Inverse problems in potential energy minimization Abhinav Kumar MIT Joint work with Henry Cohn August 19, 2012 Motivation In combinatorics, we are frequently interested in exceptional structures. 2/32 Motivation In combinatorics, we are frequently interested in exceptional structures. For example, Regular polytopes. Symmetric graphs related to geometrical structures, like 27 lines on a cubic surface (Schl¨afli graph). Exceptional lattices, like D , E , Barnes-Wall, Leech lattice. 4 8 2/32 Motivation In combinatorics, we are frequently interested in exceptional structures. For example, Regular polytopes. Symmetric graphs related to geometrical structures, like 27 lines on a cubic surface (Schl¨afli graph). Exceptional lattices, like D , E , Barnes-Wall, Leech lattice. 4 8 A natural question arises: to what extent can these structures self-assemble? i.e. form a global minimum of some potential energy problem. 2/32 Motivation In combinatorics, we are frequently interested in exceptional structures. For example, Regular polytopes. Symmetric graphs related to geometrical structures, like 27 lines on a cubic surface (Schl¨afli graph). Exceptional lattices, like D , E , Barnes-Wall, Leech lattice. 4 8 A natural question arises: to what extent can these structures self-assemble? i.e. form a global minimum of some potential energy problem. We’ll see some answers to this question, which involve a design-like property of the target structures. 2/32 Outline Spherical codes, potential energy minimization. 3/32 Outline Spherical codes, potential energy minimization. Inverse problem. 3/32 Outline Spherical codes, potential energy minimization. Inverse problem. Necessary and sufficient conditions (allowing potential wells). 3/32 Outline Spherical codes, potential energy minimization. Inverse problem. Necessary and sufficient conditions (allowing potential wells). Yudin’s LP bound. 3/32 Outline Spherical codes, potential energy minimization. Inverse problem. Necessary and sufficient conditions (allowing potential wells). Yudin’s LP bound. Nicer potentials for nicer examples. 3/32