;n which rescales columns of each A 2 Grk;n. Gives moment map ... n P Let fe1;:::; eng be basis of R ; for I ⊂ [n], let eI := i2I ei : n The moment map µ : Grk;n ! R is defined by P 2 [n] jpI (A)j eI I 2( k ) n µ(A) = P 2 ⊂ R : [n] jpI (A)j I 2( k ) Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 4 / 36 The Grassmannian and the moment map n The Grassmannian Grk;n(C) := fV j V ⊂ C ; dim V = kg Represent an element of Grk;n by a full-rank k × n matrix A. 1 0 0 −3 0 1 2 1 [n] Given I 2 k , the Pl¨ucker coordinate pI (A) is the minor of the k × k submatrix of A in column set I . ∗ n Have torus action of H = (C ) on Grk;n which rescales columns of each A 2 Grk;n. Gives moment map ... n P Let fe1;:::; eng be basis of R ; for I ⊂ [n], let eI := i2I ei : n The moment map µ : Grk;n ! R is defined by P 2 [n] jpI (A)j eI I 2( k ) n µ(A) = P 2 ⊂ R : [n] jpI (A)j I 2( k ) Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 4 / 36 The Grassmannian and the moment map n The Grassmannian Grk;n(C) := fV j V ⊂ C ; dim V = kg Represent an element of Grk;n by a full-rank k × n matrix A. 1 0 0 −3 0 1 2 1 [n] Given I 2 k , the Pl¨ucker coordinate pI (A) is the minor of the k × k submatrix of A in column set I . ∗ n Have torus action of H = (C ) on Grk;n which rescales columns of each A 2 Grk;n. Gives moment map ... n P Let fe1;:::; eng be basis of R ; for I ⊂ [n], let eI := i2I ei : n The moment map µ : Grk;n ! R is defined by P 2 [n] jpI (A)j eI I 2( k ) n µ(A) = P 2 ⊂ R : [n] jpI (A)j I 2( k ) Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 4 / 36 The Grassmannian and the moment map n The Grassmannian Grk;n(C) := fV j V ⊂ C ; dim V = kg Represent an element of Grk;n by a full-rank k × n matrix A. 1 0 0 −3 0 1 2 1 [n] Given I 2 k , the Pl¨ucker coordinate pI (A) is the minor of the k × k submatrix of A in column set I . ∗ n Have torus action of H = (C ) on Grk;n which rescales columns of each A 2 Grk;n. Gives moment map ... n P Let fe1;:::; eng be basis of R ; for I ⊂ [n], let eI := i2I ei : n The moment map µ : Grk;n ! R is defined by P 2 [n] jpI (A)j eI I 2( k ) n µ(A) = P 2 ⊂ R : [n] jpI (A)j I 2( k ) Lauren K. Williams (Harvard) The hypersimplex and the amplituhedron 2020 4 / 36 The Grassmannian and the moment map n The Grassmannian Grk;n(C) := fV j V ⊂ C ; dim V = kg Represent an element of Grk;n by a full-rank k × n matrix A. 1 0 0 −3 0 1 2 1 [n] Given I 2 k , the Pl¨ucker coordinate pI (A) is the minor of the k × k submatrix of A in column set I .