Multidisciplinary Critical Mass in Computational Algebra and Applications

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Multidisciplinary Critical Mass in Computational
Algebra and Applications

• Centre for Interdisciplinary Research in
  Computational Algebra
• University of St Andrews

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The Challenge
To deliver the power of computational algebra to
all the scientific disciplines underpinned by
algebra

• Computational Algebra Algebra Algorithms
• Algebra is the mathematics of symmetries,
  transformations and structure
• Algebra provides theoretical tools to Computer
  Science, Physics, Chemistry,
• . which in turn provide new problems and
  insights for algebra
• Now possible to add computation to this
  interaction

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The Proposal
An ambitious, coherent research programme that
will meet that challenge

• Four excellent research groups coming together
• Seven exciting interdisciplinary research
  projects
• strengthening our research where input from
  another discipline is essential
• combining to develop a world-leading
  multidisciplinary centre
• Substantial and innovative University support
• software infrastructure and secure research
  career paths

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The Centre
Meeting the challenge in the long term needs a
sustainable, multidisciplinary centre with
critical mass

• Supporting collaborations in many disciplines
• applying computational algebra to enable research
  breakthroughs
• bringing new insights and questions back
• actively reaching out to new areas
• Attracting a broad range of excellent people
• deploying them flexibly
• developing future research leaders
• International impact across disciplines
• This proposal will create that centre

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Overview of Presentation

• Introduction
• The Partners
• The Proposal
• Scientific programme
• Achieving critical mass
• Management, dissemination and outreach
• Conclusion

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The PartnersFour excellent research groups

• Ambitious long-term research plans
• Substantial external funding
• Large groups of research staff and students
• Wide links and high impact in their own community

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Centre for Interdisciplinary Research in
Computational Algebra

• Coordinates GAP (Groups, Algorithms,
  Programming).
• Developed new interactions between maths CS.
• Research success in both disciplines.
• Attracting outstanding staff, students, funding.

Est. 2000
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The Vision
External advisory group
Microsoft Research, IBM, Smith Inst. ILOG,
Gap Council
Skip
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Algebra Combinatorics Group

• Faculty Linton, Quick, Ruškuc, Robertson,
  Campbell, OConnor, two posts open
• Research Staff Elder, Huczynska, Kahrobaei,
  Mitchell, Roney-Dougal, Timochouk
• Students 12 PhD, 4 MSc
• Research groups and semigroups, algorithmic and
  computational algebra, combinatorics of words and
  permutations
• Funding 750K from EPSRC, Royal Society, LMS,
  EMS
• Publications 26 journal papers in 2003
• Links hosted 24 extended visits from 12
  countries in 2003

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Theoretical Computer Science Group

• Faculty Linton, Gent, Miguel, Hammond CS posts
  open
• Research Staff Kelsey, Roney-Dougal, Timochouk
• Students 3 PhD
• Research constraint programming, algorithms,
  medical applications
• Funding 850K from EPSRC, RAE
• Publications 24 journal conference papers in
  2003
• Links Apes, CPPod, SymNet, NETCA

• Includes two grants also counted in algebra
  combinatorics

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Complex Quantum Systems Group

• Faculty Mackenzie, Grigera, Green
• Research Staff Borzi, Kikugawa, Perry
• Students 3 PhD
• Research strongly correlated electron systems
  and unconventional superconductivity
• Funding 1.1M from EPSRC, Leverhulme
• Publications 10 journal conference papers in
  2003, including one in Nature
• Links EPSRC portfolio partnership w Bristol
  Cambridge

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Theoretical Quantum Optics Group

• Faculty Leonhardt, Koenig, Korolkova
• Research Staff Davila-Romero, Giovanazzi,
  Philbin, Vadeiko
• Students 1 PhD, 4 Masters
• Research quantum optics artificial black holes,
  geometry of optical media, quantum solitons,
  quantum polarization, quantum cryptography,
  quantum and nonlinear effects in optical fibres
• Funding 600K from EPSRC, Leverhulme, EU
• Publications 10 journal conference papers in
  2003, papers in Nature, invited surveys, many
  citations

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Overview of Presentation

• Introduction
• The Partners
• The Proposal
• Scientific programme
• Critical mass
• Management, dissemination and outreach
• Conclusion

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Scientific Programme
Seven projects - all interdisciplinary rooted in
core research of the partners together creating
critical mass

