# Professor Roger Plymen

### Pure Mathematics Group

- Room 2.120, Alan Turing Building
- School of Mathematics,
- University of Manchester
- Manchester M13 9PL, UK
- plymen[at]manchester.ac.uk
- Tel: +44 (0) 7973 698510

### Affiliations:

Emeritus professor of Pure Mathematics at Manchester University; Visiting Professor of Mathematics at Southampton University; Adjunct Professor of Mathematics at Pennsylvania State University, USA

## Page contents:

- Research
- Graduate courses
- Publications
- Lecture Notes
- Skewes Number
- Math and music
- 65th Birthday Conference

## Research

- Representation theory and number theory; the local Langlands correspondence; K-theory; Plancherel measure; Skewes' number.

## Graduate courses

- Graduate courses given at Southampton:
- (2011): "The Riemann Explicit Formula 1859 -- 2010".
- (2013): "2 by 2 matrices as stepping-stones". This included representation-theoretic aspects of SL_2(C), SL_2(R) and SL_2(Q_p).
- (2014): "Glimpses into representation theory". This included representation-theoretic aspects of SL_2(q) and SL_2(R).

## Publications

### Books

- p-Adic Methods and Their Applications. Edited by A.J. Baker and R.J. Plymen. Oxford Science Publications 1992.
- Spinors in Hilbert Space. With P.L. Robinson. Cambridge Tract in Mathematics 114 (1994).

### (2010 - 2016)

- On the symmetry of epsilon factors for GL_n. For the video, go to www.fields.utoronto.ca
- On L-packets and depth for SL_2(K). With Anne-Marie Aubert, Sergio Mendes, Maarten Solleveld.
- Conjectures about p-adic groups and their noncommutative geometry. With A-M. Aubert, P. Baum, M. Solleveld.
- The noncommutative geometry of inner forms of p-adic special linear groups. With A-M. Aubert, P. Baum, M. Solleveld.
- Extended affine Weyl groups, the Baum-Connes correspondence and Langlands duality. With G. Niblo and N. Wright.
- The principal series of p-adic groups with disconnected centre. With A.-M. Aubert, P. Baum, M. Solleveld. Submitted.
- The local Langlands correspondence for inner forms of SL(n). With A.-M. Aubert, P. Baum, M. Solleveld. Res. Math. Sciences, to appear.
- Functoriality and K-theory for GL_n(R). With S. Mendes. Munster J. Math., to appear.
- Geometric structure for the principal series of a split reductive p-adic group with connected centre. With A-M. Aubert, P. Baum, M. Solleveld. J. Noncommut. Geom. 10 (2016) 663--680.
- Hecke algebras for inner forms of p-adic special linear groups. With A-M. Aubert, P. Baum, M. Solleveld. J. Inst. Math. Jussieu. Published online 05 May 2015. Cambridge University Press.
- The local spectrum of the Dirac operator for the universal cover of SL_2(R). With J. Brodzki, G. Niblo, N. Wright. J. Functional Analysis 270 (2016) 957 - 975
- Depth and the local Langlands correspondence. With A-M. Aubert, P. Baum, M. Solleveld. Arbeitstagung Bonn 2013: In Memory of Friedrich Hirzebruch, Progress in Math. 319 (2016).
- Geometric structure in smooth dual and local Langlands conjecture. The 11th Takagi Lectures. With A.-M. Aubert, P. Baum, M. Solleveld. Japan. J. Math. 9 (2014) 99-136.
- On the local Langlands correspondence for non-tempered representations. With A.-M. Aubert, P. Baum, M. Solleveld. Munster J. of Math. 7 (2014) 27 - 50.
- K-theory and the connection index. With T. Kamran. Bull. London Math. Soc. 45 (2013) 111 - 119.
- Geometric structure in the tempered dual of SL(4). With K.F. Chao. Bull. London Math. Soc. 44 (2012) 460 - 468.
- Geometric structure in the representation theory of p-adic groups II. With A.-M. Aubert and P. Baum. Contemporary Math. 543 (2011) 71 - 90.
- Geometric structure in the principal series of the p-adic group G_2. With A.-M. Aubert and P. Baum. Represent. Theory 15 (2011) 126 - 169.
- R-groups and geometric structure in the representation theory of SL(N). With J. Jawdat. J. Noncommutative Geometry 4 (2010) 265 - 279.
- A new bound for the smallest x with \pi(x) > li(x). With K.F. Chao. Int. J. Number Theory 6 (2010) 681 - 690.

