I am an Associate Professor at Auburn University at Montgomery in the Department of Mathematics (Distinguished Research Associate Professor of Mathematics). I studied under the guidance of Tien-Yien Li and received my Ph.D. in Applied Mathematics from Michigan State University.
My main research interests are numerical analysis and scientific computing as well as their applications in physics, chemistry, biology, and engineering. My recent research focuses on numerical algebraic geometry and is supported by NSF grant DMS-1923099. You can find out more about my research here. For more information, please see my CV here (PDF version).
In the Fall of 2023 I am teaching
I am always hiring student research assistants. See detail here.
Recently, I gave a talk titled
Polynomials in economics and polyhedral homotopy
in the workshop
Invitation to Algebraic Statistics and Applications
which is a part of the long program on
Algebraic Statistics and Our Changing World
hosted by Institute of Mathematical and Statistical Innovation
at the University of Chicago
September 21, 2023
University of Chicago, Chicago, IL, USA.
[ talk slides ]
I gave a talk titled
Stratified polyhedral homotopy
in the
AMS Special Session on Polynomial Systems, Homotopy Continuation and Applications
during the Joint Mathematics Meetings
January 4, 2023
Boston, MA, USA.
[ talk slides ]
"A stratified polyhedral homotopy method for sampling positive dimensional zero sets of polynomial systems"
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{Chen2023Stratified,
year = {2023},
title = {{A stratified polyhedral homotopy method for sampling positive dimensional zero sets of polynomial systems}},
author = {Chen, Tianran},
journal = {arXiv},
eprint = {2304.08598}
}
(with Daniel Bates, Paul Breiding, Jonathan Hauenstein, Anton Leykin, and Frank Sottile)
"Numerical Nonlinear Algebra"
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{Bates2023Numerical,
year = {2023},
title = {{Numerical Nonlinear Algebra}},
author = {Bates, Daniel J and Breiding, Paul and Chen, Tianran and Hauenstein, Jonathan D and Leykin, Anton and Sottile, Frank},
journal = {arXiv},
eprint = {2302.08585}
}
(with Evgeniia Korchevskaia and Julia Lindberg)
"On the typical and atypical solutions to the Kuramoto equations"
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{Chen2022Typical,
archivePrefix = {arXiv},
arxivId = {2210.00784},
author = {Chen, Tianran and Korchevskaia, Evgeniia and Lindberg, Julia},
eprint = {2210.00784},
month = {oct},
title = {On the typical and atypical solutions to the Kuramoto equations},
url = {https://arxiv.org/abs/2210.00784},
year = {2022}
}
"GPU-accelerated path tracker for polyhedral homotopy"
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{Chen2021GPU,
year = {2021},
title = {{GPU-accelerated path tracker for polyhedral homotopy}},
author = {Chen, Tianran},
journal = {arXiv},
doi = {10.48550/arxiv.2111.14317},
eprint = {2111.14317},
}
(with Evgeniia Korchevskaia)
"On the root count of algebraic Kuramoto equations in cycle networks with uniform coupling"
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{ChenKorchevskaia2019Root,
archivePrefix = {arXiv},
arxivId = {1912.06241},
author = {Chen, Tianran and Korchevskaia, Evgeniia},
eprint = {1912.06241},
month = {dec},
title = {{On the root count of algebraic Kuramoto equations in cycle networks with uniform coupling}},
url = {http://arxiv.org/abs/1912.06241},
year = {2019}
}
(with Evgeniia Korchevskaia)
"Graph edge contraction and adjacency polytopes"
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{ChenKorchevskaia2019Graph,
archivePrefix = {arXiv},
arxivId = {1912.02841},
author = {Chen, Tianran and Korchevskaia, Evgeniia},
eprint = {1912.02841},
month = {dec},
title = {{Graph edge contraction and adjacency polytopes}},
url = {http://arxiv.org/abs/1912.02841},
year = {2019}
}
"A geometric criterion on the equality between BKK bound and intersection"
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{Chen2023Geometric,
archivePrefix = {arXiv},
arxivId = {1812.05408},
author = {Chen, Tianran},
eprint = {1812.05408},
month = {apr},
title = {{A geometric criterion on the equality between BKK bound and intersection}},
url = {http://arxiv.org/abs/1812.05408},
year = {2023}
}
(with Dhagash Mehta)
"An index-resolved fixed-point homotopy and potential energy landscapes"
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{chen_index_2015,
annote = {Comment: 7 pages, 2 figures},
author = {Chen, Tianran and Mehta, Dhagash},
journal = {arXiv:1504.06622 [cond-mat]},
month = {apr},
title = {An index-resolved fixed-point homotopy and potential energy landscapes},
