Research interests focus on noncommutative differential geometry: Sheaf cohomology, noncommutative Riemannian geometry and noncommutative complex structures. Beggs is also interested in nonassociative structures and their applications in physics and quantisation. One area of study is building a bridge between noncommutative algebraic geometry and noncommutative complex manifolds (classically this is called the algebraic / analytic correspondence). He also maintains an interest in higher dimensional integrable systems, but this seems to be very difficult!
In computer science, Beggs is interested in the theory of computability and using physical experiments as oracles. This is related to the idea of measurement in physics.
My research interests are concentrated around symmetry aspects of Noncommutative Geometry (NCG), and include ring and module theory, Hopf algebras and quantum groups, category theory, geometry as well as some aspects of mathematical physics. My main research achievements include the development of the theory of noncommutative principal bundles and connections (joint work with Shahn Majid) and the revival of the theory of corings as unifying algebraic systems that connect various branches of mathematics (algebraic and differential NCG, ring and module theory, category theory, Hopf algebras and quantum groupoids, mathematical computer science).
I have an active research programme, which currently is centred at analysis of orbifolds in noncommutative geometry and their relation to curved differential graded algebras. This programme combines methods of algebra with those of homological algebra, category theory, functional analysis (operator algebras) and mathematical physics.
My research lies on the interface of algebra, topology and combinatorics, and explores some algebraic structures which arise in these fields, notably coalgebras and Hopf algebras. In work with Clarke and Whitehouse I explored the algebras of operations associated to various flavours of K-theory, and Whitehouse and I continue to explore how these may be described in terms of free/cofree functors between appropriate categories. In many cases, the constructions are not known, or the published constructions are flawed by a failure to address some of the notoriously non-intuitive behaviour of coalgebras, such as the non-uniqueness of subcoalgebra structures. In other work, investigation into commutativity in stable homotopy led, by a circuitous route, to the study of the Leibniz-Hopf algebra, something closely related to operations in cohomology, but also central in algebraic combinatorics, and of interest in representation theory. The links between what is known about this object in these different branches of mathematics are still being established.
Together with Panin, Garkusha developed the machinery of K-motives of algebraic varieties leading to solutions of certain problems for the motivic specral sequence and motivic K-theory.
Motivated by Morel-Voevodsky's motivic homotopy theory, Garkusha developed homotopy theory of algebras with applications in algebraic K-theory. He introduced and studied various bivariant K-theories. As an application, he recovered the universal bivariant K-theory kk(A,B), the algebraic counterpart of the Kasparov KK-theory of C *-algebras.
Garkusha also gave a classification of certain localizations of quasi-coherent sheaves, which have applications for reconstruction problems in algebraic geometry.
Garkusha's early research was in non-commutative ring theory. He introduced the classes of fp-flat and fp-injective modules to get various criteria for absolutely pure rings and absolutely pure group rings. Garkusha also introduced and studied almost regular rings. This class of rings naturally occurs in the Freyd generating hypothesis as was shown by Hovey-Lockridge-Puninski.
My research began in topology and homotopy theory; I worked on the topology of moduli spaces and diffeomorphism groups using tools and ideas from algebraic K-theory and cobordism theory. This was motivated by the exciting developments relating to Madsen and Weiss's celebrated proof of the Mumford conjecture; their proof brought ideas from homotopy theory and cobordism theory to bear on algebraic geometry, and much of my work carried on this theme. I described the stable diffeomorphism groups of simply connected 4-manifolds and discovered new families of cohomology classes on the Deligne-Mumford compactified moduli space of complex algebraic curves. I have also worked with cyclic operads and Kontsevich's graph homology and questions of rational homotopy theory and formality of operads. Topological field theory is another closely related subject that I have worked in.
Recently I have started to work in the field of tropical geometry. This is a young and extremely active area closely connected to algebraic geometry and combinatorial geometry. In the 1960s Grothendieck's revolutionized algebraic geometry with his theory of schemes. Since the beginning of tropical geometry there has been a desire to realize this subject as a form of algebraic geometry, and in recent work Noah Giansiracusa and showed exactly how to do this so that the machinery of modern scheme theoretic algebraic geometry can be brought to bear on problems in tropical geometry.