Tuyen Trung Truong
I am very excited to work here at University of Oslo, with a long tradition of strong research and teaching, where Niels Henrik Abel used to be. My research is in Several Complex Variables, Dynamical Systems and related topics in Algebraic Geometry. I am also keen to applications of these fields. Originally from Vietnam, I traveled all around the world studying and working : Indiana University (USA) PhD, 2006-2012, under the supervision of Professor Eric Bedford; Syracuse University (USA), Postdoc, 2012 - 2014; Korea Institute for Advanced Study (South Korea), Postdoc, 2014-2015; and The University of Adelaide (Australia), Postdoc 2015 - 2017; before coming here to Oslo. My PhD dissertation was on pullback of positive closed currents by meromorphic maps.
Some of my current research topics are: algebraic interpolation, embedding of algebraic curves in the complex plane, geometry and dynamics of Abelian varieties and their quotients, relations between Weil's Riemann hypothesis and standard conjectures and dynamical systems. A common theme is my enthusiasm in applying computers into solving problems, both in pure theory and in real life applications.
Recently, I also do research on Gradient Descent methods and applications in Deep Learning, with help from (Random) Dynamical Systems and Geometry research. My joint work arXiv:1808.05160 demonstrated the feasibility and good performance of Backtracking Gradient Descent in Deep Neural Networks, and the results therein have been vindicated by subsequent work by other authors such as arXiv:1905.09997.
In another recent paper arXiv:2006.01512, my collaborators and I proposed a new modification of Newton's method, roughly having the following property: if the sequence {x_n}, constructed by the new method from a random initial point x_0, converges, then the limit point is a local minimum, and the rate of convergence is quadratic. The complexity of the algorithm is O(m^3) at each step, where m is the dimension.
Since I am concerned about the correctness of the proofs of claims in mathematics (in many cases - most of cases, I think - people either do not have the competence or time to check, and hence just believe the claims, in particular if the claimants are famous), I am doing research also in Automated Proof Checking. It is interesting to know that there is growing interest of applying Machine Learning techniques into Automated Proof Checking. By Curry-Howard correspondence, roughly speaking, checking the correctness of mathematical proofs are equivalent to verifying the correctness of computer programs. Therefore, if the mentioned idea works well, then it will influence enormously both whole mathematics and your daily life (given that computers and computer softwares are now universal).
Supervision:
Fall 2020 -- current: Fei Hu, Postdoc, working in pure mathematics. This position is funded from Research Council of Norway grant 300814.
Fall 2020 -- current: Viktor Balch Barth, PhD student, working in pure mathematics. I am the main advisor, co-supervising with . This position is funded from Research Council of Norway grant 300814.
Fall 2019--current: Maged Abdalla Helmy Abdou (Industrial PhD student, Informatics Department), working in Deep Learning in medical imaging. I am a co-supervisor, together with Eric Jul and Paulo Ferreira.
Fall 2018 - current: Giovanni Domenico Di Salvo (PhD student, Mathematics Department), working in Several Complex Variables. I am the main advisor, co-supervising with Erlend Fornaess Wold.
Spring semester 2019: Supervisor for 4 research undergraduate students (course MAT2000): Besmira Amiti and Max Magnus Nils Rafstedt (gradient descent methods and applications in Deep Learning), Jon Elstad Maage and Christian Schive (Grobner Basis).
Research grants:
Member of MSCA-Cofund-DP, #945371, ERC, 5 year grants funding for PhD positions, 2020-.
PI of Young Research Talents grant, #300814, Research Council of Norway, 2020 -- 2024
Supported (indirectly) by the Australian Research Council grants DP120104110 and DP150103442, 2015 -- 2017.
PI of "Young and pioneering scientist development", Ministry of Education, Science and Technology (Republic of Korea), 2014--2015.
Editorial work:
2020 -- current: The journal Experimental Results (Cambridge University Press) (link). I am Reviewing Editor (for Control systems and Optimisation in the Engineering section) and an Editor (for the section Mathematics, Statistics and Probability). Some special features of this journal: It publishes standalone experimental results (whether positive or negative). The journal is open for almost all fields in science, computer and mathematics. Its referee process is open: authors know who refereed their papers, and if a paper is accepted then referees reports for the paper is also published simultaneously - this is in harmony with my publishing philosophy, please see my more personal website (link below) for more detail. It is Open Access, you need to pay a fee, but then all people can read the paper for free. (You can check the journal webpage to see if you can get partial or total fee waiver. You can also check if your institution has some agreements with Cambridge University Press concerning the fee.)
