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| 2009 - 2013 | PhD, Statistics, University of California, Berkeley, CA, USA | | 2013 - 2015 | Simons Postdoctoral Fellow, Mathematics, University of Texas, Austin, TX, USA | | Since 2015 | Professor (W2, Bonn Junior Fellow), Mathematics, University of Bonn | | Since 2015 | Assistant Professor, Mathematics, University of Texas, Austin, TX, USA (on leave) |
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I use tools from tropical geometry and probability to create novel applications of mathematics in economics, neuroscience and other sciences. I love revisiting old problems with new geometric insights. In the immediate future, I have three main research themes: applications of tropical geometry in economics, combinatorial stochastic processes, and applications of discrete mathematics in neuroscience.
0.1. Applications of Tropical Geometry: Tropical geometry connects algebraic geometry, combinatorics and optimization theory. I use it to tackle optimization problems from a new geometric perspective. In the past I have worked on pairwise ranking problems [1,2]. This motivates the classification of tropical eigenspaces [3], commuting matrices [4], culminating in [5], where I used techniques from commutative algebra to explicitly classify polytropes in dimension 3 and 4. Recently, this tool again proves valuable in other areas of economics, namely, auction theory [6] and mechanism design [7]. I am working on extensions of the results in [7] to Bayesian mechanisms and multi-player settings, as well as related problems on tropical polynomials.
0.2. Random tropical polynomials and partitions. In [8], with Francois Baccelli, I proved a point process approximation for zeros of a random tropical polynomial with i.i.d coefficients. We obtain simpler proofs of past results on vertices of random polytopes in  , which are special cases of our setup. The higher dimension generalization of this work is particularly interesting. In particular, the power of our proof in [8] lies in a newly discovered connection between combinatorial stochastic processes (CSP) and random polytopes in , and is closely related to my work with Jim Pitman on size-biased permutations [9]. I am working on various extensions of this work [10], with applications in Bayesian clustering.
0.3. Discrete mathematics and neuroscience. In mathematical neuroscience, I use techniques from discrete mathematics to answer long-standing open questions in theoretical neuroscience, to provide theorems and algorithms for current analysis, and to inspire new developments in mathematics. Currently I have three separate projects with collaborators and master students: decoding grid cells by solving max-clique, classification of the disease Lupus using exchangeable random graph theory, and testing hypothesis on pairwise correlation in population code using lattice point enumeration algorithms.
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[ 1] Ngoc Mai Tran
Pairwise ranking: choice of method can produce arbitrarily different rank order
Linear Algebra Appl. , 438: (3): 1012--1024
2013
DOI: 10.1016/j.laa.2012.08.028[ 2] Ngoc Mai Tran
HodgeRank is the limit of Perron rank
Math. Oper. Res. , 41: (2): 643--647
2016
DOI: 10.1287/moor.2015.0744[ 3] Bernd Sturmfels, Ngoc Mai Tran
Combinatorial types of tropical eigenvectors
Bull. Lond. Math. Soc. , 45: (1): 27--36
2013
DOI: 10.1112/blms/bds058[ 4] Ralph Morrison, Ngoc M. Tran
The tropical commuting variety
Linear Algebra Appl. , 507: : 300--321
2016
DOI: 10.1016/j.laa.2016.05.039[ 5] Ngoc Mai Tran
Enumerating Polytropes
Journal of Combinatorial Theory, , Series A: (151): 1--22
2017[ 6] Ngoc Mai Tran, Josephine Yu
Product-Mix Auctions and Tropical Geometry
Mathematics of Operations Research
2019[ 7] Robert Alexander Crowell, Ngoc Mai Tran
Tropical geometry and mechanism design
arXiv preprint arXiv:1606.04880
2016[ 8] Francois Baccelli, Ngoc Mai Tran
Zeros of random tropical polynomials, random polygons and stick-breaking
Trans. Amer. Math. Soc. , 368: (10): 7281--7303
2016
DOI: 10.1090/tran/6565[ 9] Jim Pitman, Ngoc M. Tran
Size-biased permutation of a finite sequence with independent and identically distributed terms
Bernoulli , 21: (4): 2484--2512
2015
DOI: 10.3150/14-BEJ652[ 10] Francois Baccelli, Ngoc M. Tran
Iterated Gilbert Mosaics
Stochastic Processes and their Applications
2019
DOI: 10.1016/j.spa.2019.06.016 |
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| 2001 | Gold Medal U13, Hanoi Chess Competition | | 2004 | Premier’s Award for All-round Excellence, NSW, Australia | | 2006 | Margaret Pitcher Prize in Mathematics, University of Newcastle, NSW, Australia | | 2006 | Summer Scholar, Australian National University, Canberra, ACT, Australia | | 2006 | Head of College, International House, University of Newcastle, NSW, Australia | | 2007 | SAS Institute Prize in Statistics, University of Newcastle, NSW, Australia | | 2007 | Dean’s List, University of Newcastle, NSW, Australia | | 2009 - 2015 | Fellow, Vietnam Education Foundation | | 2013 - 2015 | Postdoctoral Fellow, Simons Foundation | | Since 2015 | Bonn Junior Fellow, Hausdorff Center for Mathematics, Bonn |
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| 2012 | Tropical Mathematics and its Applications, Birmingham, England, UK | | 2012 | Workshop for Women in Probability, Duke University, Durham, NC, USA | | 2013 | Workshop on Mathematics of Partially Identified Objects, Oberwolfach | | 2014 | Seminar on Stochastic Processes, San Diego, CA, USA | | 2014 | Plenary Talk, CombinaTexas, Houston, TX, USA | | 2014 | Conference on Algebraic Geometry and Optimization, Korea | | 2014 | Workshop on Mathematics and Economics, Hokkaido, Japan | | 2015 | Session on Probability and Applications, AMS Joint Meeting, San Antonio, TX, USA | | 2015 | Session on Graph Theory, Combinatorics and Discrete Geometry, AMS Joint Meeting, San Antonio, TX, USA | | 2015 | Computational and Systems Neuroscience (Cosyne), Salt Lake City, UT, USA | | 2015 | Workshop on Nonlinear Algebra, Berlin | | 2015 | Workshop on Algebraic and Stochastic Aspects in Graph Theory, Osnabrück | | 2016 | Spring School on Combinatorial Stochastic Processes, Hanoi, Vietnam |
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