# Neil Trudinger

## Contacts

Neil Trudinger was born in Ballarat, Australia in 1942. His father was a doctor at the Ballarat Base Hospital who subsequently joined the Royal Australian Air Force. Neil’s early education was in Japan and at Laverton in Victoria and his secondary education was at Richmond High School in New South Wales. He completed a BSc Honours degree in mathematics at the University of New England (UNE) in 1962, specialising in applied mathematics. During his honours year he was employed by the university as a tutor on a half time basis which became full time in 1963 until he left for postgraduate studies at Stanford University in the United States in September. He completed his PhD at Stanford in 1966 under the supervision of David Gilbarg. His thesis was concerned with the theory of nonlinear partial differential equations and this area continued as his main research speciality throughout his career.

Following his PhD, he held a prestigious Courant Instructorship at the Courant Institute for Mathematical Sciences in New York. After a short visiting appointment at the University of Pisa in Italy, he returned to Australia to a Lectureship at Macquarie University at the end of 1967. After a subsequent promotion to Senior Lecturer, he commenced a Readership at the University of Queensland in 1970 from which he took one year’s leave to undertake Visiting Associate Professorships at the University of Minnesota and Stanford University. He was awarded a Personal Chair at the University of Queensland in 1973 but left there at the end of the year to take up a Chair at the Australian National University and the Headship of the Department of Pure Mathematics in the Faculty of Arts, as the successor of Hanna Neumann. He was appointed an Emeritus Professor at ANU in 2014 and is currently also Honorary Professor at the Universities of Queensland and Wollongong.

In 1981 Trudinger led a successful proposal in the Australian government’s inaugural Centres of Excellence competition. This led to the establishment of the Commonwealth Special Research Centre for Mathematical Analysis (CMA) with Trudinger serving as Director from 1982 until 1990. During this time he also initiated the creation of the School of Mathematical Sciences (SMS) at the ANU which commenced in 1989 and was the first academic structure to combine departments from both the Institute of Advanced Studies (IAS) and The Faculties at ANU. Trudinger served as the Dean of SMS, now known as the Mathematical Sciences Institute (MSI), from 1992 until 2000. The CMA was renamed in 1991 as the Centre for Mathematics and its Applications, in conjunction with the broadening of its research programme to include statistics, and after its subsequent amalgamation in 1993 with the original IAS Departments of Mathematics and Statistics, continued for some time as a major international research centre in the mathematical sciences.

During his career Trudinger has received many honours and awards. These include election to the Australian Academy of Science in 1978 and to the Royal Society of London in 1997. His prizes include the first award of the Australian Mathematical Society Medal in 1981 and the first award of its Gavin Brown Prize for best paper in 2011, the Hannan Medal of the Australian Academy of Science in 1996, (first award in Pure Mathematics), and the Steele Prize of the American Mathematical Society in 2008. He has also given many prestigious international named lectures and lecture series. Among his significant overseas appointments, he was the principal organiser of the half year programme in Nonlinear Partial Differential Equations at the Isaac Newton Institute at Cambridge in 2001, he was Chancellor’s Professor at the University of California at Berkeley in 2005 and a Member of the Institute for Advanced Study at Princeton in 2008. He has also been heavily involved in professional service, including being President of the Australian Mathematical Society from 1986 to 1988, Chief Judge in the Singapore National Science Talent Search in 2002 and a member of the prestigious Abel Prize Committee from 2008 to 2010. In 2013, he received an honorary doctorate from his “alma mater”, UNE, and gave the Occasional Address at the Graduation ceremony.

Neil Trudinger’s research has spanned a large terrain of mathematics. Although largely focused on the theory and applications of nonlinear elliptic partial differential equations it has spread through functional analysis, differential geometry and computational mathematics, with recent work in optimal transportation and more recently in geometric optics. His papers have appeared in the leading international journals and his early work from the 1960s is still highly cited. He is most famous for his book on elliptic equations, written jointly with David Gilbarg, which grew out of his lectures at Stanford in 1971. This book was first published in 1977 and has gone though various editions, also being translated into Chinese and Russian. Taking account of its various reprintings, it has been the most cited work on the database of the American Mathematical Society, MathSciNet, since its inception in 2000, and accounted for the Steele Prize award in 2008,

