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4. Stephen Wills

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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.

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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

	Project	Funding Body	Start Date	End Date	Award
	Supersymmetric Quantum Stochastic Analysis	Enterprise Ireland	01-AUG-03	30-SEP-06	€64,600.00
	Operator theory & Operator Algebras in Cork: A Conference in memory of Gerard J. Murphy	Science Foundation of Ireland	01-APR-08	31-JUL-08	€3,000.00
	IRCSET "A random-walk approach to quantum stochastic processes"	Irish Research Council for Science, Engineering & Technology (IRCSET)	10-OCT-08	30-SEP-07	€96,300.00

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Publications

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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):349-375 [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   [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]

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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) , 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]

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Professional Activities

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Professional Associations

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

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Conference Contributions

	Year	Publication
	(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]

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Employment

	Employer	Position	From / To
	UCC	Lecturer	01-OCT-01 /
	University of Nottingham	Lloyds of London Tercentenary Research Fellow	01-NOV-99 / 30-SEP-01
	Université de Paris VI	Visiting Postdoctoral Researcher	01-APR-99 / 31-OCT-99
	Indian Statistical Institute, Delhi Centre	Royal Society Research Fellow	01-OCT-98 / 31-MAR-99

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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

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Journal Activities

	Journal	Role	To / From
	Mathematical Proceedings Of The Royal Irish Academy	Member of Editorial Board	01-JAN-10 -

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Other Activities

	Description
	Visits Abroad Year Destination of Visit Purpose Period 2008 University of Lancaster, UK Research collaboration 11/8/08-13/8/08 2006 University of Lancaster, UK Research collaboration 23/5/06-24/6/08 2006 Politecnico di Milano, Italy Research collaboration 9/4/06-6/5/06 2006 Universidad Catolica, Santiago, Chile Research collaboration 5/1/08-25/3/06   2005 University of Nottingham, UK Research collaboration 10/4/05-16/4/05 2004 Indian Statistical Institute, Kolkata, India Research collaboration 10/12/04-23/12/04 2004 University of Nottingham, UK Research collaboration 6/9/04-11/9/04 2003 Università di Genova, Italy Research collaboration 4/9/03-10/9/03 2003 University of Nottingham, UK Research collaboration 13/2/03-16/2/03
	Research Talks:   Details of invited talks Year Title Location Date 2008 Completely positive semigroups and dilations University of Glasgow, UK 20/5/08 2006 Quantisation of probability and applications London School of Economics 8/6/06 2005 Operator cocycles on Fock space TCD 20/9/05 2004 Commutation relations and stochastic integration TCD 7/5/04 2003 Endomorphism semigroups and Markovian cocycles Università di Genova, Italy 8/9/03 2003 Introduction to Quantum Probability DCU 9/4/03
	Visiting Academics Year Name Institute of origin Purpose Period 2006 Lingaraj Sahu Indian Statistical Institute Research collaboration 30/8/06-16/9/06

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Teaching Activities

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Teaching Interests

2001-02: MA1015, MA3056, MS2001, MS3001
2002-03: MA2052, MA3056, MS2001, MS3001
2003-04: MA1008, MA3056, MS2001, MS3001
2004-05: MA1008, MA1053, MS2001, MS3001
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
2015-16: MA1011, MA1012, MA3051, MA4059, MS1002
2016-17: MA1011, MA1012, MA3051, MA4403, MS1002, MS3019
2017-18: MA1011, MA1012, MA3051, MA3056, MS1002
2018-19: MA1011, MA1012, MA2051, MA3056, MS1002
2019-20: MA1011, MA1012, MA2051, MA3056, MS1002