Algebraic Topology (5 lectures)
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.
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.
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.
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
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