Conferences and meetings organising:
Others:
My hobbies include: reading, swimming and diving, listening to music, hanging out with friends, travelling, playing the game of Go, and solving mazes.
Here is my personal webpage:
https://sites.google.com/site/tuyentruongswebpage/home
I also have accounts on Quora and Reddit and answered some questions about the gradient descent method (the most popular and effective method used in practice, in particular machine learning), there.
Publications
Selected preprints:
Tuyen Trung Truong, Unconstrained optimisation on Riemannian manifolds, arXiv:2008.11091.
Tuyen Trung Truong, Tat Dat To, Tuan Hang Nguyen, Thu Hang Nguyen, Hoang Phuong Nguyen and Maged Helmy, A modification of quasi-Newton's methods helping to avoid saddle points, arXiv:2006.01512. Source code is available on GitHub (Link)
Tuyen Trung Truong, Convergence to minima for the continuous version of Backtracking Gradient Descent, arXiv: 1911.04221.
Tuyen Trung Truong, Strong submeasures and applications to non-compact dynamical systems, arXiv:1910.06394. This is extracted and developed from the more dynamical part of arXiv: 1712.02490.
Tuyen Trung Truong and Tuan Hang Nguyen, Backtracking gradient descent method for general C^1 functions with applications in Deep Learning, 37 pages. Preprint: arXiv: 1808.05160v2. Accompanying source codes are available on GitHub: [Link]
Tuyen Trung Truong, Bounded birationality problem is computable, 17 pages. Preprint: arXiv: 1801.00901. (An extended version, 20 pages, which contains additional results compared to the arXiv version, is available on my personal webpage.)
Tuyen Trung Truong, Sub-measures and applications to wedge intersection of positive closed currents and complex dynamics, 29 pages. Preprint: arXiv: 1712.02490.
Tuyen Trung Truong, Relations between dynamical degrees, Weil's Riemann hypothesis and the standard conjectures, 20 pages. Preprint arXiv: 1611.01124.
Selected published papers:
Tuyen Trung Truong and Tuan Hang Nguyen, Backtracking Gradient Descent method and some applications in Large scale optimisation. Part 2: algorithms and experiments. The main part of the paper is based on the more experimental part of arXiv:1808.05160, together with arXiv:2001.02005 and arXiv:2007.03618. Accompanying source codes are available on GitHub: [Link] Published online in Applied Mathematics and Optimization. The paper is Open Access. doi:10.1007/s00245-020-09718-8. [Link to PDF]
Tuyen Trung Truong, Some new theoretical and computational results around the Jacobian conjecture, developed and revised from preprint arXiv: 1503.08733. Accepted in International Journal of Mathematics.
Finnur Larusson and Tuyen Trung Truong, Approximation and interpolation of regular maps from affine varieties to algebraic manifolds, 9 pages. Preprint: arXiv: 1706.00519. Accepted in Mathematica Scandinavia.
Tuyen Trung Truong, Etale dynamical systems and topological entropy, 13 pages. Preprint arXiv: 1607.07412. Accepted in Proceedings of the American Mathematical Society.
Shalom Kaliman, Frank Kutzschebauch and Tuyen Trung Truong, On subelliptic manifolds, 12 pages. Preprint arXiv: 1611.01311v3. Accepted in Israel Journal of Mathematics.
Tuyen Trung Truong, Relative dynamical degrees of correspondences over a field of arbitrary characteristic, 41 pages. Preprint arXiv: 1605.05049. Accepted in Journal fur die Reine und Agnewandte Mathematik (Crelle's journal). DOI: 10.1515/crelle-2017-0052.
Tien-Cuong Dinh, Viet-Anh Nguyen and Tuyen Trung Truong, Growth of the number of periodic points of meromorphic maps, 18 pages. Preprint arXiv: 1601.03910. Accepted in Bulletin of the London Mathematical Society.
Finnur Larusson and Tuyen Trung Truong, Algebraic subellipticity and dominability of blowups of affine spaces, 10 pages. Preprint: arXiv: 1606.08115. Accepted in Documenta Mathematica.
Tuyen Trung Truong, Comments on Sampson's approach toward Hodge conjecture on Abelian varieties, accepted in Annali di Matematica Pura ed Applicata. Preprint arXiv: 1409.0495.
