# Professor Roger Plymen

### Pure Mathematics Group

- Room 1.211, Alan Turing Building
- School of Mathematics,
- University of Manchester
- Manchester M13 9PL, UK
- plymen[at]manchester.ac.uk
- Tel: +44 (0) 7973 698510

### Affiliation:

Emeritus professor of Pure Mathematics at Manchester University, Visiting professor at Southampton University

## Page contents:

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

## Research

- Representation theory of p-adic groups; complex topological K-theory.

## 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).
- The Skewes Number. The Skewes Number is the smallest number for which the prime number theorem undercounts the number of primes. The best upper bound for the Skewes Number is around exp(727.9513). See my Open Math Notes on the AMS website.

### (2010 - 2018)

- On depth in the local Langlands correspondence for tori. With A.-M. Aubert.
- Smooth duals of inner forms of GL_n and SL_n . With A-M. Aubert, P. Baum, M. Solleveld.
- On the spectra of finite type algebras. With A-M. Aubert, P. Baum, M. Solleveld.
- Some invariance properties of local constants. For the video, go to www.fields.utoronto.ca
- Stratified Langlands duality in the A_n tower. With Graham Niblo, Nick Wright. J. Noncommutative Geometry (2018) to appear.
- Poincare' duality and Langlands duality for extended affine Weyl groups . With G. Niblo and N. Wright. Annals of K-Theory 3-3 (2018) 491--522.
- On L-packets and depth for SL_2(K) and its inner form. With Anne-Marie Aubert, Sergio Mendes, Maarten Solleveld. Int.J. Number Theory 13 (2017) 2545--2568
- Conjectures about p-adic groups and their noncommutative geometry. With A-M. Aubert, P. Baum, M. Solleveld. In "Around Langlands Correspondences", Contemporary Math. 691 (2017) 15 - 51.
- Hecke algebras for inner forms of p-adic special linear groups. With A-M. Aubert, P. Baum, M. Solleveld. J. Inst. Math. Jussieu 16 (2017) 351 - 419.
- The principal series of p-adic groups with disconnected center. With A.-M. Aubert, P. Baum, M. Solleveld. Proc. London Math. Soc. 114 (2017) 798 - 854.
- Functoriality and K-theory for GL_n(R). With S. Mendes. Munster J. Math. 10 (2017) 39--58.
- The local Langlands correspondence for inner forms of SL(n). With A.-M. Aubert, P. Baum, M. Solleveld. Res. Math. Sci. (2016) 3:32
- 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.
- 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. W. Ballmann et al.(eds.), Arbeitstagung 2013, Progress in Math. 319 (2016) 17 - 41.
- 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

For my recent Online Math Notes on the Skewes Number, please click here. The Skewes Number is the least number for which the prime number theorem undercounts the number of primes. Two papers on this subject were published in July 2010: Chao-Plymen and Saouter-Demichel. The best upper bound for the Skewes Number is around exp(727.9513), a number with 316 digits, see Stefanie Zegowitz . For a popular lecture on this topic, please click here. The article of Stoll and Demichel 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.