• The projects include
• New and extended applications of computational
  algebra
• Fundamental theory on the borders of mathematics
  and computer science
• Core enabling projects in computational algebra
• They are chosen
• To solve real problems in physics, CS or
  mathematics
• To strengthen interdisciplinary relationships
• To bring the right mix of skills into the group
• Because they cant be done outside a proposal of
  this kind

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Project 1 Computational Algebra and Artificial
Intelligence

• AI search
• constraints, planning, scheduling,
• symmetry in search space a major issue
• we pioneered use of algebraic methods
• We will
• extend notion of symmetry, using new algebra
• bring strengths of constraint programming to
  algebra.

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Project 2 Solid State Physics

• Fundamental physics of new materials
• High temperature superconductors
• Novel magnetic materials
• Tools to untangle interplay of subtle
  measurements and crystal symmetries
• simplify measurement of fundamental properties
• Computationally study quantum symmetries
• guide search for novel phase changes and new
  physics

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Project 3 Applications in Advanced Optics

• New metamaterials with precisely controlled
  optical properties
• Unprecedented freedom in optical design
• Assist study of optical geometries through groups
  of complex functions
• Invisibility cloak
• Assist modelling of highly symmetrical photonic
  crystal fibres
• Optical black hole

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Project 4 Pattern Classes of Permutations

• A rapidly growing area in combinatorics
• Multiple connections to CS
• Data structures,
• Network routing
• Formal languages
• Study power of general sorting networks
• Address new decidability questions

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Project 5 Automatic Structures in Algebra

• Deep links between theoretical CS and algebra
• Many fundamental open questions
• Recently generalised from groups to monoids
• Practical and theoretical investigation of
  finding automatic structures for monoids
• Computability questions, via monoids to groups

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Project 6 Improving Algebraic Algorithms

• Driven by needs of other projects
• Adapting to new circumstances
• Hardware advances
• Problems from application areas
• Meataxe revised
• Advanced linear algebra
• Recognition-based algorithms
• Adapted to our applications

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Project 7 GAP Development and Support

• University-funded software infrastructure work
• Memory management
• Software linking
• T hreads
• User support maintenance and documentation
• Driven by needs of other projects
• Benefits GAP users in mathematics, CS, physics,

Map Showing Known GAP user sites
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The Virtuous Circle
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Scientific Programme and CIRCA
Combinatorics Algebra
Complex Quantum Systems
AA Algebraic Algorithms AI Artificial
Intellignce AO Advanced Optics AS Automatic
Structures FS Fermi Surfaces GAP
Development PC Pattern Classes
CIRCA
AI
GAP
FS
PC
AA
AO
AS
Theoretical Computer Science
Quantum Optics
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Critical Mass
This proposal will give us the scale, stability
and flexibility to realise our multidisciplinary
vision and the management and infrastructure to
sustain it.

• Broad range of skills group theory, Lie
  algebras, formal languages, algorithms, AI,
  physics, .
• Stability from 5 year funding and tenure
  review, enabling us to retain excellent research
  staff and to make long term plans
• Flexibility to allocate to each project exactly
  the mix of staff knowledge and skills that it
  needs and to respond to new opportunities
• Support for Management through the appointment of
  senior RAs
• Resources for common software and algorithmic
  infrastructure

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University Alignment and Support

• Strategic commitment to research
• Most research intensive in Scotland
• by proportion of research income
• Excellent environment for interdisciplinary and
  multidisciplinary research
• Investment in maths, CS Physics
• 4m in buildings
• Recent and forthcoming appointments
• Strong financial support linked to proposal
• Full time programmer 320K
• Underwrite overseas fees 42K
• Tenure review for key research staff

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Management
University Management
Head of Maths
Head of CS
GAP Council
CIRCA Management Board
Assoc. Director
Director
External Advisory Board
Director
Assoc. Director
Training Coordinator
CIRCA Academic Board
Senior RAs
Academic Staff
Leader
Leader
Leader
Leader
Leader
RAs Students Other staff Support staff
RAs Students Other staff Support staff
RAs Students Other staff Support staff
RAs Students Other staff Support staff
RAs Students Other staff Support staff
Seven Projects from this Proposal
Other Projects
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Dissemination and Outreach