### Research interests

### Some conference talks and notes

3. On convexity notions in optimal transportation, February, 2008

5. Recent developments in elliptic partial differential equations, ICM,Madrid, 3 (2006), 291-302.

### Groups

- Emeritus, Applied & nonlinear analysis

### Projects

- Supervisor, Viscosity solutions

- Trudinger, N 2021, 'On the local theory of prescribed Jacobian equations revisited', Mathematics in Engineering, vol. 3, no. 6, pp. 1-17.
- Trudinger, N 2020, 'Remarks on the Pucci Conjecture', Indiana University Mathematics Journal, vol. 69, no. 1, pp. 109-118.
- Jiang, F & Trudinger, N 2020, 'On the Dirichlet problem for general augmented Hessian equations', Journal of Differential Equations, vol. 269, no. 6, pp. 5204-5227.
- Jiang, F & Trudinger, N 2019, 'Oblique boundary value problems for augmented Hessian equations III', Communications in Partial Differential Equations, vol. 44, no. 8, pp. 708-748.
- Jiang, F & Trudinger, N 2018, 'On the Second Boundary Value Problem for Monge-AmpÃ¨re Type Equations and Geometric Optics', Archive for Rational Mechanics and Analysis, vol. 229, pp. 547-567pp.
- Jiang, F & Trudinger, N 2018, 'Oblique boundary value problems for augmented Hessian equations I', Bulletin of Mathematical Sciences, vol. 8, no. 2, pp. 353-411pp.
- Jiang, F & Trudinger, N 2017, 'Oblique boundary value problems for augmented Hessian equations II', Nonlinear Analysis, vol. 154, pp. 148-173pp.
- Jiang, F, Trudinger, N & Xiang, N 2016, 'On the Neumann problem for Monge-AmpÃ¨re type equations', Canadian Journal of Mathematics, vol. 68, no. 6, pp. 1334-1361.
- Liu, J & Trudinger, N 2016, 'On the classical solvability of near field reflector problems', Discrete and Continuous Dynamical Systems, Series A, vol. 36, no. 2, pp. 895-916pp.
- Jiang, F, Trudinger, N & Yang, X 2015, 'On the Dirichlet problem for a class of augmented Hessian equations', Journal of Differential Equations, vol. 258, no. 5, pp. 1548-1576.
- Liu, J, Trudinger, N & Wang, X 2015, 'On Asymptotic Behaviour and W-2,W-p Regularity of Potentials in Optimal Transportation', Archive for Rational Mechanics and Analysis, vol. 215, no. 3, pp. 867-905.
- Jiang, F & Trudinger, N 2014, 'On Pogorelov estimates in optimal transportation and geometric optics', Bulletin of Mathematical Sciences, vol. 4, no. 3, pp. 407-431.
- Jiang, F, Trudinger, N & Yang, X 2014, 'On the Dirichlet problem for Monge-AmpÃ¨re type equations', Calculus of Variations and Partial Differential Equations, vol. 49, no. 3-4, pp. 1223-1236.
- Trudinger, N & Zhang, W 2014, 'Weak continuity of the complex k-hessian operators with respect to local uniform convergence', Bulletin of the Australian Mathematical Society, vol. 89, no. 2, pp. 227-233.
- Trudinger, N 2014, 'On the local theory of prescribed Jacobian equations', Discrete and Continuous Dynamical Systems, Series A, vol. 34, no. 4, pp. 1663-1681.
- Trudinger, N & Zhang, W 2013, 'Hessian measures on the Heisenberg group', Journal of Functional Analysis, vol. 264, no. 10, pp. 2335-2355.
- Trudinger, N 2013, 'A note on global regularity in optimal transportion', Bulletin of Mathematical Sciences, vol. 3, no. 3, pp. 551-557.
- Sheng, W, Trudinger, N & Wang, X 2012, 'Prescribed Weingarten Curvature Equations', in Yuxin Dong; Jixiang Fu; Guozhen Lu; Weimin Sheng; Xiaohua Zhu (ed.),
*Recent Developments in Geometry and Analysis*, International Press, USA, pp. 359-386. - Sheng, W, Trudinger, N & Wang, X 2012, 'The k-Yamabe problem', in Huai-Dong Cao; Shing-Tung Yau (ed.),
*Surveys in Differential Geometry Volume 17*, International Press, USA, pp. 427-457. - Dai, Q, Trudinger, N & Wang, X-J 2012, 'The mean curvature measure', Journal of European Mathematical Society, vol. 14, no. 3, pp. 779-800.
- Trudinger, N 2011, 'On generated prescribed Jacobian equations',
