Geometry of Gaussoids
|
|
- Stella Lindsey
- 5 years ago
- Views:
Transcription
1 Geometry of Gaussoids Bernd Sturmfels MPI Leipzig and UC Berkeley p 3 p 13 p 23 a 12 3 p 123 a 23 a 13 2 a 23 1 a 13 p 2 p 12 a 12 p p 1 Figure 1: With The vertices Tobias andboege, 2-faces ofalessio the 3-cube D Ali, are labeled and Thomas by the set of Kahle unknowns P [ A.
2 Matroids Figure The hypersimplex 2,4 is a regular octahedron. Its matroid subdivisions correspond to tropical lines in 3-space. A matroid is a combinatorial structure that encodes independence in linear algebra and geometry. The basis axioms reflect the ideal of homogeneousexample relations among We explain all the minors concepts ofina this rectangular section for thematrix simplest nontrivial case, r = 2 and m = 4. Fix the uniform matroid M = U 2,4. Here, ( ) ( ) 1 the Dressian equals the tropical Grassmannian, by the four-point-condition. Writing 0 for the space of phylogenetic 1 0 trees p23 p = on four taxa, 24 we have p 13 p 14 Dr M =Gr M =Dr(2, 4) = Gr(2, 4) =, A matroid is an assignment of 0 or to these minors so that the quadratic Plücker relations have a chance of vanishing: p 12 p 34 p 13 p 24 + p 14 p 23 = 0. We also like oriented matroids, positroids and valuated matroids. 34
3 Gaussoids A gaussoid is a combinatorial structure that encodes independence in probability and statistics. The gaussoid axioms reflect the ideal of homogeneous relations among the principal and almost-principal minors of a symmetric matrix ( ) ( ) p1 a = a 12 p 2 A gaussoid is an assignment of 0 or to these minors so that the quadratic Plücker relations have a chance of vanishing: p p 12 p 1 p 2 + a 2 12 = 0. Ditto: oriented gaussoids, positive gaussoids, valuated gaussoids. The gaussoid axioms were introduced in [R. Lněnička and F. Matúš: On Gaussian conditional independence structures, Kybernetika (2007)]
4 Principal and almost-principal minors A symmetric n n-matrix Σ has 2 n principal minors p I one for each subset I of [n] = {1, 2,..., n}. The matrix Σ has 2 n 2( n 2) almost-principal minors aij K. This is the subdeterminant of Σ with row indices {i} K and column indices {j} K, where i, j [n] and K [n]\{i, j}. Principal minors are in bijection with the vertices of the n-cube. Almost-principal minors are in bijection with the 2-faces of the n-cube. Σ = p 1 a 12 a 13 a 12 p 2 a 23 a 13 a 23 p 3 p 3 p 23 a 12 3 p 123 p 13 a 23 a 13 2 a 23 1 a 13 p 2 p 12 a 12 p p 1
5 Why Gauss? If Σ is positive definite then it is the covariance matrix of a Gaussian distribution on R n. In statistics: p I > 0 for all I [n]. Study n random variables X 1, X 2,..., X n, with the aim of learning how they are related. (Yes, data science) Almost-principal minors a ij K measure partial correlations. We have a ij K = 0 if and only if X i and X j are conditionally independent given X K. The inequalities a ij K > 0 and a ij K < 0 indicate whether conditional correlation is positive or negative.
6 Ideals, Varieties,... Write J n for the homogeneous prime ideal of relations among the principal and almost-principal minors of a symmetric n n-matrix. It lives in a polynomial ring R[p, a] with N = 2 n + 2 n 2( ) n 2 unknowns, and defines an irreducible subvariety of P N 1. Proposition The projective variety V (J n ) is a coordinate projection of the Lagrangian Grassmannian. They share dimension and degree: dim(v (J n )) = ( ) n+1 2 degree(v (J n )) = ( n+1 ) 2! 1 n 3 n 1 5 n 2 (2n 1) 1. The elimination ideal J n R[p] was studied by Holtz-St and Oeding. They found hyperdeterminantal relations of degree 4.
7 3-cube The ideal J 3 is generated by 21 quadrics. There are 9 quadrics associated with the facets of the 3-cube: S 200 (2, 0, 0) a pp 23 p 2 p 3 (0, 0, 0) 2a 23 a pp p 1 p 23 p 2 p 13 p 12 p 3 ( 2, 0, 0) a p 1p 123 p 12 p and two other such weight components There are 12 trinomials associated with the edge of the 3-cube: (1, 1, 0) a 13 a 23 + a 12 3 p a 12 p 3 S 110 (1, 1, 0) a 13 2 a 23 + a 12 3 p 2 a 12 p 23 ( 1, 1, 0) a 13 a a 12 3 p 1 a 12 p 13 ( 1, 1, 0) a 13 2 a a 12 3 p 12 a 12 p and two other such weight components The variety V (J 3 ) is the Lagrangian Grassmannian in P 13, which has dimension 6 and degree 16. It is arithmetically Gorenstein. Intersections with subspaces P 8 are canonical curves of genus 9.
8 3-cube Of most interest are the 12 edge trinomials: p 1 a 23 pa 23 1 a 12 a 13 p 2 a 13 pa 13 2 a 12 a 23 p 3 a 12 pa 12 3 a 23 a 13 p 12 a 13 p 1 a 13 2 a 12 a 23 1 p 12 a 23 p 2 a 23 1 a 12 a 13 2 p 13 a 12 p 1 a 12 3 a 13 a 23 1 p 13 a 23 p 3 a 23 1 a 13 a 12 3 p 23 a 12 p 2 a 12 3 a 23 a 13 2 p 23 a 13 p 3 a 13 2 a 23 a 12 3 p 123 a 12 p 12 a 12 3 a 23 1 a 13 2 p 123 a 13 p 13 a 13 2 a 23 1 a 12 3 p 123 a 23 p 23 a 23 1 a 12 3 a 13 2 p 3 p 13 p 23 a 12 3 p 123 a 23 a 13 2 a 23 1 a 13 p 2 p 12 a 12 p p 1
9 Gaussoid Axioms Let A be the set of ( n 2) 2 n 2 symbols a ij K. Following Lněnička and Matúš, a subset G of A is a gaussoid on [n] if it satisfies: 1. {a ij L, a ik jl } G implies {a ik L, a ij kl } G, 2. {a ij kl, a ik jl } G implies {a ij L, a ik L } G, 3. {a ij L, a ik L } G implies {a ij kl, a ik jl } G, 4. {a ij L, a ij kl } G implies ( a ik L G or a jk L G ). These axioms are known as 1. semigraphoid, 2. intersection, 3. converse to intersection, 4. weak transitivity.
