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Andrew Baker: Future research topics for PhD students

Below are some topics on which I would be willing to supervise future PhD students. You can get an idea of my past research activities by looking at my lists of publications (pdf file) and recent preprints. My main research area is Algebraic Topology but I also dabble extensively in algebra and number theory often with a view to applications.

Please email me if you want to discuss the possibilities for PhD work. For details of my department and how to apply for postgraduate degrees in it see our departmental home page and postgraduate studies page.

• The homotopy theory of brave new rings and modules: In the past decade a major revolution has revitalised the foundations of stable homotopy theory, leading to new stable homotopy categories based on modules over S-algebras. Each of these gives rise to questions analogous to classical ones for the sphere spectrum. In recent work with A. Jeanneret and A. Lazarev, I have been exploring some of the basic objects such as cooperation Hopf algebroids and Adams spectral sequences. In other work with B. Richter I have investigated the existence of commutative S-algebras structures on spectra. Important examples such as those related to the spectra of complex cobordism MU and K-theory lead to interesting calculations. Other topics that I am currently thinking about include J. Rognes's Galois theory for commutative S-algebras and topological André-Quillen cohomology. I have several ideas for further work, all of which would involve lots of algebra including homological algebra, (commutative) ring theory, etc. Work in this area would suite someone who is interested in learning and applying such algebra. For my recent work see
  • A. Baker & A. Jeanneret: Brave new Hopf algebroids and extensions of MU-algebras, Homology, Homotopy and Applications 4 (2002) 163-173. (pdf file)
  • A. Baker & A. Lazarev: On the Adams Spectral Sequence for R-modules, Algebraic & Geometric Topology 1 (2001), 173-99. (pdf file)
  • A. Baker & A. Lazarev: Topological Hochschild cohomology and generalized Morita equivalence, Algebraic & Geometric Topology 4 (2004), 623-645.
  • A. Baker & B. Richter: Gamma-cohomology of rings of numerical polynomials and E-infinity structures on K-theory, Glasgow University Mathematics Department preprint 02/50. (pdf file; math.AT/0304473)
  • A. Baker & A. Jeanneret: Brave new Bockstein operations, Glasgow University Mathematics Department preprint 03/18. (pdf file)
  • A. Baker & B. Richter: Invertible modules for commutative S-algebras with residue fields, Glasgow University Mathematics Department preprint 04/20. (pdf file; math.AT/0405525)
  • A. Baker & B. Richter, Realizability of algebraic Galois extensions by strictly commutative ring spectra, Glasgow University Mathematics Department preprint 04/31. (pdf file; math.AT/0406314)

• Formal groups in algebraic topology: A major theme in algebraic topology for the past 30 years has been the application of ideas centred around the appearance of formal group laws in complex oriented cohomology theories. The most interesting aspects of this tend to involve number theoretic ideas and their consequences. However, even working over the rational numbers can lead to important results. Much of my research has involved formal group laws over finite characteristic fields and in particular those associated to elliptic curves and abelian varieties. There is considerable scope for further developments involving input from algebraic geometry to study operations in periodic cohomology theories. This would be a good area for someone wanting to learn about number theory or algebraic geometry and their applications to geometric questions. Some of my relevant papers are
  • A. Baker: On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537-48. (pdf file)
  • A. Baker: Elliptic genera of level N and elliptic cohomology, Jour. Lond. Math. Soc. 49 (1994), 581-93. (pdf file)
  • A. Baker: Operations and cooperations in elliptic cohomology, Part I: Generalized modular forms and the cooperation algebra, New York J. Math. 1 (1995), 39-74. (pdf file)
  • A. Baker: A supersingular congruence for modular forms, Acta Arithmetica, 85 (1998), 91-100. (pdf file)
  • A. Baker: Isogenies of supersingular elliptic curves over finite fields and operations in elliptic cohomology, Glasgow University Mathematics Department preprint 98/39. (pdf file)
  • A. Baker & C.B. Thomas: Classifying spaces, Virasoro equivariant bundles, elliptic cohomology and Moonshine, Glasgow University Mathematics Department preprint 99/39. (pdf file)
  • A. Baker: On the Adams E_2-term for elliptic cohomology, in Proceedings of the 1999 Boulder Conference in Contemp. Math.271 (2001), 1-15. (pdf file)
  • A. Baker: On the cohomology of some Hopf algebroids and Hattori-Stong theorems, Homology, Homotopy and Applications 2 (2000), 29-40. (pdf file)
  • A. Baker: I_n-local Johnson-Wilson spectra and their Hopf algebroids, Documenta Mathematica 5 (2000), 351--364. (pdf file)

• Interesting finite group actions and their topology: This is a fairly recent interest which formed the basis of my student Mark Brightwell's 1999 PhD thesis. There are many possible ways to follow up his work and in particular its relationships with fashionable topics such as Moonshine, Mirror Symmetry and Elliptic Cohomology. This area also involves number theory and algebraic geometry as well as finite group theory, particularly simple groups and their realisation as symmetry groups. I am particularly interested in calculating topological invariants of desingularised orbifolds making use of equivariant Atiyah-Singer Index theory and this would be a good topic for a PhD thesis.

• Quillen's geometric interpretation of cobordism: cobordism is traditionally viewed as a relation on manifolds of finite dimension and usually applied to compact manifolds. However, Quillen gave a description of complex cobordism in which compactness was replaced by the use of proper maps. The PhD thesis of Cenap Özel began the investigation of this and showed that much of Quillen's description works for infinite dimensional manifolds and proper Fredholm maps. However, there are technical difficulties resulting from a lack of good transversality theory for Fredholm maps and there are many intriguing questions about this idea which may need modifying to yield its full potential. This topic would be very geometric and probably requires a study of the analysis of maps between infinite dimensional manifolds as well as `classical' cobordism theory. For details of what we already know see
  • A. Baker & C. Özel: Complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups, in Geometry and Topology: Ĺrhus, Contemp. Math. 258 (2000), 1-19. (pdf file)
  There are other interesting geometric questions about even the finite dimensional theory that appear not to have been studied and these would also provide a good area to work on.

As well as the above topics, I have dabbled in many other aspects of cobordism and the machinery of algebraic topology and if you have any suggestions for a thesis topic based on my earlier work please let me know.

[last updated 02/09/2004]