*Partial Differential Equations*, ed. G Huisken, EMS Publishing House, Zurich Switzerland, pp. 2194-2189. - Liu, J & Trudinger, N 2010, 'On Pogorelov estimates for Monge-AmpÃ¨re type equations', Discrete and Continuous Dynamical Systems, Series A, vol. 28, no. 3, pp. 1121-1135.
- Liu, J, Trudinger, N & Wang, X-J 2010, 'Interior C 2,alpha regularity for potential functions in optimal transportation', Communications in Partial Differential Equations, vol. 35, no. 1, pp. 165-184.
- Maugeri, A, Venkatesha Murthy, M & Trudinger, N 2008, 'Journal of Global Optimization: Preface', Journal of Global Optimization, vol. 40, no. 1-3, pp. 1-5.
- Trudinger, N & Wang, X-J 2010, 'The intermediate case of the Yamabe problem for higher order curvatures', International Mathematics Research Notices, vol. 227, pp. 1-22.
- Kuo, H & Trudinger, N 2009, 'On the Krylov maximum principle for discrete parabolic schemes', Tamkang Journal of Mathematics, vol. 40, no. 4, pp. 437-450.
- Guan, P, Ma, X, Trudinger, N et al 2009, 'A Form of Alexandrov-Fenchel Inequality', Pure and Applied Mathematics Quarterly, vol. 6, no. 3, pp. 999-1012.
- Trudinger, N & Wang, X-J 2009, 'On Harnack inequalities and singularities of admissible metrics in the Yamabe problem', Calculus of Variations and Partial Differential Equations, vol. 35, no. 3, pp. 317-338.
- Trudinger, N & Wang, X-J 2009, 'On strict convexity and continuous differentiability of potential functions in optimal transportation', Archive for Rational Mechanics and Analysis, vol. 192, no. 3, pp. 403-418.
- Trudinger, N & Wang, X-J 2009, 'On the second boundary value problem for Monge-Ampere type equations and optimal transportation', Annali Della Scuola Normale Superiore di Pisa, vol. 8, no. 1, pp. 143-174.
- Trudinger, N & Wang, X-J 2009, 'Quasilinear Elliptic Equations with Signed Measure Data', Discrete and Continuous Dynamical Systems, Series A, vol. 23, no. 1 & 2, pp. 477-494.
- Kuo, H & Trudinger, N 2008, 'On the maximum principle for linear parabolic equations', Journal of Global Optimization, vol. 40, no. 3, pp. 495-500.
- Trudinger, N 2008, 'On the Prescribed Jacobian Equation',
*International Conference for the 25th Anniversary of Viscosity Solutions*, ed. Y Giga; K Ishii; S Koike; T Ozawa; N Yamada, Gakuto International Series, Tokyo, Japan, pp. 243-255. - Trudinger, N & Wang, X-J 2008, 'Boundary regularity for the Monge-Ampere and affine maximal surface equations', Annals of Mathematics, vol. 167, no. 3, pp. 993-1028.
- Trudinger, N & Wang, X-J 2008, 'The Monge-Ampere equation and its geometric applications', in Lizhen Ji; Peter Li; Richard Schoen; Leon Simon (ed.),
*Handbook of Geometric Analysis (Volume 1)*, Higher Education Press, Somerville, USA, pp. 467-524. - Trudinger, N 2007, 'Recent developments in the theory of nonlinear elliptic partial differential equations', in M.Sanz-Sole, J.Soria, J.L.Varona, J.Verdera (ed.),
*Proceedings of the International Congress of Mathematicians, Madrid 2006 - Invited Lectures*, American Mathematical Society, Germany, pp. 291-302. - Kuo, H & Trudinger, N 2007, 'New maximum principles for linear elliptic equations', Indiana University Mathematics Journal, vol. 56, no. 5, pp. 2439-2452.
- Sheng, W, Trudinger, N & Wang, X-J 2007, 'The Yamabe problem for higher order curvatures', Journal of Differential Geometry, vol. 77, pp. 515-553.
- Trudinger, N 2006, 'On Hessian Measures for Non-Commuting Vector Fields', Pure and Applied Mathematics Quarterly, vol. 2, no. 1, pp. 147-161.
- Sheng, W, Trudinger, N & Wang, X-J 2006, 'Enclosed convex hypersurfaces with maximal affine area', Mathematische Zeitschrift, vol. 252, pp. 497-510.
- Trudinger, N 2006, 'Recent developments in elliptic partial differential equations of Monge-Ampere type', International Congress of Mathematicians, Madrid, European Mathematical Society, vol. 3, pp. 291-312.