Tuyen Trung Truong, Automorphisms of blowups of threefolds being Fano or having Picard number 1, accepted in Ergodic Theory and Dynamical Systems. Preprint arXiv: 1501.01515.
Tuyen Trung Truong, Some dynamical properties of pseudo-automorphisms in dimension 3. Transactions of the American Mathematical Society 368 (2016), no 1, 727--753.
Keiji Oguiso and Tuyen Trung Truong, Explicit examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy. Kodaira Centennial Volume, J. Math. Sci. Univ. Tokyo 22 (2015), no 1, 361--385.
Tien-Cuong Dinh, Viet-Anh Nguyen and Tuyen Trung Truong, Equidistribution for meromorphic maps with dominant topological degree. Indiana University Journal of Mathematics 64, No 6 (2015), 1805--1828.
Dan Coman and Tuyen Trung Truong, Geometric properties of upper level sets of Lelong numbers on projective spaces. Mathematische Annalen 361 (2015), no 3--4, 981--994.
Fabrizio Catanese, Keiji Oguiso and Tuyen Trung Truong, Unirationality of the Ueno-Campana's threefold. Manuscripta Mathematica 145 (2014), no 3--4, 399--406.
Kenji Oguiso and Tuyen Trung Truong, Salem numbers in dynamics of Kahler 3-folds and complex tori. Mathematische Zeitschrift 278 (2014), no 1--2, 93--117.
Tuyen Trung Truong, The simplicity of the first spectral radius of a meromorphic map. Michigan Mathematics Journal 63 (2014), no 3, 623--633.
Dang Duc Trong, Cao Xuan Phuong, Dinh Ngoc Thanh and Tuyen Trung Truong, Tikhonov's regularization to deconvolution problem, Communications in Statistics: Theory and Methods, vol 43, issue 20, 2014, pp. 4384-4400.
Tuyen Trung Truong, Pullback of currents by meromorphic maps, Bulletin de la Societe Mathematique de France 141 (2013), no 4, 517--555.
Tien-Cuong Dinh, Viet-Anh Nguyen and Tuyen Trung Truong, On the dynamical degrees of meromorphic maps preserving a fibration, Communications in Contemporary Mathematics 14, 1250042, 2012.
Tuyen Trung Truong, Degree complexities of birational maps related to matrix inversions: Symmetric case, Mathematische Zeitschrift 270 (2012), no 3--4, 725--738.
Eric Bedford and Tuyen Trung Truong, Degree complexities of birational maps related to matrix inversion, Communications in Mathematical Physics 298 (2010), no. 2, 357--368.
- Truong, Tuyen Trung (2020). Etale dynamical systems and topological entropy. Proceedings of the American Mathematical Society. ISSN 0002-9939. . doi: 10.1090/proc/14100
- Truong, Tuyen Trung (2020). Some new theoretical and computational results around the Jacobian conjecture. International Journal of Mathematics. ISSN 0129-167X. 31(7) . doi: 10.1142/S0129167X20500500
- Truong, Tuyen Trung & Nguyen, Hang-Tuan (2020). Backtracking Gradient Descent Method and Some Applications in Large Scale Optimisation. Part 2: Algorithms and Experiments. Applied Mathematics and Optimization. ISSN 0095-4616. . doi: 10.1007/s00245-020-09718-8 Full text in Research Archive.
- Larusson, Finnur & Truong, Tuyen Trung (2019). Approximation and interpolation of regular maps from affine varieties to algebraic manifolds. Mathematica Scandinavica. ISSN 0025-5521. 125(2), s 199- 209 . doi: 10.7146/math.scand.a-114893
- Kaliman, Shulim; Kutzschebauch, Frank & Truong, Tuyen Trung (2018). On subelliptic manifolds. Israel Journal of Mathematics. ISSN 0021-2172. 228(1), s 229- 247 . doi: 10.1007/s11856-018-1760-7
- Truong, Tuyen Trung (2018). Bounded birationality problem is computable. arXiv.org. ISSN 2331-8422. Show summary
- Truong, Tuyen Trung (2018). Relative dynamical degrees of correspondences over a field of arbitrary characteristic. Journal für die Reine und Angewandte Mathematik. ISSN 0075-4102. . doi: 10.1515/crelle-2017-0052 Full text in Research Archive.
- Dinh, Tien Cuong; Nguyen, Viet Anh & Truong, Tuyen Trung (2017). Growth of the number of periodic points for meromorphic maps. Bulletin of the London Mathematical Society. ISSN 0024-6093. . doi: 10.1112/blms.12082