• Usual publication channels
• GAP system and packages
• package refereeing
• Workshops
• annual event, variety of focus
• Visitor programme
• Outreach activities in project 8
• Senior RAs tasked to actively seek new
  application areas and new collaborations
• Resources set aside for initial contacts and
  proof of concept

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Sustainability

• This proposal doubles our research income
• Need to double capacity to obtain and manage
  research funds to sustain activity after 2010
• Develop a new generation of research leaders
• 5 new faculty posts assured in CIRCA
• By 2010 they will be established
• As research leaders in our multidisciplinary
  culture
• Trained and experienced
• Ongoing research funding in many disciplines and
  from many sources
• Possibilities for funding stability and
  flexibility
• EU projects
• Portfolio Partnership
• Platform Grants
• Fellowships

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Conclusion
We can deliver the power of computational algebra
to all the scientific disciplines underpinned by
algebra

• Four top research groups coming together to
  achieve critical mass in a multidisciplinary
  centre
• Seven exciting initial projects across the whole
  range of our interests
• Exciting and adventurous science with wide impact
• Building and shaping the centre
• Outstanding University support

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Project 1 Artificial Intelligence

• Symmetry major feature of AI search problems
• UK clearly world leader in studying symmetry in
  search
• St Andrews clearly leading UK site
• Gent, Linton, Miguel, Kelsey, Roney-Dougal
• Constraint programming is a powerful problem
  solving technique
• problems modelled declaratively
• industrial applications collaborations
• IBM, Microsoft, ILOG
• St Andrews has pioneered the application of
  computational algebra to symmetry breaking

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WP1.1 Monoids Groupoids

• Search using computational monoids groupoids
• monoids describe structure of dominances
• dominances commonly arise in optimisation
  problems
• St Andrews identified conditional symmetry
• e.g. in steel mill scheduling
• corresponds to groupoid structure
• apply our successful methodology for exploiting
  symmetry
• large impact on constraints and AI in academia
  industry
• provide a strong focus for algebraic development
  of algorithms for monoids groupoids

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WP1.2 Algebraic Constraint Solver

• Algebraic constraint solver
• ambitious programme with potentially dramatic
  impact
• giving computational algebra advantages of
  constraint programming
• declarative statements of algebraic search
  problems
• advanced search mechanisms
• propagation algorithms, backjumping,
  conflict-recording,
• applicable to whole algebraic community
• e.g. counterexample generation

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Project 2 Solid State Physics

• Understanding new materials for twenty-first
  century applications
• High temperature superconductors
• Novel magnetic materials
• Fundamental approximations from traditional
  metals and semiconductors no longer hold
• Strong complex quantum interactions between
  electrons
• New inherently quantum phase transitions
• Computational algebra can help by working with
  symmetries of crystals of correlated quantum
  electron states

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WP 2.1 Fermi Surfaces

• Fermi surface controls electrical and magnetic
  properties
• Strongly related to crystal structure
• Difficult experiments (deHvA) yield partial
  information
• Interpreted in the light of crystal symmetries
• Hard to determine at low temperatures and high
  magnetic fields
• Mackenzie's group world experts
• Novel materials unconventional superconductors
• We will develop tools to support this work
• Simultaneously explore possible Fermi surfaces
  and crystal structures.

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WP 2.2 New Emergent Symmetries

• Phase transitions are often associated with the
  appearance or disappearance of symmetry
• Crystallization of water, magnetism at the Curie
  point
• New phenomena in advanced materials associated
  with similar phase transitions in complex quantum
  systems
• We will develop computational tools to understand
  and predict these transitions from the symmetry
  group
• Medium term project, deepening collaboration
• Draw on new work in computing with Lie groups,
  Lie algebras and quantum groups

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Project 3 Advanced Optics

• Modern technology makes it possible to produce
  artificial materials whose EM properties at
  each point can be very precisely controlled.
• In microwave frequencies using artificial
  dielectrics
• In IR and optical frequencies using photonic
  crystals
• These will enable extremely new and exciting EM
  and optical devices
• Invisibility cloak
• Optical black holes
• We will develop new algebraic methods and
  computational tools for the design of such devices

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WP 3.1 Geometries

• Light follows a straight path in optical space
• Refractive index of medium relates optical to
  real space
• Try to design media so tha the straight paths in
  optical space map to something interesting in
  real space
• Paths which are asymptotically straight but
  avoid a bounded region in middle
• Invisibility cloak
• Computationally exploit the algebraic structure
  of suitable classes of complex functions to
  support this design

