## Biography

Research interests: Complex Analysis, Potential Theory, links with Probability Theory, especially Brownian motion.## Research Interests

My main research area is Complex Analysis, in particular the interplay between analytic quantities associated with a planar domain (such as the first Dirichlet eigenvalue of the Laplacian, the torsion function, the hyperbolic metric, lifetime of Brownian motion) and geometric quantities associated with the domain (such as its area, its inradius). Generalised isoperimetric problems in this area often have counterparts in higher dimensions, where potential theoretic techniques apply, or for special classes of planar regions, for example convex regions. Such problems are of interest within pure and applied mathematics. A wide variety of techniques from conformal mapping and geometric function theory, complex analysis in general, potential theory and probability theory are used.

## Publications

### Book Chapters

Year | Publication | |
---|---|---|

(2013) | 'An isoperimetric inequality for extremal Sobolev functions' Tom Carroll and Jesse Ratzkin (2013) 'An isoperimetric inequality for extremal Sobolev functions' In: K. Hirata (eds). Potential Theory and its Related Fields. Kyoto, Japan: RIMS Kôkyûroku Bessatsu. [Details] | |

(2006) | 'Brownian motion and harmonic measure in conic sections' Tom Carroll (2006) 'Brownian motion and harmonic measure in conic sections' In: Advanced Studies in Pure Mathematics 44: Potential Theory in Matsue. Tokyo, Japan: Mathematical Society of Japan. [Details] |

### Peer Reviewed Journals

Year | Publication | |
---|---|---|

(2015) | 'A Reverse Hölder Inequality for Extremal Sobolev Functions' Tom Carroll and Jesse Ratzkin (2015) 'A Reverse Hölder Inequality for Extremal Sobolev Functions'. Potential Analysis, 42 (1):283-292 [DOI] [Details] | |

(2014) | 'The maximum voltage drop in an on-chip power distribution network: analysis of square, triangular and hexagonal power pad arrangements' Tom Carroll and Joaquim Ortega-Cerda (2014) 'The maximum voltage drop in an on-chip power distribution network: analysis of square, triangular and hexagonal power pad arrangements'. The European Journal of Applied Mathematics, 25 (5):531-551 [DOI] [Details] | |

(2012) | 'Two isoperimetric inequalities for the Sobolev constant' Tom Carroll and Jesse Ratzkin (2012) 'Two isoperimetric inequalities for the Sobolev constant'. Zeitschrift Fur Angewandte Mathematik Und Physik, 63 (5):855-863 [DOI] [Details] | |

(2012) | 'On Lundh's percolation diffusion' Tom Carroll, Julie O'Donovan and Joaquim Ortega-Cerdà (2012) 'On Lundh's percolation diffusion'. Stochastic Processes and Their Applications, [DOI] [Details] | |

(2011) | 'Interpolating between torsional rigidity and principal frequency' Tom Carroll and Jesse Ratzkin; (2011) 'Interpolating between torsional rigidity and principal frequency'. Journal of Mathematical Analysis and Applications, 379 (2):818-826 [DOI] [Details] | |

(2011) | 'The maximal expected lifetime of Brownian motion' Rodrigo Banuelos and Tom Carroll; (2011) 'The maximal expected lifetime of Brownian motion'. Mathematical Proceedings of the Royal Irish Academy, 111A (1):1-11 [DOI] [Details] | |

(2010) | 'Stretching convex domains and the hyperbolic metric' Rodrigo Banuelos, Tom Carroll; (2010) 'Stretching convex domains and the hyperbolic metric'. Quarterly Jnl. Mathematics, 61 (3):265-273 [DOI] [Details] | |

(2009) | 'Hardy inequality and Lp estimates for the torsion function' M. van den Berg, Tom Carroll; (2009) 'Hardy inequality and Lp estimates for the torsion function'. Bulletin of the London Mathematical Society, 41 (6):980-986 [DOI] [Details] | |

(2009) | 'The univalent Bloch-Landau constant, harmonic symmetry and conformal glueing' Tom Carroll, Joaquim Ortega-Cerda; (2009) 'The univalent Bloch-Landau constant, harmonic symmetry and conformal glueing'. Journal de Mathematiques Pures et Appliquees, 92 (4):396-406 [DOI] [Details] | |

(2008) | 'An extension of Jenkins' condition for extremal domains associated with the univalent Bloch-Landau constant' Carroll, Tom; (2008) 'An extension of Jenkins' condition for extremal domains associated with the univalent Bloch-Landau constant'. Computational Methods and Function Theory, 8 (1-2):159-165 [Details] | |

