Biography
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 Esemigroups, and beginning to study quantum random walks on quantum groups.Research Grants
Project  Funding Body  Start Date  End Date  Award  

Supersymmetric Quantum Stochastic Analysis  Enterprise Ireland  01AUG03  30SEP06  €64,600.00  
Operator theory & Operator Algebras in Cork: A Conference in memory of Gerard J. Murphy  Science Foundation of Ireland  01APR08  31JUL08  €3,000.00  
IRCSET "A randomwalk approach to quantum stochastic processes"  Irish Research Council for Science, Engineering & Technology (IRCSET)  10OCT08  30SEP07  €96,300.00 
Publications
Peer Reviewed Journals
Year  Publication  

(2015)  'An algebraic construction of quantum flows with unbounded generators' Belton, Alexander C.R.; Wills, Stephen J. (2015) 'An algebraic construction of quantum flows with unbounded generators'. Annales De L'Institut Henri Poincaré (B) Probabilités Et Statistiques, 51 (1):349375 [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):30963139 [DOI] [Details]  
(2010)  'Esemigroups subordinate to CCR flows' Wills, Stephen J.; (2010) 'Esemigroups subordinate to CCR flows'. Communications on Stochastic Analysis, 4 (4):579592 [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):535556 [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):191211 [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):519529 [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):279310 [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):221247 [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):255272 [Details]  
(2001)  'A stochastic Stinespring theorem' Goswami, Debashish; Lindsay, J. Martin; Wills, Stephen J. ; (2001) 'A stochastic Stinespring theorem'. Mathematische Annalen, 319 (4):647673 [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):613621 [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 :158171 [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):269305 [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):505543 [Details] 
Conference Publications
Year  Publication  

(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) , 15MAR01  20MAR01 , pp.181193 [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.455470 [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.109119 [Details] 
Professional Activities
Professional Associations
Association  Function  From / To  

Irish Mathematical Society  Member (2002); Ordinary Committee Member (2008); Vice President (20092010); President (20112012)  01JAN03 / 
Conference Contributions
Year  Publication  

(2014)  Operator & Spectral theory, Operator algebras, Noncommutative Geometry & Probability, Stephen Wills (2014) Dilation of quantum subMarkov semigroups. [Invited Lectures (Conference)], Operator & Spectral theory, Operator algebras, Noncommutative Geometry & Probability, Kerala School of Mathematics, India , 07FEB14  14FEB14. [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 , 23SEP13  24SEP14. [Details]  
(2012)  Quantum Probability, Noncommutative Geometry, Quantum Information, Stephen Wills (2012) Subordinate Esemigroups. [Invited Lectures (Conference)], Quantum Probability, Noncommutative Geometry, Quantum Information, Edinburgh, UK , 16JAN12  18JAN12. [Details]  
(2011)  Product Systems and Independence in Quantum Dynamics, Stephen Wills (2011) ESemigroups Subordinate to CCR Flows. [Invited Lectures (Workshops)], Product Systems and Independence in Quantum Dynamics, Greifswald, Germany , 14MAR11  18MAR11. [Details]  
(2009)  Joint Meeting of the 61st British Mathematical Colloquium and the 22nd Annual Meeting of the Irish Mathematical Society, Stephen Wills (2009) Projectionvalued 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 , 06APR09  09APR09. [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 , 07MAY08  09MAY08. [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 , 12JAN06  12JAN06. [Details]  
(2006)  27th QPConference: Quantum Probability and Infinite Dimensional Analysis, Wills, Stephen J; (2006) Strong solutions to the right HudsonParthasarathy quantum stochastic differential equation. [Invited Oral Presentation], 27th QPConference: Quantum Probability and Infinite Dimensional Analysis, Nottingham, UK , 18JUL06  22JUL06. [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 , 20FEB05  26FEB05. [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 , 12JAN06  12JAN06. [Details]  
(2005)  NonCommutative Analysis, Quantum Theory and Stochastic Analysis, Wills, Stephen J; (2005) Generators of operator Markovian cocycles. [Oral Presentation], NonCommutative Analysis, Quantum Theory and Stochastic Analysis, Lancaster, UK , 20OCT05  21OCT05. [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 , 14JUN05  17JUN05. [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 , 01JUN05  04JUN05. [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 , 29MAY03  31MAY03. [Details] 
Employment
Employer  Position  From / To  

UCC  Lecturer  01OCT01 /  
University of Nottingham  Lloyds of London Tercentenary Research Fellow  01NOV99 / 30SEP01  
Université de Paris VI  Visiting Postdoctoral Researcher  01APR99 / 31OCT99  
Indian Statistical Institute, Delhi Centre  Royal Society Research Fellow  01OCT98 / 31MAR99 
Education
Year  Institution  Qualification  Subject  

1993  University of Oxford, UK  BA (Hons)  Mathematics  
1994  King's College London, UK  MSc  Mathematics  
1997  University of Nottingham, UK  PHD  Mathematics 
Journal Activities
Journal  Role  To / From  

Mathematical Proceedings Of The Royal Irish Academy  Member of Editorial Board  01JAN10  
Other Activities
Description  

Visits Abroad
 
Research Talks: Details of invited talks
 
Visiting Academics

Teaching Activities
Teaching Interests
200102: MA1015, MA3056, MS2001, MS3001200203: MA2052, MA3056, MS2001, MS3001
200304: MA1008, MA3056, MS2001, MS3001
200405: MA1008, MA1053, MS2001, MS3001
200506: MA4403, MS2001, MS3001
200607: MA1008, MA2055, MA4403, MS2001
200708: MA1008, MA4052, MA4403, MS2005
200809: MA1008, MA4052, MA4403, MS2005
200910: MA1008, MA2051, MA4052, MS2005/MS5005
201011: MA1008, MA2051, MA4052, MS2005/MS5005
201112: MA1008, MA2051, MA4052, MS2005/MS5005
201213: MA1008, MA2051, MA4052
201314: MA1008, MA3051, MA4052, MS2002
201415: MA1008, MA3051, MS2002
201516: MA1011, MA1012, MA3051, MA4059, MS1002
201617: MA1011, MA1012, MA3051, MA4403, MS1002, MS3019
201718: MA1011, MA1012, MA3051, MA3056, MS1002
201819: MA1011, MA1012, MA2051, MA3056, MS1002
201920: MA1011, MA1012, MA2051, MA3056, MS1002