Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Aspects of Group Theory in Stochastic Problems Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia November 18, 2008 Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments ◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact Results? ◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments ◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact Results? ◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments ◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact Results? ◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments ◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact Results? ◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments ◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact Results? ◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments ◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact Results? ◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments ◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact Results? ◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments Outline of what is (would be nice) to come... ◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous Groups. Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments Outline of what is (would be nice) to come... ◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous Groups. Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments Outline of what is (would be nice) to come... ◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous Groups. Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments Outline of what is (would be nice) to come... ◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous Groups. Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments Outline of what is (would be nice) to come... ◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous Groups. Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
Outline Groups Scene Probability on Finite Groups Finite Groups crash course outline Minkowski functionals and Valuations Motivational Papers Applications Algebraic and Numeric Programming environments Outline of what is (would be nice) to come... ◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous Groups. Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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