Second de Brún Workshop on Computational Algebra

30 June - 9 July 2009

Scope

The workshop will be based around the following four lecture courses aimed at mathematicians with a general interest in algebra.

Gunnar Carlsson


Applied Algebraic Topology (5 lectures)

Course outline
1) Persistent homology
2) Applications to vision and neuroscience
3) Clustering, homotopy colimits and applications in medical statistics
4) Persistent homology in ziz-zag diagrams
5) Multi-dimensional persistence and future work.





Marian  Mrozek

Computational Homology (5 lectures)

Certain difficult existence problems in the theory of differential equations, in particular the existence of chaotic dynamics, can be reduced to questions about the homology of some sets in R^n and continuous maps acting on them. The sets and some information about the maps may be derived algorithmically via rigorous numerical enclosures of the trajectories of the system based on interval arithmetic. The problem is that although the topology of the sets is in general simple, the sets are often huge. For most problems the classical homology algorithms are not fast enough for the computer assisted proofs to succeed. Also, there are  no standard algorithms which can find the homology maps on the basis of the information available from the rigorous numerics of differential equations. However, some non standard techniques based on reduction algorithms and cubical homology of multivalued maps make such computations possible.

Surprisingly, the same method may be applied to the analysis of dynamical systems generated by time series data. This is because the data obtained from the numerical study of a differential equation and data gathered from an experiment share a common feature, which is the lack of precision coming from roudings and truncations in the case of numerics and finitness and measurement errors in the case of experiments.

Course Outline
1) Rigorous numerics of dynamical systems via interval arithmetic.
2) Homological invariants of dynamical systems.
3) Computer assisted proofs in dynamics based on rigorous numerics and homological invariants.
4) Representation of sets: simpicial, cubical, Cech.
5) Cubical homology.
6) Homology algorithms: limitations of classical approach, reduction algorithms.
7) Homology of maps: multivalued approach, chain selector algorithm, graph decomposition algorithm, Cech type approach.
8) Applications.





Eva Riccomagno


Henry Wynn

Algebraic Statistics (10 lectures)

Worked examples will be given out with solutions for students to try between lectures.

The course is part introductory part on more advanced research topics. It will cover:
1. Experimental design, including the use of indicator functions, corner
cuts and the algebraic fan.
2. Algebraic statistical models, toric ideals
3. Exact tests and Markov bases
4. Algebraic methods in causal models
5. Links between algebraic statistical models and Information geometry
6. Gaussian models, projections and time series
7. Uses of monomial ideals and Hilbert functions in probability



There will also be a full programme of afternoon research talks.

Programme & Registration Details

There is no registration fee. Those wishing to attend the workshop should e-mail graham.ellis@nuigalway.ie as soon as possible.

Schedule

The first lecture will be at 10am on Tuesday 30 June, and the last lecture will end at 5.30pm on Thursday 9 July. There will be a Workshop trip to the Burren, Co. Clare in the first week. There will be no lectures on the Saturday afternoon and Sunday. Lecture courses will take place in the mornings, and  participants will have the opportunity to contribute talks on their research in the afternoons.

The programme and list of  participants are available online.

Accommodation

The cost of campus accommodation in Corrib Village is 43 euro per night (bed and breakfast).

Contribute a talk

If you would like to give a talk, please send a provisional title as soon as possible. For these talks our interpretation of "computational algebra" is very catholic!

Travel

There are regular rail connections from Dublin to Galway, and bus connections from all Irish cities and towns.
There are direct flights to Galway Airport from Belfast, Dublin, Luton, Manchester, Edinburgh, Liverpool, Lorient, Cardiff, Leeds, Cork and Prague. The Airport is 7 kms from the NUI Galway campus and is unfortunately not served by public transport. Taxis typically cost €25.
Directions to NUI Galway by road can be found here .
NUI Galway has a number of pay-and-display parking places for visitors. Cars parked in other spaces on the NUI Galway campus and not displaying a valid parking permit will be clamped.