- Chaudhuri, N & Trudinger, N 2005, 'An Alexandrov type theorem for k-convex functions', Bulletin of the Australian Mathematical Society, vol. 71, no. 2, pp. 305-314.
- Kuo, H & Trudinger, N 2005, 'Estimates for solutions of fully nonlinear discrete schemes', in J Rodrigues et al. (ed.),
*Trends in partial differential equations of mathematical physics*, Birkhauser Verlag, Switzerland, pp. 275-282. - Trudinger, N & Wang, X-J 2005, 'The affine Plateau problem', Journal of the American Mathematical Society, vol. 18, no. 2, pp. 253-289.
- Trudinger, N, Ma, X & Wang, X-J 2005, 'Regularity of Potential Functions of the Optimal Transportation Problem', Archive for Rational Mechanics and Analysis, vol. 177, pp. 151-183.
- Trudinger, N 2004, 'The Chern Conjecture in Affine geometry',
*International Congress of Chinese Mathematicians (ICCM 2001)*, ed. CS Lin, Lo Yang and ST Yau, International Press, USA, pp. 25-30. - Ivochkina, N, Trudinger, N & Wang, X-J 2004, 'The Dirichlet Problem for degenerate Hessian Equations', Communications in Partial Differential Equations, vol. 29, pp. 219-235.
- Ivochkina, N, Trudinger, N & Wang, X-J 2005, 'The Dirichlet Problem for Degenerate Hessian Equations', Communications in Partial Differential Equations, vol. 29, no. 1&2, pp. 219-235.
- Sheng, W, Trudinger, N & Wang, X-J 2004, 'Convex hypersurfaces of prescribed Weingarten curvatures', Communications in Analysis and Geometry, vol. 12, no. 1, pp. 213-232.
- Trudinger, N & Sheng, W 2003, 'On the affine diameter of closed convex hypersurfaces', Bulletin of the Australian Mathematical Society, vol. 68, no. 3, pp. 431-437.
- Kuo, H & Trudinger, N 2002, 'Schauder estimates for fully nonlinear elliptic difference operators', Proceedings of the Royal Society of Edinburgh, vol. 132A, pp. 1395-1406.
- Trudinger, N & Wang, X-J 2002, 'Affine complete locally convex hypersurfaces', Inventiones Mathematicae, vol. 150, pp. 45-60.
- Trudinger, N & Wang, X-J 2002, 'Bernstein-Jorgens Theorem for a Fourth Order Partial Differential Equation', Journal of Partial Differential Equations, vol. 15, no. 1, pp. 78-88.
- Trudinger, N & Wang, X-J 2002, 'Hessian Measures III', Journal of Functional Analysis, vol. 193, pp. 1-23.
- Trudinger, N & Wang, X-J 2002, 'On locally convex hypersurfaces with boundary', Journal fur Reine und Angewandte Mathematik, vol. 551, pp. 11-32.
- Trudinger, N & Wang, X-J 2002, 'On the weak continuity of Elliptic Operators and applications to Potential Theory', American Journal of Mathematics, vol. 124, no. 2, pp. 369-410.
- Trudinger, N 2001, 'Glimpses of nonlinear partial differential equations in the Twentieth century. A Priori estimates and the Bernstein problem', in Chen L H Y, Jesudason J P, Lai C H, Oh C H, Phua K K, Tan E-C (ed.),
*Challenges for the 21st Century: International Conference on Fundamental Sciences: Mathematics and Theoretical Physics*, World Scientific Publishing Company, Singapore, pp. 196-212. - Trudinger, N & Wang, X-J 2001, 'On the Monge mass transfer problem', Calculus of Variations and Partial Differential Equations, vol. 13, pp. 19-31.
- Kuo, H & Trudinger, N 2000, 'A note on the discrete Aleksandrov-Bakelman maximum principle', Taiwanese Journal of Mathematics, vol. 4, no. 1, pp. 55 - 64.
- Trudinger, N & Wang, X-J 2000, 'The Bernstein problem for affine maximal hypersurfaces', Inventiones Mathematicae, vol. 140, no. 2, pp. 399 - 422.
- Guan, P., Trudinger, N, & Wang, X.-J 1999, 'On the Dirichlet problem for degenerate Monge-Amp\`ere equations', Acta Mathematica, vol. 182, no. 1, pp. 87-104.
- Trudinger, N & Wang, X-J 1999, 'Hessian Measures II', Annals of Mathematics, vol. 150, no. 2, pp. 579-604.
- Trudinger, N 1995, 'On the Dirichlet problem for Hessian equations', Acta Mathematica, vol. 175, pp. 151-164.