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Details of Grant 

EPSRC Reference: EP/V009877/1
Title: DMS-EPSRC Topology of automated motion planning
Principal Investigator: Farber, Professor M
Other Investigators:
Researcher Co-Investigators:
Project Partners:
University of Chicago
Department: Sch of Mathematical Sciences
Organisation: Queen Mary University of London
Scheme: Standard Research
Starts: 01 January 2021 Ends: 31 December 2023 Value (£): 368,192
EPSRC Research Topic Classifications:
Algebra & Geometry Logic & Combinatorics
Statistics & Appl. Probability
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
31 Aug 2020 EPSRC Mathematical Sciences Prioritisation Panel September 2020 Announced
Summary on Grant Application Form
Algebraic topology uses various techniques allowing to reduce problems about flexible geometric objects to questions about rigid algebraic structures. Cohomological methods are currently among the most effective techniques which are used in situations motivated by problems of mathematical analysis, algebraic geometry, engineering and mathematical physics.

In this research we shall use topological tools (and in particular methods of cohomology theory) to analyse algorithms of making decisions in situations when the outcome is a choice made from a continuum of possibilities rather than from a discrete set of values. In many such situations an algorithms can be interpreted as a section of a fibration and the topological concept of Schwartz genus of a fibration is a natural reflection of complexity of the task.

In this research we shall develop mathematical methods to study algorithms for motion planning in engineering, algorithms for coordination of computations in distributed computing and algorithms for aggregation of personal preferences and reaching consensus in negotiations and their relations to social choice theory. Although these problems may appear to be very different at the first glance, they involve quite similar underlying mathematics and can be analysed by analogous methods.

We shall develop theory of parametrised topological complexity which will be applicable

in a variety of real life situations. The fundamental novelty in our approach consists in the assumption that the configuration space is "large" and is known to us only partially and, besides, the motion planning algorithm will operate without prior knowledge of the positions of the obstacles. Mathematically this will be reflected in the assumption that the actual configuration space is one of a large family of spaces parametrised by a base of a fibration.

We shall also study motion planning algorithms in situations when motion of the system is constrained by technical limitations. This theory will be applicable in the context of distributed computing and will help to design computational algorithms for multiple computers performing simultaneous computations and sharing common resources.

We shall tackle the Rationality Conjecture which predicts behaviour of the sequential topological complexity in the case of a large number of intermediate destinations. Besides, we shall further develop theory of geodesic motion planning where one requires that motion planning algorithms produce geodesic paths of minimal lengths.

In the classical theory of social choice the preferences are assumed to be given on discrete sets of alternatives. We shall study a topological approach aiming to devise methods allowing to quantify the necessary failure of aggregation methods of various kind, along the lines of topological complexity and its quantitative variants, and to produce some positive results in this direction.

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