Lecturer
School of Mathematical Sciences
1.60
Western Gateway Building
School of Mathematical Sciences
University College Cork
Cork

T: +353 (0)21 420 5859
F: +353 (0)21 420 5364
E: s.wills@ucc.ie
W: http://euclid.ucc.ie/pages/staff/wills/stephenwills.html

I am a quantum probabilist, with most of my work focusing on quantum stochastic differential equations (QSDEs), for example giving existence and uniqueness results, as well as studying properties of the solutions. I make use of many areas of functional analysis (semigroups, operator algebras and operator spaces) in order to discuss processes on C*-algebras and operator spaces, and QSDEs with unbounded coefficients. Such equations are used in quantum optics, and quantum control and statistics.

Research Interests

Quantum probability Quantum stochastic calculus Operator algebras Operator spaces

My main interest is in the field of quantum probability. This is a noncommutative generalisation of traditional probability theory in which the commutative algebra of (bounded) random variables is replaced by a more general operator algebra. The application of traditional probability to quantum mechanics is problematic in the sense that it cannot adequately deal with phenomenon such as Heisenberg's Uncertainty Principle; the quantised version of probability seeks to rectify this.

The passage of any mathematical theory from its traditional form to a quantised version throws up a number of questions. Ones that are relevant for quantum probability include

  • Does the law of large numbers still work?
  • Are there conditional expectations?
  • What is independence?

The answers for the above are yes, not always, and you have a choice, respectively. Once you make a choice for independence it is then possible to discuss central limit theorems and deduce what should be considered to be a normal random variable. With this then comes the chance to talk about quantised versions of Brownian motion (and other stochastic processes), and related quantum stochastic calculi.

My research to date has been on matters concerned with quantum stochastic calculus, in particular on the theory of quantum stochastic differential equations, and algebraic characterisations of their solutions through cocycles. I use techniques from operator theory, operator algebras and operator spaces, semigroup theory etc. I am investigatings links with E-semigroups, and beginning to study quantum random walks on quantum groups.

Research Grants

 ProjectFunding
Body
Start DateEnd DateAward
Supersymmetric Quantum Stochastic AnalysisEnterprise Ireland01-AUG-0330-SEP-06€64,600.00
Operator theory & Operator Algebras in Cork: A Conference in memory of Gerard J. MurphyScience Foundation of Ireland01-APR-0831-JUL-08€3,000.00
IRCSET "A random-walk approach to quantum stochastic processes"Irish Research Council for Science, Engineering & Technology (IRCSET)10-OCT-0830-SEP-07€96,300.00

Peer Reviewed Journals

YearPublication
(2014)'An algebraic construction of quantum flows with unbounded generators'
Belton, Alexander C.R.; Wills, Stephen J. (2014) 'An algebraic construction of quantum flows with unbounded generators'. Annales De L'Institut Henri Poincaré (B) Probabilités Et Statistiques, [Details]
(2014)'Quantum stochastic cocycles and completely bounded semigroups on operator spaces'
Lindsay, J. Martin; Wills, Stephen J. (2014) 'Quantum stochastic cocycles and completely bounded semigroups on operator spaces'. International Mathematics Research Notices, 2014 (11):3096-3139 [DOI] [Details]
(2010)'E-semigroups subordinate to CCR flows'
Wills, Stephen J.; (2010) 'E-semigroups subordinate to CCR flows'. Communications on Stochastic Analysis, 4 (4):579-592 Globe icon [Details]
(2007)'Quantum stochastic operator cocycles via associated semigroups'
Lindsay, J. Martin; Wills, Stephen J. ; (2007) 'Quantum stochastic operator cocycles via associated semigroups'. Mathematical Proceedings of the Cambridge Philosophical Society, 142 (3):535-556 [Details]
(2007)'On the generators of quantum stochastic operator cocycles'
Wills, Stephen J. ; (2007) 'On the generators of quantum stochastic operator cocycles'. Markov Processes and Related Fields, 13 (1):191-211 [Details]
(2006)'Construction of some quantum stochastic operator cocycles by the semigroup method'
Lindsay, J. Martin; Wills, Stephen J. ; (2006) 'Construction of some quantum stochastic operator cocycles by the semigroup method'. Proceedings of the Indian Academy of Sciences: Mathematical Sciences, 116 (4):519-529 [Details]
(2003)'Solving quantum stochastic differential equations with unbounded coefficients'
Fagnola, Franco; Wills, Stephen J. ; (2003) 'Solving quantum stochastic differential equations with unbounded coefficients'. Journal of Functional Analysis, 198 (2):279-310 [Details]
(2003)'Dilation of Markovian cocycles on a von Neumann algebra'
Goswami, Debashish; Lindsay, J. Martin; Sinha, Kalyan B.; Wills, Stephen J. ; (2003) 'Dilation of Markovian cocycles on a von Neumann algebra'. Pacific Journal of Mathematics, 211 (2):221-247 [Details]
(2003)'Homomorphic Feller cocycles on a C*-algebra'
Lindsay, J. Martin; Wills, Stephen J. ; (2003) 'Homomorphic Feller cocycles on a C*-algebra'. Journal of the London Mathematical Society, 68 (2) (1):255-272 [Details]
(2001)'A stochastic Stinespring theorem'
Goswami, Debashish; Lindsay, J. Martin; Wills, Stephen J. ; (2001) 'A stochastic Stinespring theorem'. Mathematische Annalen, 319 (4):647-673 [Details]
(2001)'Existence of Feller cocycles on a C*-algebra'
Lindsay, J. Martin; Wills, Stephen J. ; (2001) 'Existence of Feller cocycles on a C*-algebra'. Bulletin of the London Mathematical Society, 33 (5):613-621 [Details]
(2000)'Mild solutions of quantum stochastic differential equations'
Fagnola, Franco ; Wills, Stephen J. ; (2000) 'Mild solutions of quantum stochastic differential equations'. Electronic Communications in Probability, 5 :158-171 [Details]
(2000)'Markovian cocycles on operator algebras adapted to a Fock filtration'
Lindsay, J. Martin; Wills, Stephen J.; (2000) 'Markovian cocycles on operator algebras adapted to a Fock filtration'. Journal of Functional Analysis, 178 (2):269-305 [Details]
(2000)'Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise'
Lindsay, J. Martin; Wills, Stephen J. ; (2000) 'Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise'. Probability Theory and Related Fields, 116 (4):505-543 [Details]