(2008) | 'Lipschitz continuity of the Green function in Denjoy domains' Carroll, Tom and Gardiner, Stephen J; (2008) 'Lipschitz continuity of the Green function in Denjoy domains'. Arkiv för matematik, 46 (2):271-283 [Details] | |

(2007) | 'Configurations of balls in Euclidean space that Brownian motion cannot avoid' Carroll Tom, Ortega-Cerda Joaquim; (2007) 'Configurations of balls in Euclidean space that Brownian motion cannot avoid'. Annales Academiæ Scientiarum Fennicæ, 32 (1):223-234 [Details] | |

(2005) | 'Sharp integrability for Brownian motion in parabola-shaped regions' Carroll Tom Banuelos R; (2005) 'Sharp integrability for Brownian motion in parabola-shaped regions'. Journal of Functional Analysis, 218 (1):219-253 [Details] | |

(2004) | 'Conformal mapping of parabola-shaped domains' Carroll Tom Hayman W.K.; (2004) 'Conformal mapping of parabola-shaped domains'. Computational Methods and Function Theory, 4 (1):111-126 [Details] | |

(2002) | 'Hermite polynomials and Brownian motion' Carroll Tom; (2002) 'Hermite polynomials and Brownian motion'. The Mathematical Scientist, 27 (2):91-101 [Details] | |

(2002) | 'Boundary behaviour of positive harmonic functions on Lipschitz domains' Carroll, Tom; (2002) 'Boundary behaviour of positive harmonic functions on Lipschitz domains'. Annales Academiæ Scientiarum Fennicæ, 27 (no. 1):231-236 [Details] | |

(2002) | 'Torsional rigidity and expected lifetime of Brownian motion' Bañuelos, Rodrigo; van den Berg, M.; Carroll, Tom J. London Math. Soc.; (2002) 'Torsional rigidity and expected lifetime of Brownian motion'. Journal of The London Mathematical Society, 66 (no. 2):499-512 [Details] | |

(2001) | 'Old and new on the bass note, the torsion function and the hyperbolic metric' Carroll, Tom; (2001) 'Old and new on the bass note, the torsion function and the hyperbolic metric'. Irish Mathematical Society Bulletinetin, 47 (*):41-65 [Details] | |

(2001) | 'On the Bieberbach and Koebe constants of a simply connected domain' Carroll Tom Rugeihyamu Sylvester; (2001) 'On the Bieberbach and Koebe constants of a simply connected domain'. Complex Variables. Theory and Application, 46 (1):31-49 [Details] | |

(2000) | 'Extremal problems for conditioned Brownian motion and the hyperbolic metric' Carroll Tom Banuelos R; (2000) 'Extremal problems for conditioned Brownian motion and the hyperbolic metric'. Annales de l'Institut Fourier Univ. Grenoble, 50 (5):1507-1532 [Details] | |

(1998) | 'Inradius and integral means for Green's functions and conformal mappings' Bañuelos, Rodrigo; Carroll, Tom; Housworth, Elizabeth; (1998) 'Inradius and integral means for Green's functions and conformal mappings'. Proceedings of the American Mathematical Society, 126 (no. 2):577-585 [Details] | |

(1997) | 'Conformal mappings of close-to-convex domains' Carroll, Tom and Twomey, J. B.: ; (1997) 'Conformal mappings of close-to-convex domains'. Journal of The London Mathematical Society-Second Series, 55 (3):489-498 [Details] | |

(1994) | 'Brownian motion and the fundamental frequency of a drum' Bañuelos, Rodrigo and Carroll, Tom: ; (1994) 'Brownian motion and the fundamental frequency of a drum'. Duke Mathematical Journal, 82 (1):87-87 [Details] | |

(1993) | 'Conditioned Brownian motion and hyperbolic geodesics in simply connected domains' Bañuelos, Rodrigo and Carroll, Tom; (1993) 'Conditioned Brownian motion and hyperbolic geodesics in simply connected domains'. The Michigan Mathematical Journal, 40 (2):321-332 [Details] | |

(1991) | 'Compact composition operators not in the Schatten classes' Carroll, Tom; Cowen, Carl C.; (1991) 'Compact composition operators not in the Schatten classes'. Journal of Operator Theory, 26 (no. 1):109-120 [Details] | |