10 Gaussoid Axioms Let A be the set of ( n 2) 2 n 2 symbols a ij K. Following Lněnička and Matúš, a subset G of A is a gaussoid on [n] if it satisfies: 1. {a ij L, a ik jl } G implies {a ik L, a ij kl } G, 2. {a ij kl, a ik jl } G implies {a ij L, a ik L } G, 3. {a ij L, a ik L } G implies {a ij kl, a ik jl } G, 4. {a ij L, a ij kl } G implies ( a ik L G or a jk L G ). These axioms are known as 1. semigraphoid, 2. intersection, 3. converse to intersection, 4. weak transitivity. Theorem The following are equivalent for a set G of 2-faces of the n-cube: (a) G is a gaussoid, i.e. the four axioms above are satisfied for G. (b) G is compatible with the quadratic edge trinomials in J n.
11 Duality and Minors Let G be any gaussoid on [n]. The dual of G is G = { a ij L : a ij K G and L = [n]\({i, j} K) }. Fix an element u [n]. The marginalization equals The conditioning equals G\u = { a ij K G : u {i, j} K }. G/u = { a ij K\{u} : a ij K G and u K }. Think of operations on sets of 2-faces of the n-cube. Proposition If G is a gaussoid on [n], and u [n], then G, G\u and G/u are gaussoids on [n]\{u}. The following duality relation holds: ( G\u ) = G /u and ( G/u ) = G \u. If G is realizable (with Σ positive definite) then so are G, G\u, G/u.
12 A Pinch of Representation Theory Fix the Lie group G = (SL 2 (C)) n. Write V i C 2 for the defining representation of the i-th factor. The irreducible G-modules are S d1d 2 d n = n Sym di (V i ), Proposition G acts on the space W pr spanned by the principal minors and the spaces W ij ap spanned by almost-principal minors. As G-modules, i=1 W pr n i=1v i and W ij ap k [n]\{i,j} V k for 1 i < j n. This defines the G-action and Z n -grading on our polynomial ring C[p, a]. The formal character of C[p, a] 1 = W pr i,j W ij ap is the sum of weights: n i=1 (x i + x 1 i ) + 1 i<j n k [n]\{i,j} (x k + x 1 k )
13 Commutative Algebra The number of linearly independent quadrics in the ideal J n equals ( ) n n 3 ( ) n n 2 ( ) n 3 n k (n k)(n k 1) + 2 k3 n 2k 2 k 2k k=0 Derived via the lowering and raising operators in the Lie algebra g. Conjecture These quadrics generate J n. Proposition The number of face trinomials and edge trinomials equals ( ) ( ) n n 2 n n 3 = 2 n 3 n(n 1)(2n 3). 2 3 k=2 These trinomials generate the image of J n in C[p, a ± ].
14 Page 1 of 1 4-cube p3 p23 a12 3 p13 p123 a23 a13 2 a23 1 a13 p12 p2 a12 p p1 There are 16 principal and 24 almost principal minors. They span Figure 1: The vertices and 2-faces of the 3-cube are labeled by the set of unknowns P [ A. Example 2.2 (n = 3). The polynomial ring R[P [ A] has eight unknowns pi and six unknowns aij K. These are identified respectively with the vertices and facets of the 3-cube. This is shown in Figure 1. The variety V (J3) hasdimension6anddegree6!/( ) = 16 in P 13. The ideal J3 is minimally generated by the 21 quadrics shown in the following tables: 2 3 (2, 0, 0) a pp23 p2p3 S200 4 (0, 0, 0) 2a23a pp123 + p1p23 p2p13 p12p3 5 ( 2, 0, 0) a p1p123 p12p (0, 2, 0) a pp13 p1p3 S020 4 (0, 0, 0) 2a13a pp123 + p2p13 p1p23 p12p3 5 (0, 2, 0) a p2p123 p12p23 2 (0, 0, 2) a pp12 p1p2 3 S002 4 (0, 0, 0) 2a12a pp123 + p3p12 p1p23 p13p2 5 (0, 0, 2 2) a p3p123 p13p23 3 (1, 1, 0) a13a23 + a12 3p a12p3 S ( 1, 1, 0) a13a a12 3p1 a12p13 ( 1, 2 1, 0) a13 2a a12 3p12 a12p123 3 (1, 0, 1) a12a23 + a13 2p a13p2 S101 6 (1, 0, 1) a12 3a23 + a13 2p3 a13p ( 1, 0, 1) a12a a13 2p1 a13p12 ( 1, 0, 2 1) a12 3a a13 2p13 a13p123 3 (0, 1, 1) a12a13 + a23 1p a23p1 S011 6 (0, 1, 1) a12 3a13 + a23 1p3 a23p (0, 1, 1) a12a a23 1p2 a23p12 (0, 1, 1) a12 3a a23 1p23 a23p123 C[p, a] 1 = S 1111 S 1100 S 1010 S 1001 S 0110 S 0101 S The space of quadrics has dimension 820. As G-module, C[a, p] 2 S 2222 S 2211 S 2121 S 2112 S 1221 S 1212 S S S S S S S S S S S S S S S S S S S S S S S S S S S The 226-dimensional submodule (J 4 ) 2 of quadrics in our ideal equals S 2200 S 2020 S 2002 S 0220 S 0202 S 0022 S 2110 S 2101 S 2011 S 1210 S 1201 S 0211 S 1120 S 1021 S 0121 S 1102 S 1012 S 0112 S 1100 S 1010 S 1001 S 0110 S 0101 S S Of these, 120 are trinomials: 96 edge trinomials and 24 face trinomials. 3
15 Enumeration of Gaussoids Theorem The number of gaussoids for n = 3, 4, 5 equals: n all gaussoids orbits for S n Z/2Z S n (Z/2Z) n S n , 212, , , , 981 For n = 3, all 11 gaussoids are realizable: {}, {a 12 }, {a 13 }, {a 23 }, {a 12 3 }, {a 13 2 }, {a 23 1 }, {a 12, a 12 3, a 13, a 13 2 }, {a 12, a 12 3, a 23, a 23 1 }, {a 13, a 13 2, a 23, a 23 1 }, {a 12, a 12 3, a 13, a 13 2, a 23, a 23 1 }.
