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Multidisciplinary Critical Mass in Computational Algebra and Applications

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Title: Multidisciplinary Critical Mass in Computational Algebra and Applications


1
Multidisciplinary Critical Mass in Computational
Algebra and Applications
  • Centre for Interdisciplinary Research in
    Computational Algebra
  • University of St Andrews

2
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

3
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

4
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

5
Overview of Presentation
  • Introduction
  • The Partners
  • The Proposal
  • Scientific programme
  • Achieving critical mass
  • Management, dissemination and outreach
  • Conclusion

6
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

7
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
8
The Vision
External advisory group
Microsoft Research, IBM, Smith Inst. ILOG,
Gap Council
Skip
9
Algebra Combinatorics Group
  • Faculty Linton, Quick, RuÅ¡kuc, 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

10
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

11
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

12
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

13
Overview of Presentation
  • Introduction
  • The Partners
  • The Proposal
  • Scientific programme
  • Critical mass
  • Management, dissemination and outreach
  • Conclusion

14
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

15
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.

16
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

17
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

18
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

19
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

20
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

21
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
22
The Virtuous Circle
23
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
24
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

25
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

26
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
27
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

28
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

29
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

30
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

31
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

32
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

33
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

34
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.

35
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

36
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

37
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

38
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

39
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

40
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.

41
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.

42
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?

43
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.)

44
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.

45
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

46
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

47
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

48
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

49
Workplan
50
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
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