(1990) | 'A monotone slit mapping with large logarithmic derivative' Carroll, Tom.; (1990) 'A monotone slit mapping with large logarithmic derivative'. Michigan Mathematical Journal, 37 (no. 3):409-426 [Details] | |

(1988) | 'A classical proof of Burdzy's theorem on the angular derivative' Carroll, Tom (1988) 'A classical proof of Burdzy's theorem on the angular derivative'. Journal of The London Mathematical Society, 38 (3):423-441 [Details] |

## Professional Activities

### Conference Contributions

Year | Publication | |
---|---|---|

(2008) | Annual One Day Function Theory Conference (Main invited speaker),T. Carroll; (2008) Annual One Day Function Theory Conference (Main invited speaker). [Invited Lectures (Conference)], Annual One Day Function Theory Conference (Main invited speaker), The Open University (Milton Keynes, UK) , 08-SEP-08 - 08-SEP-08. [Details] | |

(2008) | 3rd Irish Mathematics Service Teaching Conference,T. Carroll; (2008) 3rd Irish Mathematics Service Teaching Conference. [Invited Lectures (Conference)], 3rd Irish Mathematics Service Teaching Conference, Waterford IT , 26-MAY-08 - 28-MAY-08. [Details] | |

(2007) | Workshop on the hyperbolic metric in Complex Analysis,T. Carroll; (2007) Workshop on the hyperbolic metric in Complex Analysis. [Invited Lectures (Conference)], Workshop on the hyperbolic metric in Complex Analysis, University of Bristol , 03-SEP-07 - 07-SEP-07. [Details] | |

(2007) | New trends in Complex Analysis and Harmonic Analysis,T. Carroll; (2007) New trends in Complex Analysis and Harmonic Analysis. [Invited Oral Presentation], New trends in Complex Analysis and Harmonic Analysis, Voss, Norway , 07-MAY-07 - 12-MAY-07. [Details] | |

(2006) | ISUME 3 (Irish Symposium for Undergraduate Mathematics Education),T. Carroll; (2006) ISUME 3 (Irish Symposium for Undergraduate Mathematics Education). [Invited Lectures (Conference)], ISUME 3 (Irish Symposium for Undergraduate Mathematics Education), NUI Galway , 15-DEC-06 - 16-DEC-06. [Details] | |

(2006) | Barcelona Analysis Conference,T .Carroll; (2006) Barcelona Analysis Conference. [Invited Lectures (Conference)], Barcelona Analysis Conference, Barcelona , 04-SEP-06 - 08-SEP-06. [Details] | |

(2004) | ISUME 2 (Irish Symposium for Undergraduate Mathematics Education),T. Carroll; (2004) ISUME 2 (Irish Symposium for Undergraduate Mathematics Education). [Invited Lectures (Conference)], ISUME 2 (Irish Symposium for Undergraduate Mathematics Education), UCD , 10-DEC-04 - 11-DEC-04. [Details] | |

(2011) | Diffusion through random configurations of spherical obstacles in a ball,Tom Carroll (2011) Invited lecture at The Open University. [Invited Seminars/Guest Lectures], Diffusion through random configurations of spherical obstacles in a ball, Milton Keynes, UK , 11-NOV-11 - 11-NOV-11. [Details] | |

(2011) | Diffusion through random spherical obstacles,Tom Carroll; (2011) Invited research visit and lecture at University of Bristol. Hosts: Prof. M. van den Berg & Dr E. Crane. [Invited Seminars/Guest Lectures], Diffusion through random spherical obstacles, University of Bristol, U.K , 07-MAR-11 - 09-MAR-11. [Details] | |

(2011) | Torsional Rigidity and Principal Frequency,Tom Carroll; (2011) Invited research visit and lecture at CRM Barcelona as part of 'Research Programme on Complex Analysis and Spectral Problems'. Host: Prof. J. Ortega-Cerda. [Invited Seminars/Guest Lectures], Torsional Rigidity and Principal Frequency, Centre de Recerca Matemàtica, Universita Autonoma de Barcelona, Spain , 10-APR-11 - 15-APR-11. [Details] |

### Committees

Committee | Function | From / To | |
---|---|---|---|

School of Mathematical Sciences APCD | Chair | 2012 / | |

College of SEFS APCD | Representative of the School of Mathematical Sciences | 2010 / |

### Employment

Employer | Position | From / To | |
---|---|---|---|

Dept of Mathematics, UCC | Senior Lecturer | / | |

Purdue University, Indiana, U.S.A. | Research Assistant Prof. | 01-AUG-98 / 02-JUN-90 | |