16 Enumeration of Gaussoids Theorem The number of gaussoids for n = 3, 4, 5 equals: n all gaussoids orbits for S n Z/2Z S n (Z/2Z) n S n , 212, , , , 981 For n = 3, all 11 gaussoids are realizable: {}, {a 12 }, {a 13 }, {a 23 }, {a 12 3 }, {a 13 2 }, {a 23 1 }, {a 12, a 12 3, a 13, a 13 2 }, {a 12, a 12 3, a 23, a 23 1 }, {a 13, a 13 2, a 23, a 23 1 }, {a 12, a 12 3, a 13, a 13 2, a 23, a 23 1 }. For n = 4, five of the 42 gaussoid classes are non-realizable. For instance, G = {a 12 3, a 13 4, a 14 2 } is not realizable. Real Nullstellensatz certificate: a 14 ( a 2 34 p 2 p 4 p 23 + a 2 23 a2 34 p 24 + p 2 2 p 3p 4 p 34 ) (a 23 a 24 a 34 + p 2 p 3 p 4 )(a 24 p 4 a a 24 a 23 a p 3 p 4 a 14 2 ) J 4.
17 SAT Solvers Current software for the satisfiability problem is very impressive, and useful for enumerating combinatorial structures like gaussoids. The input is a Boolean formula in conjunctive normal form (CNF). One can specify one of the following three output options: SAT: Is the formula satisfiable? #SAT: How many satisfying assignments are there? AllSAT: Enumerate all satisfying assignments. We found the 60, 212, 776 gaussoids for n = 5 in about one hour using Thurley s software bdd minisat all. The input was a SAT formulation of the gaussoid axioms using 1680 clauses in the CNF. We then analyzed the output with respect to the symmetry groups.
18 Oriented gaussoids An oriented gaussoid is a map A {0, ±1} such that, for each edge trinomial, after setting each p I to +1 and each a ij K to its image, the set of signs of terms is {0} or { 1, +1} or { 1, 0, +1}. Analogous to oriented matroids. A positive gaussoid is an assignment A {0, +1} with the same compatibility requirement. Analogous to positroids.
19 Oriented gaussoids An oriented gaussoid is a map A {0, ±1} such that, for each edge trinomial, after setting each p I to +1 and each a ij K to its image, the set of signs of terms is {0} or { 1, +1} or { 1, 0, +1}. Analogous to oriented matroids. A positive gaussoid is an assignment A {0, +1} with the same compatibility requirement. Analogous to positroids. Example Let n = 3. Each singleton gaussoid, like G = {a 12 } or {a 12 3 } supports four oriented gaussoids, related by reorientation. We display these 24 = 6 4 oriented gaussoids by listing the six signs for A in the order a 12, a 13, a 23, a 12 3, a 13 2, a 23 1 :
20 3-Cube and Beyond Proposition For n=3 there are 51 oriented gaussoids in seven symmetry classes. All are realizable. This includes 20 uniform gaussoids A {±1}. The following table exhibits the seven classes. The first column gives a covariance matrix Σ that realizes the first oriented gaussoid in the class: (p 1,p 2,p 3, a 12,a 13,a 23 ) Symmetry class of oriented gaussoids (2, 2, 2, 1, 1, 1) , + +, + +, + + (3, 5, 1, 1, 1, 2) , +, ++ +,..., + (6, 9, 6, 1, 1, 7), ++ ++, ++ ++, (4, 3, 3, 2, 2, 1) , ,... previous page (2, 2, 2, 0, 1, 1) 0, ,... previous page (3, 2, 2, 0, 0, 1) , 00 00, 00 00,..., (1, 1, 1, 0, 0, 0)
21 3-Cube and Beyond Proposition For n=3 there are 51 oriented gaussoids in seven symmetry classes. All are realizable. This includes 20 uniform gaussoids A {±1}. The following table exhibits the seven classes. The first column gives a covariance matrix Σ that realizes the first oriented gaussoid in the class: (p 1,p 2,p 3, a 12,a 13,a 23 ) Symmetry class of oriented gaussoids (2, 2, 2, 1, 1, 1) , + +, + +, + + (3, 5, 1, 1, 1, 2) , +, ++ +,..., + (6, 9, 6, 1, 1, 7), ++ ++, ++ ++, (4, 3, 3, 2, 2, 1) , ,... previous page (2, 2, 2, 0, 1, 1) 0, ,... previous page (3, 2, 2, 0, 0, 1) , 00 00, 00 00,..., (1, 1, 1, 0, 0, 0) Theorem The number of oriented gaussoids is 34,873 for n = 4, and it is for n = 5. Among these, are uniform.
