Profile
Profile

Prof. Dr. Ngoc Tran

Bonn Junior Fellow

E-mail: ngoc.tran(at)hcm.uni-bonn.de
Phone: +49 228 73 62390
Homepage: https://www.ma.utexas.edu/users/ntran/
Room: 4.034
Location: Mathematics Center
Institute: Institute for Applied Mathematics
Research Area: Research Area G

Academic Career

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)

Research Profile

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 R^2, 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 R^2, 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.

Selected Publications

[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
arXiv preprint arXiv:1310.2012
2013
[6] Ngoc Mai Tran, Josephine Yu
Product-Mix Auctions and Tropical Geometry
arXiv preprint arXiv:1505.05737
2015
[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 and Poisson Tropical Plane Curves
arXiv preprint arXiv:1610.08533
2016

Publication List

Awards

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

Selected Invited Lectures

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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