Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results k -point semidefinite programming bounds for equiangular lines Fabr´ ıcio C. Machado (joint work with D. de Laat, F.M. de Oliveira Filho, and F. Vallentin) arXiv:1812.06045 Instituto de Matem´ atica e Estat´ ıstica Universidade de S˜ ao Paulo, Brasil SIAM Conference on Applied Algebraic Geometry Bern, July 9-13, 2019 1/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Acknowledgements F.C.M was supported by grant #2017/25237-4, from the S˜ ao Paulo Research Foundation (FAPESP) and was financed in part by the Coordena¸ c˜ ao de Aperfei¸ coamento de Pessoal de N´ ıvel Superior — Brasil (CAPES). 2/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Contents 1 Problem: equiangular lines and spherical codes 2 A hierarchy of k -point bounds for packing problems Independent sets The bound ∆ k ( G ) ∗ 3 Parameterizing invariant kernels on the sphere T ( x, y, Q ) and O( n ) -invariance Gegenbauer polynomials P n l Fixing a set Q ∈ I k − 2 4 Results a = 1 / 11 , 1 / 9 , 1 / 7 , 1 / 5 3/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results 1 Problem: equiangular lines and spherical codes 4/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Equiangular lines 5/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Spherical codes x · y := � n i =1 x i y i S n − 1 := { x ∈ R n : x · x = 1 } D ⊆ [ − 1 , 1) - set of allowable inner products Examples: D = {− a, a } equiangular lines with com- mon angle arccos a | D | = s spherical s -distance sets D = [ − 1 , a ] packing of spherical caps | C | : C ⊆ S n − 1 , x · y ∈ D for all distinct x, y ∈ C � � A ( n, D ) := max 6/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Spherical codes x · y := � n i =1 x i y i S n − 1 := { x ∈ R n : x · x = 1 } D ⊆ [ − 1 , 1) - set of allowable inner products Examples: D = {− a, a } equiangular lines with com- mon angle arccos a | D | = s spherical s -distance sets D = [ − 1 , a ] packing of spherical caps | C | : C ⊆ S n − 1 , x · y ∈ D for all distinct x, y ∈ C � � A ( n, D ) := max 6/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Spherical codes x · y := � n i =1 x i y i S n − 1 := { x ∈ R n : x · x = 1 } D ⊆ [ − 1 , 1) - set of allowable inner products Examples: D = {− a, a } equiangular lines with com- mon angle arccos a | D | = s spherical s -distance sets D = [ − 1 , a ] packing of spherical caps | C | : C ⊆ S n − 1 , x · y ∈ D for all distinct x, y ∈ C � � A ( n, D ) := max 6/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Spherical codes x · y := � n i =1 x i y i S n − 1 := { x ∈ R n : x · x = 1 } D ⊆ [ − 1 , 1) - set of allowable inner products Examples: D = {− a, a } equiangular lines with com- mon angle arccos a | D | = s spherical s -distance sets D = [ − 1 , a ] packing of spherical caps | C | : C ⊆ S n − 1 , x · y ∈ D for all distinct x, y ∈ C � � A ( n, D ) := max 6/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Spherical codes x · y := � n i =1 x i y i S n − 1 := { x ∈ R n : x · x = 1 } D ⊆ [ − 1 , 1) - set of allowable inner products Examples: D = {− a, a } equiangular lines with com- mon angle arccos a | D | = s spherical s -distance sets D = [ − 1 , a ] packing of spherical caps | C | : C ⊆ S n − 1 , x · y ∈ D for all distinct x, y ∈ C � � A ( n, D ) := max 6/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Spherical codes x · y := � n i =1 x i y i S n − 1 := { x ∈ R n : x · x = 1 } D ⊆ [ − 1 , 1) - set of allowable inner products Examples: D = {− a, a } equiangular lines with com- mon angle arccos a | D | = s spherical s -distance sets D = [ − 1 , a ] packing of spherical caps | C | : C ⊆ S n − 1 , x · y ∈ D for all distinct x, y ∈ C � � A ( n, D ) := max 6/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results Spherical codes x · y := � n i =1 x i y i S n − 1 := { x ∈ R n : x · x = 1 } D ⊆ [ − 1 , 1) - set of allowable inner products Examples: D = {− a, a } equiangular lines with com- mon angle arccos a | D | = s spherical s -distance sets D = [ − 1 , a ] packing of spherical caps | C | : C ⊆ S n − 1 , x · y ∈ D for all distinct x, y ∈ C � � A ( n, D ) := max 6/23 Fabr´ ıcio C. Machado k -point SDP bounds for equiangular lines
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