22 From Positroids to Statistics Positroids are oriented matroids whose bases are positive. These are important in representation theory and algebraic combinatorics, and they have desirable topological properties. Positive gaussoids correspond to distributions that are of current interest in statistics: S.Fallat, S.Lauritzen, K.Sadeghi, C.Uhler, N.Wermuth and P.Zwiernik: Total positivity in Markov structures, Annals of Statistics 45 (2017) F. Mohammadi, C. Uhler, C. Wang and J. Yu: Generalized permutohedra from probabilistic graphical models, arxiv: S. Lauritzen, C. Uhler and P. Zwiernik: Maximum likelihood estimation in Gaussian models under total positivity, arxiv:
23 From Positroids to Statistics Positroids are oriented matroids whose bases are positive. These are important in representation theory and algebraic combinatorics, and they have desirable topological properties. Positive gaussoids correspond to distributions that are of current interest in statistics: S.Fallat, S.Lauritzen, K.Sadeghi, C.Uhler, N.Wermuth and P.Zwiernik: Total positivity in Markov structures, Annals of Statistics 45 (2017) F. Mohammadi, C. Uhler, C. Wang and J. Yu: Generalized permutohedra from probabilistic graphical models, arxiv: S. Lauritzen, C. Uhler and P. Zwiernik: Maximum likelihood estimation in Gaussian models under total positivity, arxiv: Ardila, Rincón and Williams (2017) proved a 1987 conjecture of Da Silva by showing that all positroids are realizable. We derive the analogue for gaussoids: all positive gaussoids are realizable and their realization spaces are very nice.
24 Positive Gaussoids are Graphical Every graph Γ = ([n], E) defines a gaussoid G Γ via CI statements that hold for the graphical model Γ. Here, a ij K lies in G Γ iff every path from i to j in Γ passes through K. Thus a ij G Γ when i and j are disconnected in Γ, and a ij [n]\{i,j} G Γ when {i, j} E. Theorem For n 2, there are precisely 2 (n 2) positive gaussoids. All are realizable from graphs as above. The space of covariance matrices Σ that realize G Γ is homeomorphic to a ball of dimension E + n. The concentration matrices Σ 1 are M-matrices with support Γ. [S. Karlin and Y. Rinott: M-matrices as covariance matrices of multinormal distributions, Linear Algebra Appl. (1983)] Positive gaussoids satisfy the axiomatic requirements in [K. Sadeghi: Faithfulness of probability distributions and graphs, arxiv:1701..]
25 Conclusion Matroids are cool. And so are gaussoids. Positivity is crucial in algebraic combinatorics. And in statistics. On this journey, let the quadratic equations be your guide. Hitch a fast ride using SAT solvers and representation theory. p 1 a 23 pa 23 1 a 12 a 13, p 2 a 13 pa 13 2 a 12 a 23, p 3 a 12 pa 12 3 a 23 a 13, p 12 a 13 p 1 a 13 2 a 12 a 23 1, p 12 a 23 p 2 a 23 1 a 12 a 13 2,...
26 Conclusion Matroids are cool. And so are gaussoids. Positivity is crucial in algebraic combinatorics. And in statistics. On this journey, let the quadratic equations be your guide. Hitch a fast ride using SAT solvers and representation theory. p 1 a 23 pa 23 1 a 12 a 13, p 2 a 13 pa 13 2 a 12 a 23, p 3 a 12 pa 12 3 a 23 a 13, p 12 a 13 p 1 a 13 2 a 12 a 23 1, p 12 a 23 p 2 a 23 1 a 12 a 13 2,... p 3 p 13 p 23 a 12 3 p 123 a 23 a 13 2 a 23 1 a 13 p 2 p 12 a 12 p p 1 Thank You Figure 1: The vertices and 2-faces of the 3-cube are labeled by the set of unknowns P [ A. Stay tuned for valuated gaussoids via tropical Lagrangian Grassmannian. Example 2.2 (n = 3). The polynomial ring R[P [ A] has eight unknowns p I and six
The Geometry of Gaussoids
The Geometry of Gaussoids arxiv:1710.07175v2 [math.co] 24 May 2018 Tobias Boege, Alessio D Alì, Thomas Kahle and Bernd Sturmfels Dedicated to the memory of František Matúš Abstract A gaussoid is a combinatorial
More informationA Grassmann Algebra for Matroids
Joint work with Jeffrey Giansiracusa, Swansea University, U.K. Matroid (Whitney, 1935) A collection B of subsets of [n] = {1,2,...,n}, called bases, such that the basis exchange property holds: Matroid
More informationOpen Problems in Algebraic Statistics
Open Problems inalgebraic Statistics p. Open Problems in Algebraic Statistics BERND STURMFELS UNIVERSITY OF CALIFORNIA, BERKELEY and TECHNISCHE UNIVERSITÄT BERLIN Advertisement Oberwolfach Seminar Algebraic
More informationTotal positivity in Markov structures
1 based on joint work with Shaun Fallat, Kayvan Sadeghi, Caroline Uhler, Nanny Wermuth, and Piotr Zwiernik (arxiv:1510.01290) Faculty of Science Total positivity in Markov structures Steffen Lauritzen
More informationDissimilarity maps on trees and the representation theory of GL n (C)
Dissimilarity maps on trees and the representation theory of GL n (C) Christopher Manon Department of Mathematics University of California, Berkeley Berkeley, California, U.S.A. chris.manon@math.berkeley.edu
More informationHodge theory for combinatorial geometries
Hodge theory for combinatorial geometries June Huh with Karim Adiprasito and Eric Katz June Huh 1 / 48 Three fundamental ideas: June Huh 2 / 48 Three fundamental ideas: The idea of Bernd Sturmfels that
More informationTutorial: Gaussian conditional independence and graphical models. Thomas Kahle Otto-von-Guericke Universität Magdeburg
Tutorial: Gaussian conditional independence and graphical models Thomas Kahle Otto-von-Guericke Universität Magdeburg The central dogma of algebraic statistics Statistical models are varieties The central
More informationThe Geometry of Semidefinite Programming. Bernd Sturmfels UC Berkeley
The Geometry of Semidefinite Programming Bernd Sturmfels UC Berkeley Positive Semidefinite Matrices For a real symmetric n n-matrix A the following are equivalent: All n eigenvalues of A are positive real
More information5. Grassmannians and the Space of Trees In this lecture we shall be interested in a very particular ideal. The ambient polynomial ring C[p] has ( n
