Rational Hyperplane Arrangements and Counting Independent Sets of Symmetric Graphs

Transcription

1 Rational Hyperplane Arrangements and Counting Independent Sets of Symmetric Graphs MIT PRIMES Conference Nicholas Guo Mentor: Guangyi Yue May 21, 2016 Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

2 Hyperplane Arrangements Definition Begin with the Euclidean space R n = {(x 1, x 2,..., x n ) : x i R}. An affine hyperplane is the set of points in R n satisfying an equation of the form a 1 x 1 + a 2 x a n x n = b. A hyperplane arrangement is simply a finite collection of affine hyperplanes. A hyperplane arrangement is central if all of its hyperplanes pass through at least one point. Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

3 Familiar Examples in R 2 Hyperplanes are lines in R 2 11 pieces 8 pieces Cake cut for maximum # pieces Hyperplane arrangement in a general position Pizza cut through a center A central hyperplane arrangement Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

4 An Example in R 3 A central hyperplane arrangement of 6 planes in a 3-dimensional space. Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

5 An Arrangement and Its Intersection Poset c d c d b a b a a b a c a d b c b d c d a b c d a b c d a b c d ô = V = R 2 Poset for Cake cut ô = V = R 2 Poset for Pizza cut Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

6 The Möbius Function for Posets Definition Let P be a locally finite poset. Define a function µ = µ P : Int(P) Z, called the Möbius function of P, by the conditions: µ(x, y) = µ(x, x) = 1, x in P x z<y µ(x, z), x < y in P. The second condition can be be written: µ(x, z) = 0, x < y in P. x z y Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

7 Möbius Function Values and the Characteristic Polynomial Definition The characteristic polynomial of a hyperplane arrangement is defined by: χ A (t) = µ(ˆ0, x)t dim(x) x L(A) χ A (t) = t 2 4t + 6 χ A (t) = t 2 4t + 3 Cake-cut arrangement Pizza-cut arrangement Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

8 Regions Definition A region of the arrangement A is a connected component of the complement R n H A H. r(a) denotes the total number of regions, and b(a) denotes the number of bounded regions. Theorem (Zaslavsky) The number of regions and bounded regions can be found as: r(a) = χ A ( 1) b(a) = χ A (1) Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

9 Region Counting Again! 11 pieces 8 pieces χ A (t) = t 2 4t + 6 χ A (t) = t 2 4t + 3 r(a) = ( 1) 2 4( 1) + 6 = 11 r(a) = ( 1) 2 4( 1) + 3 = 8 b(a) = 1 2 4(1) + 6 = 3 b(a) = 1 2 4(1) + 3 = 0 Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

10 Another Example of a Hyperplane Arrangement Let A n be the arrangement in R n with hyperplanes x i = 0 x i = x j x i = 2x j x i = 3x j i i < j i j i j Find χ An (t). χ A2 (t) = (t 1)(t 6) χ A3 (t) = (t 1)(t 2 17t + 78) χ A4 (t) = (t 1)(t 3 33t t 1608) χ A5 (t) = (t 1)(t 4 54t t t )... Calculating χ An (t) becomes more difficult for higher dimensions. Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

11 The Finite Field Method The characteristic polynomial of a Rational Arrangement can be found alternatively: Theorem Let A be any subspace arrangement in R n defined over the integers and q be a large enough prime number, then: χ A (q) = #(F n q H A H) = q n H A H. Equivalently, identifying F n q with [0, 1,..., q 1] n, χ A (q) = # of points with integer coordinates in [0, q 1] n which do not satisfy mod q the defining equations of any of the subspaces in A. Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

12 The Finite Field Method : An Example For the hyperplane arrangement in R n x i = 0 x i = x j x i = 2x j i i < j i j x x 2 0 X X X X X X X X X X X 1 X X X X 2 X X X X 3 X X X X 4 X X X X 5 X X X X 6 X X X X 7 X X X X 8 X X X X 9 X X X X 10 X X X X χ A (p) = (p 1)(p n 2) n 1 F 2 p = [0, p 1] 2, p = 11 where (x) m = x(x 1) (x m 1). Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

13 Applications of Hyperplane Arrangements Hyperplane arrangements have increasing applications in: biology, mathematical physics, statistical economics, topology of collision-free robot motion planning, machine learning and deep neural networks for Artificial Intelligence, combinatorics and graph theory,... Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

14 Independent Sets of G(V, E) Definition In a graph, an independent set is a set of vertices, no two of which are adjacent, or connected by an edge. Example: 10 Hyperplane arrangement A n in R n x i = 0 x i = x j x i = 2x j x i = 3x j i i < j i j i j How many 3-element independent sets of G on vertices [10] with edges ij: j = 2i mod 11 (red line) and j = 3i mod 11 (blue line)? Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, /

15 Counting Independent Sets of Symmetric Graphs A simple problem Choose n labeled points from a circular arrangement of p 1 points (cycle graph C p 1 ). What is the number of n-element independent sets? Example: C 10 x i = 0 x i = x j x i = x j + 1 i i < j i j Solution The number of n-element independent sets is the characteristic polynomial: χ n (p) = (p 1)(p n 2) n 1 where (x) m = x(x 1) (x m 1). Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

16 Independent Sets of Graph with Disjoint Union of Orbits Theorem (Prior Conjecture) Let a = {a 1, a 2,..., a m } be a set of coprime integers. For an integer k 1, let G(k) be the graph with vertex set Z/kZ and edges ij if i a r j mod (k + 1) for some r. Let G be the disjoint union G(n 1 ) G(n 2 ) G(n s ) (n 1 + 1, n 2 + 1,..., n s + 1 all primes 1), then the number of n-element independent sets of G depends only on n, m, and n i. Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

17 Invariance of Characteristic Polynomial Lemma Let A n be the arrangement in R n with hyperplanes x i = 0 x i = x j x i = a 1 x j x i = a 2 x j... x i = a m x j i i < j i j i j i j For a fixed m, χ An (t) is independent of a i s as long as they are coprime. We proved this by using generating functions. Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

18 Graph with Disjoint Union of Orbits Theorem Let a = {a 1, a 2,..., a k } be a set of positive integers. For an integer n 1, Let G(a, n) be the graph with vertex set Z/nZ and edges ij if i j a r mod n for some r. For some n 1, n 2,..., n s 0, let G be the disjoint union G(a, n 1 ) + G(a, n 2 ) + + G(a, n s ). Then the number of l-element independent sets of G depends only on a, l, and n i. We proved this similarly by employing its corresponding characteristic polynomial of hyperplane arrangements. Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

19 Future Research Generalization: G = G n1 + G n2 + + G nk or Z/n 1 Z Z/n 2 Z Z/n k Z be a graph which is the disjoint union of k graphs G ni which has n i vertices. What kind of such graph has the property that the number of n-element independent sets depends solely on n and i n i? Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20

20 Acknowledgments Many thanks to : Guangyi Yue, my mentor Prof. Richard P. Stanley for suggesting this project Dr. Tanya Khovanova, Prof. Pavel Etingof and Dr. Slava Gerovitch My parents MIT PRIMES USA Nicholas Guo (Mentor: Guangyi Yue) PRIMES Presentation May 21, / 20