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Matroids and Tropical Geometry matroids unimodality and log concavity strategy: geometric models tropical models directions Matroids and Tropical Geometry Federico Ardila San Francisco State University (San Francisco, California) Mathematical Sciences Research Institute (Berkeley, California) Universidad de Los Andes (Bogotá, Colombia) Introductory Workshop: Geometric and Topological Combinatorics MSRI, September 5, 2017 (3,-2,0) Who is here? • This is the Introductory Workshop. • Focus on accessibility for grad students and junior faculty. • # (questions by students + postdocs) # (questions by others) • ≥ matroids unimodality and log concavity strategy: geometric models tropical models directions Preface. Thank you, organizers! • This is the Introductory Workshop. • Focus on accessibility for grad students and junior faculty. • # (questions by students + postdocs) # (questions by others) • ≥ matroids unimodality and log concavity strategy: geometric models tropical models directions Preface. Thank you, organizers! • Who is here? • matroids unimodality and log concavity strategy: geometric models tropical models directions Preface. Thank you, organizers! • Who is here? • This is the Introductory Workshop. • Focus on accessibility for grad students and junior faculty. • # (questions by students + postdocs) # (questions by others) • ≥ matroids unimodality and log concavity strategy: geometric models tropical models directions Summary. Matroids are everywhere. • Many matroid sequences are (conj.) unimodal, log-concave. • Geometry helps matroids. • Tropical geometry helps matroids and needs matroids. • (If time) Some new constructions and results. • Joint with Carly Klivans (06), Graham Denham+June Huh (17). Properties: (B1) B = /0 6 (B2) If A;B B and a A B, 2 2 − then there exists b B A 2 − such that (A a) b B. − [ 2 Definition. A set E and a collection B of subsets of E are a matroid if they satisfies properties (B1) and (B2). matroids unimodality and log concavity strategy: geometric models tropical models directions Matroids Motivation Matroids Tutte polynomials Hyperplane arrangements Computing Tutte polynomials Goal: Capture the combinatorial essence of independence. MOTIVATING EXAMPLES: 2. Linear Algebra. d E= set of vectors spanning R .Goal: Choose a minimal set of vectors that spans R3. d B = collection of subsets of E whichNo 3 on aare plane, bases no 2 on of a line,R no. 1 at the origin. a b d e f c E = abcde Solutions: abc, abd, abe, acd, ace {B = abc;abd;abe};acd;ace (The bases of thef vector configuration.) g Wednesday, October 2, 13 Definition. A set E and a collection B of subsets of E are a matroid if they satisfies properties (B1) and (B2). matroids unimodality and log concavity strategy: geometric models tropical models directions Matroids Motivation Matroids Tutte polynomials Hyperplane arrangements Computing Tutte polynomials Goal: Capture the combinatorial essence of independence. MOTIVATING EXAMPLES: 2. Linear Algebra. d E= set of vectors spanning R .Goal: Choose a minimal set of vectors that spans R3. d B = collection of subsets of E whichNo 3 on aare plane, bases no 2 on of a line,R no. 1 at the origin. a Properties: b (B1) B = /0 d e 6 f (B2) If A;B B and a A B, 2 2 − c then there exists b B A 2 − E = abcde such that (A a) b B. Solutions: abc, abd, abe, acd, ace − [ 2 {B = abc;abd;abe};acd;ace (The bases of thef vector configuration.) g Wednesday, October 2, 13 matroids unimodality and log concavity strategy: geometric models tropical models directions Matroids Motivation Matroids Tutte polynomials Hyperplane arrangements Computing Tutte polynomials Goal: Capture the combinatorial essence of independence. MOTIVATING EXAMPLES: 2. Linear Algebra. d E= set of vectors spanning R .Goal: Choose a minimal set of vectors that spans R3. d B = collection of subsets of E whichNo 3 on aare plane, bases no 2 on of a line,R no. 1 at the origin. a Properties: b (B1) B = /0 d e 6 f (B2) If A;B B and a A B, 2 2 − c then there exists b B A 2 − E = abcde such that (A a) b B. Solutions: abc, abd, abe, acd, ace − [ 2 {B = abc;abd;abe};acd;ace (The bases of thef vector configuration.) g Definition. A set E and a collection B of subsets of E are a matroid if they satisfies properties (B1) and (B2). Wednesday, October 2, 13 Motivation Matroids Tutte polynomials Hyperplane arrangements Computing Tutte polynomials MOTIVATING EXAMPLES: 1. Graph Theory. Goal: Build internet connections that will connect the 4 cities. To lower costs, build the minimum number of connections. 2. Graphical matroids b d E= edges of a connected graph G. e f a B = spanning trees of G. c 3. Algebraic matroids Solutions:Motivation abcMatroids, abd, abeTutte polynomials, acd, aceHyperplane arrangements Computing Tutte polynomials { } 3 E = set of elements in a field extension L=K . (Thea =spanningz MOTIVATING; b = treesx EXAMPLES:of+ they; graph.)c 3. Matching= x Theory.y − B = transcendence bases for L=K in E Goal: Marryd as= manyxy people; e as= possible.x2y2. No gay marriage in Texas.