5. Grassmannians and the Space of Trees In this lecture we shall be interested in a very particular ideal. The ambient polynomial ring C[p] has ( n d) variables, which are called Plücker coordinates: C[p]
More informationDissimilarity Vectors of Trees and Their Tropical Linear Spaces
Dissimilarity Vectors of Trees and Their Tropical Linear Spaces Benjamin Iriarte Giraldo Massachusetts Institute of Technology FPSAC 0, Reykjavik, Iceland Basic Notions A weighted n-tree T : A trivalent
More informationVector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle
Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,
More informationNoetherianity up to symmetry
1 Noetherianity up to symmetry Jan Draisma TU Eindhoven and VU Amsterdam Singular Landscapes in honour of Bernard Teissier Aussois, June 2015 A landscape, and a disclaimer 2 # participants A landscape,
More informationPositroids, non-crossing partitions, and a conjecture of da Silva
Positroids, non-crossing partitions, and a conjecture of da Silva Federico Ardila M. San Francisco State University, San Francisco, California. Universidad de Los Andes, Bogotá, Colombia. Discrete Models
More informationarxiv: v3 [math.st] 7 Dec 2017
GENERALIZED PERMUTOHEDRA FROM PROBABILISTIC GRAPHICAL MODELS FATEMEH MOHAMMADI, CAROLINE UHLER, CHARLES WANG, AND JOSEPHINE YU arxiv:1606.0181v [math.st] 7 Dec 2017 Abstract. A graphical model encodes
More informationCombinatorics and geometry of E 7
Combinatorics and geometry of E 7 Steven Sam University of California, Berkeley September 19, 2012 1/24 Outline Macdonald representations Vinberg representations Root system Weyl group 7 points in P 2
More informationSign variation, the Grassmannian, and total positivity
Sign variation, the Grassmannian, and total positivity arxiv:1503.05622 Slides available at math.berkeley.edu/~skarp Steven N. Karp, UC Berkeley April 22nd, 2016 Texas State University, San Marcos Steven
More informationarxiv: v1 [math.st] 6 Jun 2016
GENERALIZED PERMUTOHEDRA FROM PROBABILISTIC GRAPHICAL MODELS FATEMEH MOHAMMADI, CAROLINE UHLER, CHARLES WANG, AND JOSEPHINE YU arxiv:606.08v [math.st] 6 Jun 206 Abstract. A graphical model encodes conditional
More informationTwisted commutative algebras and related structures
Twisted commutative algebras and related structures Steven Sam University of California, Berkeley April 15, 2015 1/29 Matrices Fix vector spaces V and W and let X = V W. For r 0, let X r be the set of
More informationCommuting birth-and-death processes
Commuting birth-and-death processes Caroline Uhler Department of Statistics UC Berkeley (joint work with Steven N. Evans and Bernd Sturmfels) MSRI Workshop on Algebraic Statistics December 18, 2008 Birth-and-death
More informationGorenstein rings through face rings of manifolds.
Gorenstein rings through face rings of manifolds. Isabella Novik Department of Mathematics, Box 354350 University of Washington, Seattle, WA 98195-4350, USA, novik@math.washington.edu Ed Swartz Department
More informationPolynomials, Ideals, and Gröbner Bases
Polynomials, Ideals, and Gröbner Bases Notes by Bernd Sturmfels for the lecture on April 10, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra We fix a field K. Some examples of fields
More informationCombinatorics for algebraic geometers
Combinatorics for algebraic geometers Calculations in enumerative geometry Maria Monks March 17, 214 Motivation Enumerative geometry In the late 18 s, Hermann Schubert investigated problems in what is
More informationAn Algebraic and Geometric Perspective on Exponential Families
An Algebraic and Geometric Perspective on Exponential Families Caroline Uhler (IST Austria) Based on two papers: with Mateusz Micha lek, Bernd Sturmfels, and Piotr Zwiernik, and with Liam Solus and Ruriko
More informationGrassmann Coordinates
Grassmann Coordinates and tableaux Matthew Junge Autumn 2012 Goals 1 Describe the classical embedding G(k, n) P N. 2 Characterize the image of the embedding quadratic relations. vanishing polynomials.
More informationTotal binomial decomposition (TBD) Thomas Kahle Otto-von-Guericke Universität Magdeburg
Total binomial decomposition (TBD) Thomas Kahle Otto-von-Guericke Universität Magdeburg Setup Let k be a field. For computations we use k = Q. k[p] := k[p 1,..., p n ] the polynomial ring in n indeterminates
More informationLecture 1. Toric Varieties: Basics
Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture
More informationFaithful Tropicalization of the Grassmannian of planes
Faithful Tropicalization of the Grassmannian of planes Annette Werner (joint with Maria Angelica Cueto and Mathias Häbich) Goethe-Universität Frankfurt am Main 2013 1 / 28 Faithful Tropicalization 2013
More informationADVANCED TOPICS IN ALGEBRAIC GEOMETRY
ADVANCED TOPICS IN ALGEBRAIC GEOMETRY DAVID WHITE Outline of talk: My goal is to introduce a few more advanced topics in algebraic geometry but not to go into too much detail. This will be a survey of
More informationNotes on the Matrix-Tree theorem and Cayley s tree enumerator
Notes on the Matrix-Tree theorem and Cayley s tree enumerator 1 Cayley s tree enumerator Recall that the degree of a vertex in a tree (or in any graph) is the number of edges emanating from it We will
More informationALGEBRA: From Linear to Non-Linear. Bernd Sturmfels University of California at Berkeley
ALGEBRA: From Linear to Non-Linear Bernd Sturmfels University of California at Berkeley John von Neumann Lecture, SIAM Annual Meeting, Pittsburgh, July 13, 2010 Undergraduate Linear Algebra All undergraduate
More informationON THE RANK OF A TROPICAL MATRIX
ON THE RANK OF A TROPICAL MATRIX MIKE DEVELIN, FRANCISCO SANTOS, AND BERND STURMFELS Abstract. This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring.