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WP 3.2 Photonic Crystal Fibre

• Modern optical fibres can be engineered with very
  precise and unusual properties
• Symmetrical micro-structuring of the fiber
• Develop tools to predict the optical
  consequences, using the symmetry
• Optimise fibre design for experiments and
  applications
• Optical event horizon
• Quantum black holes

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Project 4 Pattern Classes of Permutations

• An active research area at the interface of
  combinatorics, algebra and CS
• Twofold role within the project
• High potential to benefit from computational
  tools, due to CS connections - data structures,
  information networks and formal languages
• A test case for exporting computational algebra
  techniques into other suitable areas of
  combinatorics

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Project 4 Pattern Classes

• WP 4.1 General sorting networks What happens
  when in a communication network some nodes are
  allowed be simple data structures such as stacks,
  queues, etc? Use the existing computational tools
  to generate examples, develop new algorithms.

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Project 4 Pattern Classes

• WP 4.2 Structure Links to algebraic methods, via
  constructions and decompositions, paving the way
  for extended application of computational
  algebraic methods.
• WP 4.3 Decidability Which properties of pattern
  classes can be tested algorithmically? Implement
  those that can and have efficient (practical)
  algorithms.

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Project 5 Automatic Structures in Algebra

• Defining algebraic structures (groups,
  semigroups, monoids, ) by using finite state
  automata.
• These structures should have good computational
  properties.
• Fundamental question how good?

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Project 5 Automatic Structures

• Role within the proposal New exciting area in
  the computational group/semigroup theory, which
  is one of the traditional cohesive,
  interdisciplinary areas within CIRCA.
• WP 5.1 Structure and Properties Develop
  algorithms which allow one to compute with
  elements of automatic (semi)groups and determine
  their algebraic properties.
• WP 5.2 Computing Automatic Structure If an
  automatic monoid is given by a (finite)
  presentation, is there an algorithm which
  computes an automatic structure for it? (Yes for
  groups.)

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Project 5 Automatic Structures

• WP 5.3 Computing Presentations If a monoid is
  given by its automatic structure, is there an
  algorithm which computes a presentation for it
  and decides whether it is finitely presented?
• WP 5.4 Undecidability By finding appropriate
  encodings of Turing Machines, find some natural
  properties of automatic monoids which are
  undecidable. By encoding, in turn, these into
  groups, find undecidable properties of automatic
  groups.

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Project 6 Algorithms

• Algebraic algorithm development so far motivated
  by
• applications within algebra
• complexity theoretic questions
• New applications impose new demands
• practically effective algorithms for overlooked
  types of group
• Eg large-base, imprimitive permutation groups
• Existing algorithms and implementations being
  overtaken by hardware developments
• cache management
• Scaling
• This project responds to these developments as
  required for other projects

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WP 6.1 Linear Algebra

• Specialized software tools (meat-axe) for
  linear algebra in large dimensions over small
  fields
• key step in many algorithms
• Revisit meat-axe design, considering
• Cache
• Parallelism
• Strassen type algorithms
• Sparse methods

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WP 6.2 Recognition-based Algorithms

• Abstract permutation group
• Black box group black box permutation action
• Can encode a very high degree permutation action
• New algorithms recognise Sn in action on k-sets
  in sub-linear time
• Effective isomorphism to much smaller
  representation
• Solve problem there and pull back solution
• This WP will extend and exploit this approach

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Project 7 GAP Development

• GAP Groups, Algorithms, Programming
• 1000 known user sites
• 580 publications
• Kernel, libraries, databases, packages
• International development collaboration
• University-funded developer will undertake
  projects that update GAPs infrastructure to
  better support research
• Memory management
• Software Linking
• Threads
• Share in user support maintenance and
  documentation

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Workplan
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Monoid Algorithms

• Recent developments in EPSRC GR/S56085
• Algorithms to determine properties of a monoid of
  transformations without listing all the elements
• implemented through an core data structure
• a labeled digraph of orbits of images and kernels
• Examples
• tests for standard properties inverse,
  completely regular, completely simple, regular,
  orthodox, etc.
• determining Green's relations
• properties of Green's classes, such as size,
  elements, or structure