Conference Publications

YearPublication
(2003)Quantum probability and infinite dimensional analysis (Burg, 2001)
Lindsay, J. Martin; Wills, Stephen J. ; (2003) Multiplicativity via a hat trick . In: Freudenberg, Wolfgang eds. Quantum probability and infinite dimensional analysis (Burg, 2001) , 15-MAR-01 - 20-MAR-01 , pp.181-193 [Details]
(2000)Stochastic processes, physics and geometry: new interplays
Lindsay, J. Martin; Wills, Stephen J. ; (2000) Fock space Markovian cocycles: their representation, generation, and dilation Stochastic processes, physics and geometry: new interplays Leipzig, , pp.455-470 [Details]
(1998)Stochastic analysis and mathematical physics
Lindsay, J. Martin; Wills, Stephen J. ; (1998) Completely positive quantum stochastic flows with infinite degrees of freedom . In: World Scientific Publishing, River Edge, NJ eds. Stochastic analysis and mathematical physics Viña del Mar, , pp.109-119 [Details]

Professional Associations

 AssociationFunctionFrom / To
Irish Mathematical Society Member (2002-); Ordinary Committee Member (2008); Vice President (2009-2010); President (2011-2012)01-JAN-03 /

Conference Contributions

YearPublication
(2014)Operator & Spectral theory, Operator algebras, Non-commutative Geometry & Probability,
Stephen Wills (2014) Dilation of quantum sub-Markov semigroups. [Invited Lectures (Conference)], Operator & Spectral theory, Operator algebras, Non-commutative Geometry & Probability, Kerala School of Mathematics, India , 07-FEB-14 - 14-FEB-14. [Details]
(2013)Quantum (Semi)groups and (Co)actions,
Stephen Wills (2013) An algebraic construction of quantum flows with unbounded generators. [Invited Lectures (Conference)], Quantum (Semi)groups and (Co)actions, University of Leeds, UK , 23-SEP-13 - 24-SEP-14. [Details]
(2012)Quantum Probability, Noncommutative Geometry, Quantum Information,
Stephen Wills (2012) Subordinate E-semigroups. [Invited Lectures (Conference)], Quantum Probability, Noncommutative Geometry, Quantum Information, Edinburgh, UK , 16-JAN-12 - 18-JAN-12. [Details]
(2011)Product Systems and Independence in Quantum Dynamics,
Stephen Wills (2011) E-Semigroups Subordinate to CCR Flows. [Invited Lectures (Workshops)], Product Systems and Independence in Quantum Dynamics, Greifswald, Germany , 14-MAR-11 - 18-MAR-11. [Details]
(2009)Joint Meeting of the 61st British Mathematical Colloquium and the 22nd Annual Meeting of the Irish Mathematical Society,
Stephen Wills (2009) Projection-valued operator stochastic cocycles. [Invited Lectures (Conference)], Joint Meeting of the 61st British Mathematical Colloquium and the 22nd Annual Meeting of the Irish Mathematical Society, National University of Ireland Galway, Ireland , 06-APR-09 - 09-APR-09. [Details]
(2008)Operator Theory and Operator Algebras in Cork: A Conference in Memory of Gerard J Murphy,
Wills, Stephen J; (2008) Operator Theory and Operator Algebras in Cork: A Conference in Memory of Gerard J Murphy. [Conference Organising Committee Chairperson], Operator Theory and Operator Algebras in Cork: A Conference in Memory of Gerard J Murphy, Cork , 07-MAY-08 - 09-MAY-08. [Details]
(2006)Quantum Probability and its Applications,
Wills, Stephen J; (2006) On the generators of operator Markovian cocycles. [Oral Presentation], Quantum Probability and its Applications, Universidad de Santiago de Chile, Chile , 12-JAN-06 - 12-JAN-06. [Details]
(2006)27th QP-Conference: Quantum Probability and Infinite Dimensional Analysis,
Wills, Stephen J; (2006) Strong solutions to the right Hudson-Parthasarathy quantum stochastic differential equation. [Invited Oral Presentation], 27th QP-Conference: Quantum Probability and Infinite Dimensional Analysis, Nottingham, UK , 18-JUL-06 - 22-JUL-06. [Details]
(2005)Quantum Probability and Infinite Dimensional Analysis,
Wills, Stephen J; (2005) Dilations of completely positive semigroups. [Invited Oral Presentation], Quantum Probability and Infinite Dimensional Analysis, Levico, Italy , 20-FEB-05 - 26-FEB-05. [Details]
(2006)Second Workshop on Functional Analysis and Evolution Equations,
Stephen Wills (2006) On the generators of operator Markovian cocycles. [Invited Lectures (Conference)], Second Workshop on Functional Analysis and Evolution Equations, Universidad de Santiago de Chile, Chile , 12-JAN-06 - 12-JAN-06. [Details]
(2005)Non-Commutative Analysis, Quantum Theory and Stochastic Analysis,
Wills, Stephen J; (2005) Generators of operator Markovian cocycles. [Oral Presentation], Non-Commutative Analysis, Quantum Theory and Stochastic Analysis, Lancaster, UK , 20-OCT-05 - 21-OCT-05. [Details]
(2005)J.T. Lewis Memorial Conference,
Wills, Stephen J; (2005) On the construction of quantum stochastic processes. [Invited Oral Presentation], J.T. Lewis Memorial Conference, DIT, Dublin , 14-JUN-05 - 17-JUN-05. [Details]
(2005)Operator Algebras and Applications,
Wills, Stephen J; (2005) http://euclid.ucc.ie/pages/staff/wills/OpAlg2005/home.html. [Conference Organising Committee Chairperson], Operator Algebras and Applications, UCC , 01-JUN-05 - 04-JUN-05. [Details]
(2003)Noncommutative Geometry and Operator Algebras,
Wills, Stephen J; (2003) http://euclid.ucc.ie/pages/staff/wills/NCG+OA03/home.html. [Conference Organising Committee Member], Noncommutative Geometry and Operator Algebras, UCC , 29-MAY-03 - 31-MAY-03. [Details]