More informationAlgebraic Geometry. Question: What regular polygons can be inscribed in an ellipse?
Algebraic Geometry Question: What regular polygons can be inscribed in an ellipse? 1. Varieties, Ideals, Nullstellensatz Let K be a field. We shall work over K, meaning, our coefficients of polynomials
More informationSheaf cohomology and non-normal varieties
Sheaf cohomology and non-normal varieties Steven Sam Massachusetts Institute of Technology December 11, 2011 1/14 Kempf collapsing We re interested in the following situation (over a field K): V is a vector
More informationBases for Cluster Algebras from Surfaces
Bases for Cluster Algebras from Surfaces Gregg Musiker (U. Minnesota), Ralf Schiffler (U. Conn.), and Lauren Williams (UC Berkeley) Bay Area Discrete Math Day Saint Mary s College of California November
More informationSemidefinite Programming
Semidefinite Programming Notes by Bernd Sturmfels for the lecture on June 26, 208, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra The transition from linear algebra to nonlinear algebra has
More informationCombinatorial Aspects of Tropical Geometry and its interactions with phylogenetics
Combinatorial Aspects of Tropical Geometry and its interactions with phylogenetics María Angélica Cueto Department of Mathematics Columbia University Rabadan Lab Metting Columbia University College of
More informationCombinatorial Types of Tropical Eigenvector
Combinatorial Types of Tropical Eigenvector arxiv:1105.55504 Ngoc Mai Tran Department of Statistics, UC Berkeley Joint work with Bernd Sturmfels 2 / 13 Tropical eigenvalues and eigenvectors Max-plus: (R,,
More informationOeding (Auburn) tensors of rank 5 December 15, / 24
Oeding (Auburn) 2 2 2 2 2 tensors of rank 5 December 15, 2015 1 / 24 Recall Peter Burgisser s overview lecture (Jan Draisma s SIAM News article). Big Goal: Bound the computational complexity of det n,
More informationExercise Sheet 7 - Solutions
Algebraic Geometry D-MATH, FS 2016 Prof. Pandharipande Exercise Sheet 7 - Solutions 1. Prove that the Zariski tangent space at the point [S] Gr(r, V ) is canonically isomorphic to S V/S (or equivalently
More informationToric Fiber Products
Toric Fiber Products Seth Sullivant North Carolina State University June 8, 2011 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 1 / 26 Families of Ideals Parametrized by Graphs Let G be a finite
More informationSchubert Varieties. P. Littelmann. May 21, 2012
Schubert Varieties P. Littelmann May 21, 2012 Contents Preface 1 1 SMT for Graßmann varieties 3 1.1 The Plücker embedding.................... 4 1.2 Monomials and tableaux.................... 12 1.3 Straightening
More informationLikelihood Analysis of Gaussian Graphical Models
Faculty of Science Likelihood Analysis of Gaussian Graphical Models Ste en Lauritzen Department of Mathematical Sciences Minikurs TUM 2016 Lecture 2 Slide 1/43 Overview of lectures Lecture 1 Markov Properties
More informationInfinite-dimensional combinatorial commutative algebra
Infinite-dimensional combinatorial commutative algebra Steven Sam University of California, Berkeley September 20, 2014 1/15 Basic theme: there are many axes in commutative algebra which are governed by
More informationAlgebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra
Algebraic Varieties Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic varieties represent solutions of a system of polynomial
More informationBinomial Exercises A = 1 1 and 1
Lecture I. Toric ideals. Exhibit a point configuration A whose affine semigroup NA does not consist of the intersection of the lattice ZA spanned by the columns of A with the real cone generated by A.
More informationAlgebraic Statistics Tutorial I
Algebraic Statistics Tutorial I Seth Sullivant North Carolina State University June 9, 2012 Seth Sullivant (NCSU) Algebraic Statistics June 9, 2012 1 / 34 Introduction to Algebraic Geometry Let R[p] =
More informationMOTIVES ASSOCIATED TO SUMS OF GRAPHS
MOTIVES ASSOCIATED TO SUMS OF GRAPHS SPENCER BLOCH 1. Introduction In quantum field theory, the path integral is interpreted perturbatively as a sum indexed by graphs. The coefficient (Feynman amplitude)
More informationarxiv: v1 [math.co] 9 Sep 2015
ARRANGEMENTS OF MINORS IN THE POSITIVE GRASSMANNIAN AND A TRIANGULATION OF THE HYPERSIMPLEX arxiv:09.000v [math.co] 9 Sep MIRIAM FARBER AND YELENA MANDELSHTAM Abstract. The structure of zero and nonzero
More informationAlgebraic Classification of Small Bayesian Networks
GROSTAT VI, Menton IUT STID p. 1 Algebraic Classification of Small Bayesian Networks Luis David Garcia, Michael Stillman, and Bernd Sturmfels lgarcia@math.vt.edu Virginia Tech GROSTAT VI, Menton IUT STID
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationA variety of flags. Jan E. Grabowski. Mathematical Institute and Keble College.
A variety of flags Jan E. Grabowski Mathematical Institute and Keble College jan.grabowski@maths.ox.ac.uk http://people.maths.ox.ac.uk/~grabowsk Invariants Society 15th February 2011 Flags Definition A
More informationAlgebra 1 Prince William County Schools Pacing Guide (Crosswalk)
Algebra 1 Prince William County Schools Pacing Guide 2017-2018 (Crosswalk) Teacher focus groups have assigned a given number of days to each unit based on their experiences and knowledge of the curriculum.
More informationPhylogenetic Algebraic Geometry
Phylogenetic Algebraic Geometry Seth Sullivant North Carolina State University January 4, 2012 Seth Sullivant (NCSU) Phylogenetic Algebraic Geometry January 4, 2012 1 / 28 Phylogenetics Problem Given a
More informationUnless otherwise specified, V denotes an arbitrary finite-dimensional vector space.
