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

▶

Oct 09, 2023 218 likes •632 views

Moment methods in extremal geometry Laymans 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)

SLIDE 1

Moment methods in extremal geometry

Layman’s talk David de Laat

TU Delft

29 January 2016

SLIDE 2

Extremal geometry

SLIDE 3

Extremal geometry

SLIDE 4

Applications

◮ Coding theory (Example: Voyager probes)

SLIDE 5

Applications

◮ Coding theory (Example: Voyager probes) ◮ Cryptography

SLIDE 6

Applications

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

SLIDE 7

Applications

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

SLIDE 8

Applications

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

. . .

SLIDE 9

Proofs

◮ First claim: One can arrange 12 billiard balls such that all of

them kiss a 13th billiard ball

SLIDE 10

Proofs

◮ First claim: One can arrange 12 billiard balls such that all of

them kiss a 13th billiard ball

◮ Proof:

SLIDE 11

Proofs

◮ First claim: One can arrange 12 billiard balls such that all of

them kiss a 13th billiard ball

◮ Proof: ◮ Second claim: One cannot arrange 13 billiard balls such that

all of them kiss a 14th billiard ball

SLIDE 12

Proofs

◮ First claim: One can arrange 12 billiard balls such that all of

them kiss a 13th billiard ball

◮ Proof: ◮ Second claim: One cannot arrange 13 billiard balls such that

all of them kiss a 14th billiard ball

◮ Goal: Develop techniques to find proofs for claims like these

SLIDE 13

Moment methods

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

SLIDE 14

Moment methods

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

Moment Quantity 1 Mean

SLIDE 15

Moment methods

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

Moment Quantity 1 Mean 2 Variation

SLIDE 16

Moment methods

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

Moment Quantity 1 Mean 2 Variation 3 Skewness

SLIDE 17

Moment methods

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

Moment Quantity 1 Mean 2 Variation 3 Skewness 4 Kurtosis

SLIDE 18

Moment methods

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

Moment Quantity 1 Mean 2 Variation 3 Skewness 4 Kurtosis . . .

SLIDE 19

Moment methods

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

Moment Quantity 1 Mean 2 Variation 3 Skewness 4 Kurtosis . . .

◮ My thesis introduces a concept of moments for geometric

configurations

SLIDE 20

Tools

Combine moment formulation with

◮ optimization, ◮ harmonic analysis, ◮ and real algebraic geometry

to build a computer program that generates proofs

SLIDE 21

Optimization

◮ In optimization we try to find the best element from some set

f available alternatives

SLIDE 22

Optimization

◮ In optimization we try to find the best element from some set

f available alternatives

◮ Example: Dijkstra’s algorithm

SLIDE 23

Optimization

◮ In optimization we try to find the best element from some set

f available alternatives

◮ Example: Dijkstra’s algorithm ◮ Duality: Each maximization problem has a corresponding

minimization problem (and vice versa)

SLIDE 24

Harmonic analysis

t f(t)

SLIDE 25

Harmonic analysis

t f(t) ↓ ↑ ω ˆ f(ω)

SLIDE 26

Harmonic analysis

t f(t) ↓ ↑ ω ˆ f(ω)

SLIDE 27

Harmonic analysis

t f(t) ↓ ↑ ω ˆ f(ω)

SLIDE 28

Real algebraic geometry

◮ Let f(x) = x4 − 10x3 + 27x2 − 10x + 1

x f(x)

SLIDE 29

Real algebraic geometry

◮ Let f(x) = x4 − 10x3 + 27x2 − 10x + 1

x f(x)

◮ Claim: There is no x for which f(x) is negative

SLIDE 30

Real algebraic geometry

◮ Let f(x) = x4 − 10x3 + 27x2 − 10x + 1

x f(x)

◮ Claim: There is no x for which f(x) is negative ◮ Different ways of writing f:

SLIDE 31

Real algebraic geometry

◮ Let f(x) = x4 − 10x3 + 27x2 − 10x + 1

x f(x)

◮ Claim: There is no x for which f(x) is negative ◮ Different ways of writing f:

◮ f(x) = x(x3 − 10x2 + 27x − 10) + 1

SLIDE 32

Real algebraic geometry

◮ Let f(x) = x4 − 10x3 + 27x2 − 10x + 1

x f(x)

◮ Claim: There is no x for which f(x) is negative ◮ Different ways of writing f:

◮ f(x) = x(x3 − 10x2 + 27x − 10) + 1 ◮ f(x) = (x2 − 5x + 1)2

SLIDE 33

Summary

Problem in for instance coding theory

SLIDE 34

Summary

Problem in for instance coding theory ↓ Problem in extremal geometry

SLIDE 35

Summary

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

SLIDE 36

Summary

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

SLIDE 37

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

SLIDE 38

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

SLIDE 39

Thank you!

SLIDE 40

MOMENT METHODS IN EXTREMAL GEOMETRY

David de Laat

MOMENT METHODS IN EXTREMAL GEOMETRY DAVID DE LAAT