Employment

 EmployerPositionFrom / To
UCC Lecturer01-OCT-01 /
University of Nottingham Lloyds of London Tercentenary Research Fellow01-NOV-99 / 30-SEP-01
Université de Paris VI Visiting Postdoctoral Researcher01-APR-99 / 31-OCT-99
Indian Statistical Institute, Delhi Centre Royal Society Research Fellow01-OCT-98 / 31-MAR-99

Education

 YearInstitutionQualificationSubject
1993University of Oxford, UK BA (Hons)Mathematics
1994King's College London, UK MScMathematics
1997University of Nottingham, UK PHDMathematics

Journal Activities

 JournalRoleTo / From
Mathematical Proceedings Of The Royal Irish Academy Member of Editorial Board01-JAN-10 -

Other Activities

 Description

Visits Abroad 2008University of Lancaster, UKResearch collaboration11/8/08-13/8/082006University of Lancaster, UKResearch collaboration23/5/06-24/6/082006Politecnico di Milano, ItalyResearch collaboration9/4/06-6/5/062006Universidad Catolica, Santiago, ChileResearch collaboration5/1/08-25/3/06 2005University of Nottingham, UKResearch collaboration10/4/05-16/4/052004Indian Statistical Institute, Kolkata, IndiaResearch collaboration10/12/04-23/12/042004University of Nottingham, UKResearch collaboration6/9/04-11/9/042003Università di Genova, ItalyResearch collaboration4/9/03-10/9/032003University of Nottingham, UKResearch collaboration13/2/03-16/2/03
YearDestination of VisitPurposePeriod

Research Talks:   Details of invited talks 2008Completely positive semigroups and dilationsUniversity of Glasgow, UK20/5/082006Quantisation of probability and applicationsLondon School of Economics8/6/062005Operator cocycles on Fock spaceTCD20/9/052004Commutation relations and stochastic integrationTCD7/5/042003Endomorphism semigroups and Markovian cocyclesUniversità di Genova, Italy8/9/032003Introduction to Quantum ProbabilityDCU9/4/03
YearTitleLocationDate

Visiting Academics 2006Lingaraj SahuIndian Statistical InstituteResearch collaboration30/8/06-16/9/06
YearNameInstitute of originPurposePeriod

Teaching Interests

2005-06: MA4403, MS2001, MS3001
2006-07: MA1008, MA2055, MA4403, MS2001
2007-08: MA1008, MA4052, MA4403, MS2005
2008-09: MA1008, MA4052, MA4403, MS2005
2009-10: MA1008, MA2051, MA4052, MS2005/MS5005
2010-11: MA1008, MA2051, MA4052, MS2005/MS5005
2011-12: MA1008, MA2051, MA4052, MS2005/MS5005
2012-13: MA1008, MA2051, MA4052
2013-14: MA1008, MA3051, MA4052, MS2002
2014-15: MA1008, MA3051, MS2002