(!) No poligamy. 1 2 3 4. Transversal matroids E = “bottom" vertices of a bipartite graph. Wednesday, October 2, 13 B = maxl sets that can be matched to the top. a b c d e f Possible married men: abc, abd, abe, acd, ace Theorem for matroids Theorems for vectors, graphs, field exts, matchings,{ ::: } 7! (The systems of distinct representatives.) Wednesday, October 2, 13 matroids unimodality and log concavity strategy: geometric models tropical models directions E = abcde Many matroids in “nature": B = abc;abd;abe;acd;ace f g 1. Linear matroids Motivation Matroids Tutte polynomials Hyperplane arrangements Computing Tutte polynomials E= set of vectors spanning Rd . MOTIVATING EXAMPLES: 2. Linear Algebra. B = bases of Rd in E. Goal: Choose a minimal set of vectors that spans R3. No 3 on a plane, no 2 on a line, no 1 at the origin. a b d e f c Solutions: abc, abd, abe, acd, ace { } (The bases of the vector configuration.) Wednesday, October 2, 13 Motivation Matroids Tutte polynomials Hyperplane arrangements Computing Tutte polynomials MOTIVATING EXAMPLES: 1. Graph Theory. Goal: Build internet connections that will connect the 4 cities. To lower costs, build the minimum number of connections. b d e f a c 3. Algebraic matroids Solutions:Motivation abcMatroids, abd, abeTutte polynomials, acd, aceHyperplane arrangements Computing Tutte polynomials { } 3 E = set of elements in a field extension L=K . (Thea =spanningz MOTIVATING; b = treesx EXAMPLES:of+ they; graph.)c 3. Matching= x Theory.y − B = transcendence bases for L=K in E Goal: Marryd as= manyxy people; e as= possible.x2y2. No gay marriage in Texas.(!) No poligamy. 1 2 3 4. Transversal matroids E = “bottom" vertices of a bipartite graph. Wednesday, October 2, 13 B = maxl sets that can be matched to the top. a b c d e f Possible married men: abc, abd, abe, acd, ace Theorem for matroids Theorems for vectors, graphs, field exts, matchings,{ ::: } 7! (The systems of distinct representatives.) Wednesday, October 2, 13 matroids unimodality and log concavity strategy: geometric models tropical models directions E = abcde Many matroids in “nature": B = abc;abd;abe;acd;ace f g 1. Linear matroids Motivation Matroids Tutte polynomials Hyperplane arrangements Computing Tutte polynomials E= set of vectors spanning Rd . MOTIVATING EXAMPLES: 2. Linear Algebra. B = bases of Rd in E. Goal: Choose a minimal set of vectors that spans R3. 2. Graphical matroids No 3 on a plane, no 2 on a line, no 1 at the origin. E= edges of a connected graph G. B = spanning trees of G. a b d e f c Solutions: abc, abd, abe, acd, ace { } (The bases of the vector configuration.) Wednesday, October 2, 13 Motivation Matroids Tutte polynomials Hyperplane arrangements Computing Tutte polynomials MOTIVATING EXAMPLES: 1. Graph Theory. Goal: Build internet connections that will connect the 4 cities. To lower costs, build the minimum number of connections. b d e f a c Solutions: abc, abd, abe, acd, ace Motivation { Matroids Tutte polynomials Hyperplane} arrangements Computing Tutte polynomials (The spanningMOTIVATING trees EXAMPLES:of the graph.) 3. Matching Theory. Goal: Marry as many people as possible. No gay marriage in Texas.(!) No poligamy. 1 2 3 4. Transversal matroids E = “bottom" vertices of a bipartite graph. Wednesday, October 2, 13 B = maxl sets that can be matched to the top. a b c d e f Possible married men: abc, abd, abe, acd, ace Theorem for matroids Theorems for vectors, graphs, field exts, matchings,{ ::: } 7! (The systems of distinct representatives.) Wednesday, October 2, 13 matroids unimodality and log concavity strategy: geometric models tropical models directions E = abcde Many matroids in “nature": B = abc;abd;abe;acd;ace f g 1. Linear matroids Motivation Matroids Tutte polynomials Hyperplane arrangements Computing Tutte polynomials E= set of vectors spanning Rd . MOTIVATING EXAMPLES: 2. Linear Algebra. B = bases of Rd in E. Goal: Choose a minimal set of vectors that spans R3. 2. Graphical matroids No 3 on a plane, no 2 on a line, no 1 at the origin. E= edges of a connected graph G. B = spanning trees of G. a b 3. Algebraic matroids d e E = set of elements in a field extension L=K . a = z3; b = f x + y; c = x y − B = transcendence bases for L=K in E d = xy; ce = x2y2. Solutions: abc, abd, abe, acd, ace { } (The bases of the vector configuration.) Wednesday, October 2, 13 Motivation Matroids Tutte polynomials Hyperplane arrangements Computing Tutte polynomials MOTIVATING EXAMPLES: 1. Graph Theory. Goal: Build internet connections that will connect the 4 cities. To lower costs, build the minimum number of connections. b d e f a c Solutions: abc, abd, abe, acd, ace Motivation { Matroids Tutte polynomials Hyperplane} arrangements Computing Tutte polynomials (The spanningMOTIVATING trees EXAMPLES:of the graph.) 3.
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