Politecnico di Milano, Italy. | Visiting Professor | 01-SEP-94 / 01-AUG-95 |

### Education

Year | Institution | Qualification | Subject | |
---|---|---|---|---|

1988 | The Open University | PHD | Maths | |

1984 | UCC | MSc | Maths | |

1983 | UCC | BSC | Maths |

### Journal Activities

Journal | Role | To / From | |
---|---|---|---|

Bulletin, Journal And Proceedings Of The London Mathematical Society (Issn: Bulletin 0024-6093, Journal 0024-6107, Proceedings 0024-6115) | Editor | - | |

Pacific Journal Of Mathematics | Referee | - | |

Journal Of Mathematical Analysis And Applications | Referee | - | |

Journal And The Bulletin Of The London Mathematical Society | Referee | - | |

Proceedings Of The American Mathematical Society | Referee | - | |

Mathematical Proceedings Of The Royal Irish Academy | Referee | - | |

Journal Of Porous Media | Referee | - | |

Lms Journal Of Computation And Mathematics | Referee | - | |

Australian And New Zealand Journal Of Statistics | Referee | - |

### Other Activities

Description | |
---|---|

Bristol University, UK Invited research visit to work with Professor M. van den Berg (paper submitted). 12-14 May 2008 University of Barcelona Invited research visit to work with Dr Joaquim Ortega-Cerdà (paper submitted). 31 March to 4 April 2008 Purdue University, USA Invited research visit to work with Professor Rodrigo Bañuelos (paper submitted) 27 March to 10 April 2005 University of Barcelona Invited research visit to work with Dr Joaquim Ortega-Cerdà (paper published). 2-6 May 2005 | |

Configurations of balls in Euclidean space that Brownian motion cannot avoid. University of Bristol 12 May 2008 Harmonic measure in parabola-shaped regions in Rn Barcelona Analysis Seminar 31 March 2008 The univalent Bloch-Landau constant and harmonic symmetry University of Bermingham 7 Nov. 2007 Configurations of balls in Euclidean space that Brownian motion cannot avoid. University of Cambridge 15 February 2006 Harmonic measure in parabola-shaped regions in Rn Purdue University, USA 5 April 2005 Conditioned Brownian motion and hyperbolic geometry Barcelona Analysis Seminar 2 May 2005 Harmonic measure in parabola-shaped regions in Rn Aristotle University Of Thessaloniki, Greece 2 Nov. 2004 Harmonic measure in parabola-shaped regions in Rn University of Bristol 1 June 2004 Harmonic measure in parabola-shaped regions in Rn University of Edinburgh 24 May 2004
| |

Extern examiner for M337 compex Analysis The Open University, UK 2007-2010 Reviewer for research grant proposal Israeli Science Foundation March 2007 Extern examiner for Ph.D. thesis University of Nottingham, U.K. Nov. 2007 Extern examiner for Ph.D. thesis University of Cambridge, U.K. 14 Feb. 2006 Extern examiner for Ph.D. thesis University of Nottingham, U.K. Oct. 2002 |

## Teaching Activities

### Teaching Interests

My main teaching interests are in the area of Analysis, including calculus, complex analysis, real analysis, and measure theory. That said, in the past I have taught various courses on, for example, number theory, linear algebra, financial mathematics.

** Modules taught**:

__2014-2015__

MA1059 Calculus (5 credits)

MA2051 Mathematical Analysis 1 (5 credits)

MF2052 Derivatives, Securities and Option Pricing (5 credits)

MA1060 Introduction to Analysis (5 credits)

### Recent Postgraduates

Graduation Year | Student Name | Institution | Degree Type | Thesis Title | |
---|---|---|---|---|---|

2015 | Gearoid Ryan | NUI (UCC) | PHD | A study of the Hobson and Rogers volatility model |

### Current Postgraduate Students

Student | Degree Type | |
---|---|---|

Francis Rose Mary | Doctoral Degree |

### Modules Taught

Term (ID)) | Title | Link | Subject | |
---|---|---|---|---|

2015 | Calculus | MA1059 | Calculus | |

2015 | Mathematical Analysis I | MA2051 | Mathematical Analysis I | |

2015 | Introduction to Analysis | MA1060 | Introduction to Analysis | |

2015 | Derivatives, Securities and Option Pricing | MF3052 | Derivatives, Securities and Option Pricing |