### (2001-2009)

- Geometric structure in the representation theory of p-adic groups. With A.-M. Aubert and P. Baum. C. R. Acad. Sci. Paris, Ser. I, 345 (2007) 573-578. Online published version.
- Base change and K-theory for GL(n). With S. Mendes. J. Noncommutative Geometry 1 (2007) 311-331.
- The Hecke algebra of a reductive p-adic group: a geometric conjecture. With A.-M. Aubert and P. Baum, Aspects of Mathematics 37, Vieweg Verlag (2006) 1-34.
- Cycles in the chamber homology of GL(3). With A-M Aubert and S. Hasan. K-Theory 37 (2006) 341-377. Online published version.
- Entire cyclic cohomology of Schatten ideals. With J. Brodzki. Homology, Homotopy and Applications, 7(3) (2005) 37-52.
- Plancherel measure for GL(n,F) and GL(m,D): explicit formulas and Bernstein decomposition. With A-M. Aubert. J. Number Theory, 112 (2005) 26-66. Online published version.
- Explicit Plancherel formula for the p-adic group GL(n). With A-M. Aubert. C.R. Acad. Sci. Paris, Ser. I, 338 (2004) 843-848. Online published version.
- Local-global principle for the Baum-Connes conjecture with coefficients. With P. Baum and S. Millington. K-theory, 28 (2003) 1-18.
- Complex structure on the smooth dual of GL(n). With J. Brodzki. Documenta Math 7 (2002) 91-112.
- Reduced C*-algebra of the p-adic group GL(n) II, J. Functional Analysis, 196 (2002) 119-134. Online published version.
- Chern character for the Schwartz algebra of GL(n). With J. Brodzki. Bulletin London Math. Soc. 34 (2002) 219-228. Online published version.
- Proof of the Baum-Connes conjecture for reductive adelic groups. With P. Baum and S. Millington. C.R. Acad. Sci. Paris. 332 (2001) 195-201. Online published version.

