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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. I have recently begun looking at discrete time quantum Markov process and invariance principles, and their relation with representation of the Thompson group/monoid.
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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
	(2023)	'Markovianity and the Thompson monoid F+' Köstler, Claus; Krishnan, Arundhathi; Wills, Stephen J (2023) 'Markovianity and the Thompson monoid F+'. Journal of Functional Analysis, 284 (6):1-70 [DOI] [Details]
	(2021)	'Quantum Stochastic Cocycles and Completely Bounded Semigroups on Operator Spaces II' J. Martin Lindsay, Stephen J. Wills (2021) 'Quantum Stochastic Cocycles and Completely Bounded Semigroups on Operator Spaces II'. Communications in Mathematical Physics, 383 (1):153-199 [DOI] [Full Text] [Details]
	(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 [DOI] [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   [DOI] [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 [DOI] [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 [DOI] [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 [DOI] [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 [DOI] [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 [DOI] [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 [DOI] [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 [DOI] [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 [DOI] [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 [DOI] [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 [DOI] [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); Ordinary Committee Member (2013)	01-JAN-03 /

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Conference Contributions

	Year	Publication
	(2022)	BCRI Mini-Symposium: Noncommutative Probability & Quantum Information, Stephen Wills (2022) BCRI Mini-Symposium: Noncommutative Probability & Quantum Information. [Conference Organising Committee Member], BCRI Mini-Symposium: Noncommutative Probability & Quantum Information, UCC , 10-OCT-22 - 10-OCT-22. [Details]
	(2021)	Noncommutative Algebra, Probability and Analysis in Action, Stephen Wills (2021) The right Hudson-Parthasarathy QSDE. [Invited Lectures (Conference)], Noncommutative Algebra, Probability and Analysis in Action, Greifswald, Germany (Zoom) , 21-SEP-21 - 21-SEP-21. [Details]
	(2019)	KBS Fest: A Conference on the Occasion of the 75th Birthday of KB Sinha, Stephen Wills (2019) The right Hudson-Parthasarathy QSDE. [Invited Lectures (Conference)], KBS Fest: A Conference on the Occasion of the 75th Birthday of KB Sinha, ISI Bangalore, India , 14-DEC-19 - 14-DEC-19. [Details]
	(2018)	39th International Conference on Infinite Dimensional Analysis, Quantum Probability and Related Topics, Stephen Wills (2018) Constructing the quantum exclusion process. [Invited Lectures (Conference)], 39th International Conference on Infinite Dimensional Analysis, Quantum Probability and Related Topics, Levico, Italy , 15-OCT-18 - 19-OCT-18. [Details]
	(2018)	Quantum Information Theory, Stephen Wills (2018) Quantum Information Theory. [Conference Organising Committee Member], Quantum Information Theory, UCC , 15-AUG-18 - 15-AUG-18. [Details]
	(2018)	Randomness and Symmetry, Stephen Wills (2018) Randomness and Symmetry. [Conference Organising Committee Member], Randomness and Symmetry, UCC , 10-SEP-18 - 10-SEP-18. [Details]
	(2017)	Advances in Noncommutative Mathematics, Stephen Wills (2017) A brief guide to quantum stochastic differential equations. [Keynote Speaker], Advances in Noncommutative Mathematics, ISI Bangalore, India , 11-JAN-17 - 13-JAN-17. [Details]
	(2015)	George Boole Mathematical Sciences Conference, Stephen Wills (2015) George Boole Mathematical Sciences Conference. [Conference Organising Committee Member], George Boole Mathematical Sciences Conference, UCC , 17-AUG-15 - 28-AUG-15. [Details]
	(2015)	Annual Irish Mathematical Society Meeting, Stephen Wills (2015) Annual Irish Mathematical Society Meeting. [Conference Organising Committee Chairperson], Annual Irish Mathematical Society Meeting, UCC , 27-AUG-15 - 28-AUG-15. [Details]
	(2015)	Algebraic and Analytic Aspects of Quantum Levy Processes, Stephen Wills (2015) Two approaches to the construction of quantum stochastic cocycles. [Invited Lectures (Conference)], Algebraic and Analytic Aspects of Quantum Levy Processes, Greifswald, Germany , 09-MAR-15 - 13-MAR-15. [Details]
	(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 /
	Università degli Studi di Genova	CNR Visiting Researcher	10-FEB-98 / 29-MAY-98
	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

Research Talks/Seminars GIven: UCC, Cork, Ireland, January 23, 2018 (Construction of quantum stochastic cocycles) Universite Franche-Comte, June 28, 2016 (Construction of quantum stochastic cocycles) QUB, UK, October 30, 2015 (Construction of quantum stochastic cocycles) ISI Bangalore, India, February 11, 2014 (An algebraic construction of quantum flows with unbounded generators) UCC, Cork, Ireland, March 3 & 10, 2011 (Subordination and endomorphism semigroups) NUI Galway, Ireland, November 12 2009 (Markov processes and semigroups and their noncommutative counterparts) University of Oxford, UK, April 28, 2009 (Completely positive semigroups and dilations) University of Glasgow, UK, May 20, 2008 (Completely positive semigroups and dilations) London School of Economics, UK, June 8, 2006 (Quantisation of probability and applications) Pontificia Universidad Catolica de Chile, March 15, 2006 (Markovian cocycles, quantum dynamical semigroups and dilations) TCD, Dublin, Ireland, September 20, 2005 (Operator cocycles on Fock space) TCD, Dublin, Ireland, May 7, 2004 (Commutation relations and stochastic integration) Università di Genova, Genova, Italy, September 8, 2003 (Endomorphism semigroups and Markovian cocycles) DCU, Dublin, Ireland, April 9, 2003 (Introduction to Quantum Probability) UCC (Mathsoc), Cork, Ireland, March, 2003 (Does God play quantum dice?)  NUI Galway, Ireland, April, 2002 (Probabilistic Dilation Theorems) UCC, Cork, Ireland, December, 2001 (Some dilation theorems) University of Nottingham, UK, July, 2000 (Mild solutions of quantum stochastic differential equations) ISI, Bangalore, India, December, 1999 (Markovian cocycles on Hilbert space adapted to a Fock filtration) ISI, Delhi, India, March, 1999 (Strong solutions to the right quantum stochastic differential equation) Collège de France, Paris, France, June, 1998 (Markovian cocycles on operator algebras)

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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