MAT 90 // 0 points Exam Solutions Unless otherwise specified, V denotes an arbitrary finite-dimensional vector space..(0) Prove: a central arrangement A in V is essential if and only if the dual projective
More informationWest Windsor-Plainsboro Regional School District Math A&E Grade 7
West Windsor-Plainsboro Regional School District Math A&E Grade 7 Page 1 of 24 Unit 1: Introduction to Algebra Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale
More informationFirst Quarter Second Quarter Third Quarter Fourth Quarter Unit 1: Expressions and Operations 2.5 weeks/6 blocks
Algebra 1/Algebra 1 Honors Pacing Guide Focus: Third Quarter First Quarter Second Quarter Third Quarter Fourth Quarter Unit 1: Expressions and Operations 2.5 weeks/6 blocks Unit 2: Equations 2.5 weeks/6
More informationLecture 3: Tropicalizations of Cluster Algebras Examples David Speyer
Lecture 3: Tropicalizations of Cluster Algebras Examples David Speyer Let A be a cluster algebra with B-matrix B. Let X be Spec A with all of the cluster variables inverted, and embed X into a torus by
More informationThe geometry of cluster algebras
The geometry of cluster algebras Greg Muller February 17, 2013 Cluster algebras (the idea) A cluster algebra is a commutative ring generated by distinguished elements called cluster variables. The set
More informationELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS
ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need
More informationALGEBRAIC GEOMETRY I - FINAL PROJECT
ALGEBRAIC GEOMETRY I - FINAL PROJECT ADAM KAYE Abstract This paper begins with a description of the Schubert varieties of a Grassmannian variety Gr(k, n) over C Following the technique of Ryan [3] for
More informationPh.D. Qualifying Exam: Algebra I
Ph.D. Qualifying Exam: Algebra I 1. Let F q be the finite field of order q. Let G = GL n (F q ), which is the group of n n invertible matrices with the entries in F q. Compute the order of the group G
More informationQuiver mutations. Tensor diagrams and cluster algebras
Maurice Auslander Distinguished Lectures April 20-21, 2013 Sergey Fomin (University of Michigan) Quiver mutations based on joint work with Andrei Zelevinsky Tensor diagrams and cluster algebras based on
More informationOn the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem
On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem Bertram Kostant, MIT Conference on Representations of Reductive Groups Salt Lake City, Utah July 10, 2013
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More informationThe Graph Isomorphism Problem and the Module Isomorphism Problem. Harm Derksen
The Graph Isomorphism Problem and the Module Isomorphism Problem Harm Derksen Department of Mathematics Michigan Center for Integrative Research in Critical Care University of Michigan Partially supported
More informationVector spaces. EE 387, Notes 8, Handout #12
Vector spaces EE 387, Notes 8, Handout #12 A vector space V of vectors over a field F of scalars is a set with a binary operator + on V and a scalar-vector product satisfying these axioms: 1. (V, +) is
More informationMic ael Flohr Representation theory of semi-simple Lie algebras: Example su(3) 6. and 20. June 2003
Handout V for the course GROUP THEORY IN PHYSICS Mic ael Flohr Representation theory of semi-simple Lie algebras: Example su(3) 6. and 20. June 2003 GENERALIZING THE HIGHEST WEIGHT PROCEDURE FROM su(2)
More informationUncertainty and Bayesian Networks
Uncertainty and Bayesian Networks Tutorial 3 Tutorial 3 1 Outline Uncertainty Probability Syntax and Semantics for Uncertainty Inference Independence and Bayes Rule Syntax and Semantics for Bayesian Networks
More informationExtremal Problems for Spaces of Matrices
Extremal Problems for Spaces of Matrices Roy Meshulam Technion Israel Institute of Technology Optimization, Complexity and Invariant Theory Princeton, June 2018 Plan Maximal Rank and Matching Numbers Flanders
More informationMath 215B: Solutions 1
Math 15B: Solutions 1 Due Thursday, January 18, 018 (1) Let π : X X be a covering space. Let Φ be a smooth structure on X. Prove that there is a smooth structure Φ on X so that π : ( X, Φ) (X, Φ) is an
More informationThe tropical Grassmannian
arxiv:math/0304218v3 [math.ag] 17 Oct 2003 The tropical Grassmannian David Speyer and Bernd Sturmfels Department of Mathematics, University of California, Berkeley {speyer,bernd}@math.berkeley.edu Abstract
More informationMultivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry
Ann Inst Stat Math (2010) 62:603 638 DOI 10.1007/s10463-010-0295-4 Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry Bernd Sturmfels Caroline Uhler Received: 15 June
More informationGENERALIZED CAYLEY-CHOW COORDINATES AND COMPUTER VISION
GENERALIZED CAYLEY-CHOW COORDINATES AND COMPUTER VISION BRIAN OSSERMAN Abstract. A fundamental problem in computer vision is to reconstruct the configuration of a collection of cameras from the images
More informationWeak Separation, Pure Domains and Cluster Distance
Discrete Mathematics and Theoretical Computer Science DMTCS vol (subm, by the authors, 1 1 Weak Separation, Pure Domains and Cluster Distance Miriam Farber 1 and Pavel Galashin 1 1 Department of Mathematics,
More informationTriangulations and soliton graphs
Triangulations and soliton graphs Rachel Karpman and Yuji Kodama The Ohio State University September 4, 2018 Karpman and Kodama (OSU) Soliton graphs September 4, 2018 1 / 1 Introduction Outline Setting:
More informationA Characterization of (3+1)-Free Posets
Journal of Combinatorial Theory, Series A 93, 231241 (2001) doi:10.1006jcta.2000.3075, available online at http:www.idealibrary.com on A Characterization of (3+1)-Free Posets Mark Skandera Department of
More informationThe Hopf monoid of generalized permutahedra. SIAM Discrete Mathematics Meeting Austin, TX, June 2010
The Hopf monoid of generalized permutahedra Marcelo Aguiar Texas A+M University Federico Ardila San Francisco State University SIAM Discrete Mathematics Meeting Austin, TX, June 2010 The plan. 1. Species.
