Moment methods in extremal geometry Laymans talk David de Laat TU - PowerPoint PPT Presentation

Moment methods in extremal geometry Layman’s talk David de Laat TU Delft 29 January 2016

Extremal geometry

Extremal geometry

Applications ◮ Coding theory (Example: Voyager probes)

Applications ◮ Coding theory (Example: Voyager probes) ◮ Cryptography

Applications ◮ Coding theory (Example: Voyager probes) ◮ Cryptography ◮ Approximation theory

Applications ◮ Coding theory (Example: Voyager probes) ◮ Cryptography ◮ Approximation theory ◮ Modeling particle systems (Symmetry?)

Applications ◮ Coding theory (Example: Voyager probes) ◮ Cryptography ◮ Approximation theory ◮ Modeling particle systems (Symmetry?) . . .

Proofs ◮ First claim: One can arrange 12 billiard balls such that all of them kiss a 13 th billiard ball

Proofs ◮ First claim: One can arrange 12 billiard balls such that all of them kiss a 13 th billiard ball ◮ Proof:

Proofs ◮ First claim: One can arrange 12 billiard balls such that all of them kiss a 13 th billiard ball ◮ Proof: ◮ Second claim: One cannot arrange 13 billiard balls such that all of them kiss a 14 th billiard ball

Proofs ◮ First claim: One can arrange 12 billiard balls such that all of them kiss a 13 th billiard ball ◮ Proof: ◮ Second claim: One cannot arrange 13 billiard balls such that all of them kiss a 14 th billiard ball ◮ Goal: Develop techniques to find proofs for claims like these

Moment methods 30 Frequency 20 10 0 1 2 3 4 5 6 7 8 9 10 Rating

Moment methods Moment Quantity 30 1 Mean Frequency 20 10 0 1 2 3 4 5 6 7 8 9 10 Rating

Moment methods Moment Quantity 30 1 Mean Frequency 2 Variation 20 10 0 1 2 3 4 5 6 7 8 9 10 Rating

Moment methods Moment Quantity 30 1 Mean Frequency 2 Variation 20 3 Skewness 10 0 1 2 3 4 5 6 7 8 9 10 Rating

Moment methods Moment Quantity 30 1 Mean Frequency 2 Variation 20 3 Skewness 10 4 Kurtosis 0 1 2 3 4 5 6 7 8 9 10 Rating

Moment methods Moment Quantity 30 1 Mean Frequency 2 Variation 20 3 Skewness 10 4 Kurtosis 0 . . . 1 2 3 4 5 6 7 8 9 10 Rating

Moment methods Moment Quantity 30 1 Mean Frequency 2 Variation 20 3 Skewness 10 4 Kurtosis 0 . . . 1 2 3 4 5 6 7 8 9 10 Rating ◮ My thesis introduces a concept of moments for geometric configurations

Tools Combine moment formulation with ◮ optimization, ◮ harmonic analysis, ◮ and real algebraic geometry to build a computer program that generates proofs

Optimization ◮ In optimization we try to find the best element from some set of available alternatives

Optimization ◮ In optimization we try to find the best element from some set of available alternatives ◮ Example: Dijkstra’s algorithm

Optimization ◮ In optimization we try to find the best element from some set of available alternatives ◮ Example: Dijkstra’s algorithm ◮ Duality: Each maximization problem has a corresponding minimization problem (and vice versa)

Harmonic analysis f ( t ) t

Harmonic analysis f ( t ) t ↓ ↑ ˆ f ( ω ) ω

Harmonic analysis f ( t ) t ↓ ↑ ˆ f ( ω ) ω

Harmonic analysis f ( t ) t ↓ ↑ ˆ f ( ω ) ω

Real algebraic geometry ◮ Let f ( x ) = x 4 − 10 x 3 + 27 x 2 − 10 x + 1 f ( x ) x

Real algebraic geometry ◮ Let f ( x ) = x 4 − 10 x 3 + 27 x 2 − 10 x + 1 f ( x ) x ◮ Claim: There is no x for which f ( x ) is negative

Real algebraic geometry ◮ Let f ( x ) = x 4 − 10 x 3 + 27 x 2 − 10 x + 1 f ( x ) x ◮ Claim: There is no x for which f ( x ) is negative ◮ Different ways of writing f :

Real algebraic geometry ◮ Let f ( x ) = x 4 − 10 x 3 + 27 x 2 − 10 x + 1 f ( x ) x ◮ Claim: There is no x for which f ( x ) is negative ◮ Different ways of writing f : ◮ f ( x ) = x ( x 3 − 10 x 2 + 27 x − 10) + 1

Real algebraic geometry ◮ Let f ( x ) = x 4 − 10 x 3 + 27 x 2 − 10 x + 1 f ( x ) x ◮ Claim: There is no x for which f ( x ) is negative ◮ Different ways of writing f : ◮ f ( x ) = x ( x 3 − 10 x 2 + 27 x − 10) + 1 ◮ f ( x ) = ( x 2 − 5 x + 1) 2

Summary Problem in for instance coding theory

Summary Problem in for instance coding theory ↓ Problem in extremal geometry

Summary Problem in for instance coding theory ↓ Problem in extremal geometry ↓ Moment formulation of the problem

Summary Problem in for instance coding theory ↓ Problem in extremal geometry ↓ Moment formulation of the problem ↓ Use optimization, harmonic analysis, and real algebraic geometry

Summary Problem in for instance coding theory ↓ Problem in extremal geometry ↓ Moment formulation of the problem ↓ Use optimization, harmonic analysis, and real algebraic geometry ↓ Computer program

Summary Problem in for instance coding theory ↓ Problem in extremal geometry ↓ Moment formulation of the problem ↓ Use optimization, harmonic analysis, and real algebraic geometry ↓ Computer program ↓ Generate a proof that shows the geometric configuration is optimal

Thank you!

IN EXTREMAL GEOMETRY David de Laat MOMENT METHODS MOMENT METHODS IN EXTREMAL GEOMETRY DAVID DE LAAT