### Before 2001

- Geometry of the smooth dual of GL(n). With J. Brodzki. C.R. Acad. Sci. Paris 331 (2000) 213 - 218. Online published version.
- Representation theory of p-adic groups : a view from operator algebras. With P. Baum & N. Higson. The mathematical legacy of Harish-Chandra : a celebration of representation theory & harmonic analysis. Proc. Symp. Pure Math. 68 (2000) 111--149.
- Periodic cyclic homology of certain nuclear algebras. With J. Brodzki. C.R. Acad. Sci. Paris 329 (1999) 671 - 676. Online published version.
- Proof of the Baum-Connes conjecture for p-adic GL(n). With P. Baum & N. Higson. C.R. Acad. Sci. Paris 325 (1997) 171--176. Online published version.
- The representation theory of p-adic GL(n) and Deligne-Langlands parameters. With J.E. Hodgins."Analysis, Geometry and Probability", edited by R. Bhatia. Hindustan Book Agency, Texts and Readings in Mathematics 10 (1996) 54 - 72.
- Universal example for S-arithmetic groups. With J.E. Hodgins. Expo. Math. 15 (1997) 149 -- 159.
- Equivariant homology for SL(2) of a p-adic field. With P. Baum & N. Higson. Contemporary Math 148 (1993) 1-18.
- K-theory and elliptic representations of SL(l). Houston J. Math. (Herrero Memorial Issue) 18 (1992) 25-32.
- Coupling constants for p-adic groups, in ``p-Adic Methods and Their Applications'', edited by A.J. Baker & R.J. Plymen, Oxford Science Publications, Oxford University Press, 1992, p. 63-72.
- L^2-Fourier transform for reductive p-adic groups. With C.W. Leung. Bull. London Math. Soc. 23 (1991) 146-152.
- Arithmetic aspect of operator algebras. With C.W. Leung. Compositio Math. 77 (1991) 293-311.
- Equivalence bimodules in the representation theory of reductive groups. Proc. Symp. Pure Math. 51 (1990) 267-272.
- Reduced C*-algebra for reductive p-adic groups. J. Functional Analysis 88 (1990) 251-266.
- Complex conformal rescaling and spin-structure. With B.W. Westbury. Bull. London Math. Soc. 19 (1987) 363-370.
- The reduced C*-algebra of the p-adic group GL(n). J. Functional Analysis 72 (1987) 1-12.
- Strong Morita equivalence, spinors and symplectic spinors. J. Operator Theory 16 (1986) 305-324.
- K-theory of the reduced C*-algebra of SL(2,Q_p), in ``Operator algebras and their connections with topology and ergodic theory'', Lecture Notes in Mathematics 1132 (1985) 409-420.
- The Dirac operator and the principal series for complex semisimple Lie groups. With M.G. Penington. J. Functional Analysis 53 (1983) 269-286.
- The Weyl bundle. J. Functional Analysis 49 (1982) 186-197.
- Cyclotomic integers and the inner invariant of Connes. J. London Math. Soc. 22 (1980) 14-20.
- On the outer and inner invariants of Connes. With K.R. Parthasarathy. J. Functional Analysis 38 (1980) 1-15.
- The Laplacian and the Dirac operator in infinitely many variables. Compositio Math 41 (1980) 137-152.
- Automorphic group representations: the hyperfinite II_1 factor and the Weyl algebra, in "Algébres d'opérateurs", Lecture Notes in Mathematics 725 (1979) 291-306.
- Automorphic group representations: a new proof of Blattner's theorem. With de la Harpe. J. London Math. Soc. 19 (1979) 509-522.
- Some recent results on infinite-dimensional spin groups. Advances in Math., supplementary study 6 (1979) 159-171.
- Projective representations of the infinite orthogonal group. Mathematika 24 (1977) 115-121.
- Spinors in Hilbert space. Math. Proc. Camb. Phil. Soc. 80 (1976) 337-347.
- A variant of the fundamental theorem of projective geometry. With C.M. Williams. Mathematika 23 (1976) 89-93.
- On the Weyl character formula for SU(n). Int. J. Theor. Phys. 15 (1976) 201-206.
- A model of the universal covering group of SO(E)_2. With R.F. Streater. Bull. Lond. Math. Soc. 7 (1975) 283-288.
- On the spin algebra of a real Hilbert space. J. London Math. Soc. With R.M.G. Young. 9 (1974) 286-292.
- Two-state systems and the two-sphere. Nuovo Cimento 13B (1973) 55-58.
- Dispersion-free normal states. Il Nuovo Cimento 54A (1968) 862-870.
- A modification of Piron's axioms. Helv. Phys. Acta 41 (1968) 69-74.
- C*-algebras and Mackey's axioms. Commun. Math. Phys. 8 (1968) 132 - 146.
- A model of the arithmetic of alephs in the equation calculus. Math. Logic Quarterly 7 (1961) 257 - 258.

## Lecture Notes

- Base change and K-theory for GL(n).
- Noncommutative Fourier analysis.
- Noncommutative Geometry: Illustrations from the representation theory of GL(n).
- On the difference \pi(x) - li(x).

## Skewes Number

A Skewes Number \Xi is an upper bound for the smallest natural number for which \pi(x) > li(x). After a gap of 10 years, two papers on this subject were published in July 2010: Chao-Plymen and Saouter-Demichel. In addition, several students here (Christine Lee, Munibah Tahir, Stefanie Zegowitz) have made contributions to this topic. The smallest known value of \Xi is around exp(727.9513), a number with 316 digits. For a popular lecture on this topic, please click here. The recent article of Stoll and Demichel in Math. Comp. (2011) is full of lively interest.

## Math and music

Saloni Shah, a former undergraduate, has written a superlative project on math and music: please click here.

## 65th Birthday Conference

For the poster and list of speakers, please click here.