More informationCounting matrices over finite fields
Counting matrices over finite fields Steven Sam Massachusetts Institute of Technology September 30, 2011 1/19 Invertible matrices F q is a finite field with q = p r elements. [n] = 1 qn 1 q = qn 1 +q n
More informationWEAK SEPARATION, PURE DOMAINS AND CLUSTER DISTANCE
WEAK SEPARATION, PURE DOMAINS AND CLUSTER DISTANCE MIRIAM FARBER AND PAVEL GALASHIN Abstract. Following the proof of the purity conjecture for weakly separated collections, recent years have revealed a
More informationEdinburgh, December 2009
From totally nonnegative matrices to quantum matrices and back, via Poisson geometry Edinburgh, December 2009 Joint work with Ken Goodearl and Stéphane Launois Papers available at: http://www.maths.ed.ac.uk/~tom/preprints.html
More informationDISCRIMINANTS, SYMMETRIZED GRAPH MONOMIALS, AND SUMS OF SQUARES
DISCRIMINANTS, SYMMETRIZED GRAPH MONOMIALS, AND SUMS OF SQUARES PER ALEXANDERSSON AND BORIS SHAPIRO Abstract. Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed
More informationFundamental theorem of modules over a PID and applications
Fundamental theorem of modules over a PID and applications Travis Schedler, WOMP 2007 September 11, 2007 01 The fundamental theorem of modules over PIDs A PID (Principal Ideal Domain) is an integral domain
More informationHILBERT BASIS OF THE LIPMAN SEMIGROUP
Available at: http://publications.ictp.it IC/2010/061 United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL
More informationPERVERSE SHEAVES ON A TRIANGULATED SPACE
PERVERSE SHEAVES ON A TRIANGULATED SPACE A. POLISHCHUK The goal of this note is to prove that the category of perverse sheaves constructible with respect to a triangulation is Koszul (i.e. equivalent to
More informationLinear Algebra. Chapter 5
Chapter 5 Linear Algebra The guiding theme in linear algebra is the interplay between algebraic manipulations and geometric interpretations. This dual representation is what makes linear algebra a fruitful
More informationArrangements, matroids and codes
Arrangements, matroids and codes first lecture Ruud Pellikaan joint work with Relinde Jurrius ACAGM summer school Leuven Belgium, 18 July 2011 References 2/43 1. Codes, arrangements and matroids by Relinde
More informationSPECTRAHEDRA. Bernd Sturmfels UC Berkeley
SPECTRAHEDRA Bernd Sturmfels UC Berkeley Mathematics Colloquium, North Carolina State University February 5, 2010 Positive Semidefinite Matrices For a real symmetric n n-matrix A the following are equivalent:
More informationTropical Geometry Homework 3
Tropical Geometry Homework 3 Melody Chan University of California, Berkeley mtchan@math.berkeley.edu February 9, 2009 The paper arxiv:07083847 by M. Vigeland answers this question. In Theorem 7., the author
More informationA new parametrization for binary hidden Markov modes
A new parametrization for binary hidden Markov models Andrew Critch, UC Berkeley at Pennsylvania State University June 11, 2012 See Binary hidden Markov models and varieties [, 2012], arxiv:1206.0500,
More informationTwists for positroid cells
Twists for positroid cells Greg Muller, joint with David Speyer July 29, 205 Column matroids We study k n complex matrices, guided by the perspective: a k n matrix over C an ordered list of n-many vectors
More informationLINEAR SYSTEMS, MATRICES, AND VECTORS
ELEMENTARY LINEAR ALGEBRA WORKBOOK CREATED BY SHANNON MARTIN MYERS LINEAR SYSTEMS, MATRICES, AND VECTORS Now that I ve been teaching Linear Algebra for a few years, I thought it would be great to integrate
More informationAlgebraic Geometry (Math 6130)
Algebraic Geometry (Math 6130) Utah/Fall 2016. 2. Projective Varieties. Classically, projective space was obtained by adding points at infinity to n. Here we start with projective space and remove a hyperplane,
More informationIs Every Secant Variety of a Segre Product Arithmetically Cohen Macaulay? Oeding (Auburn) acm Secants March 6, / 23
Is Every Secant Variety of a Segre Product Arithmetically Cohen Macaulay? Luke Oeding Auburn University Oeding (Auburn) acm Secants March 6, 2016 1 / 23 Secant varieties and tensors Let V 1,..., V d, be
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationRevised: 1/17/14 EASTERN LEBANON COUNTY SCHOOL DISTRICT STUDENT LEARNING MAP
Course/Subject: Algebra 2 Unit 1: Equations & Inequalities Days: 10 days Grade Level: 9, 10, 11 Solve Equations in One Variable A2.1.3.2.2 Solve Inequalities in One Variable A2.2.2.1.2 Solve Combined Inequalities
More informationHigher dimensional dynamical Mordell-Lang problems
Higher dimensional dynamical Mordell-Lang problems Thomas Scanlon 1 UC Berkeley 27 June 2013 1 Joint with Yu Yasufuku Thomas Scanlon (UC Berkeley) Higher rank DML 27 June 2013 1 / 22 Dynamical Mordell-Lang
More informationMath 302 Outcome Statements Winter 2013
Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a
More informationMath 231b Lecture 16. G. Quick
Math 231b Lecture 16 G. Quick 16. Lecture 16: Chern classes for complex vector bundles 16.1. Orientations. From now on we will shift our focus to complex vector bundles. Much of the theory for real vector
More informationThe Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases
arxiv:8.976v [math.rt] 3 Dec 8 The Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases Central European University Amadou Keita (keita amadou@student.ceu.edu December 8 Abstract The most famous
More informationALGEBRAIC PROPERTIES OF BIER SPHERES
LE MATEMATICHE Vol. LXVII (2012 Fasc. I, pp. 91 101 doi: 10.4418/2012.67.1.9 ALGEBRAIC PROPERTIES OF BIER SPHERES INGA HEUDTLASS - LUKAS KATTHÄN We give a classification of flag Bier spheres, as well as
More information