url = {http://arxiv.org/abs/1504.06622},
year = {2015}
}
"Volume of convex polytopes equals mixed volume of simplices"
Archiv der Mathematik (2023)
[ link ]
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{Chen2023Volume,
year = {2023},
title = {{Volume of convex polytopes equals mixed volume of simplices}},
author = {Chen, Tianran},
journal = {Archiv der Mathematik},
issn = {0003-889X},
doi = {10.1007/s00013-023-01836-3},
pages = {1--6},
keywords = {}
}
(with Robert Davis and Evgeniia Korchevskaia)
"Facets and facet subgraphs of symmetric edge polytopes"
Discrete Applied Mathematics (2022)
[ link ]
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{ChenDavisKorchevskaia2022Facets,
title = {Facets and facet subgraphs of symmetric edge polytopes},
journal = {Discrete Applied Mathematics},
volume = {328},
pages = {139-153},
year = {2023},
issn = {0166-218X},
doi = {https://doi.org/10.1016/j.dam.2022.11.015},
url = {https://www.sciencedirect.com/science/article/pii/S0166218X22004462},
author = {Tianran Chen and Robert Davis and Evgeniia Korchevskaia},
keywords = {Symmetric edge polytope, Adjacency polytope, Kuramoto equations},
}
(with Robert Davis)
"A toric deformation method for solving Kuramoto equations on cycle networks"
Nonlinear Dynamics (2022)
[ link ]
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{ChenDavis2022Toric,
author = {Tianran Chen and Robert Davis},
doi = {10.1007/s11071-022-07550-z},
issn = {1573269X},
journal = {Nonlinear Dynamics},
keywords = {Adjacency polytope,Kuramoto model,Polyhedral homotopy,Tropical geometry},
month = {6},
pages = {1-20},
publisher = {Springer Science and Business Media B.V.},
title = {A toric deformation method for solving Kuramoto equations on cycle networks},
url = {https://link.springer.com/article/10.1007/s11071-022-07550-z},
year = {2022},
}
(with Robert Davis, (I made minor contributions))
"Computing volumes of adjacency polytopes via draconian sequences"
Electronic Journal of Combinatorics (2022)
[ link ]
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{DavisChen2022Computing,
author = {Robert Davis and Tianran Chen},
doi = {10.37236/9768},
issn = {1077-8926},
issue = {1},
journal = {The Electronic Journal of Combinatorics},
month = {3},
pages = {P1.61-P1.61},
title = {Computing Volumes of Adjacency Polytopes via Draconian Sequences},
volume = {29},
url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v29i1p61},
year = {2022},
}
(with Dhagash Mehta, Tingting Tang and Jonathan D. Hauenstein)
"The loss surface of deep linear networks viewed through the algebraic geometry lens"
IEEE Transactions on Pattern Analysis and Machine Intelligence (2021)
[ link ]
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{MehtaChenTangHauenstein2021,
archivePrefix = {arXiv},
arxivId = {1810.07716},
doi = {10.1109/TPAMI.2021.3071289},
journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
author = {Mehta, Dhagash and Chen, Tianran and Tang, Tingting and Hauenstein, Jonathan D.},
eprint = {1810.07716},
month = {oct},
title = {The loss surface of deep linear networks viewed through the algebraic geometry lens},
url = {http://arxiv.org/abs/1810.07716},
year = {2021}
}
(with Jakub Mareček, Dhagash Mehta, and Matthew Niemerg)
"Three formulations of the Kuramoto model as a system of polynomial equations"
2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Monticello, IL, USA, 2019, pp. 810-815
[ link ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@INPROCEEDINGS{ChenMarecekMehtaNiemerg2019Three,
author={Chen, Tianran and Mareček, Jakub and Mehta, Dhagash and Niemerg, Matthew},
booktitle={2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton)},
title={Three Formulations of the Kuramoto Model as a System of Polynomial Equations},
year={2019},
volume={},
number={},
pages={810-815},}
"Directed acyclic decomposition of Kuramoto equations"
Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol.29, Issue 9 2019
[ link ]
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{Chen2019Directed,
author = {Chen, Tianran},
doi = {10.1063/1.5097826},
issn = {1054-1500},
journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
month = {Sep},
number = {9},
pages = {093101},
publisher = {AIP Publishing LLC},
title = {{Directed acyclic decomposition of Kuramoto equations}},
url = {http://aip.scitation.org/doi/10.1063/1.5097826},
volume = {29},
year = {2019}
}
"Unmixing the mixed volume computation"
Discrete & Computational Geometry (2019) 62:55–86
[ link ]
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{Chen2019Unmixing,
author="Chen, Tianran",
title="Unmixing the Mixed Volume Computation",
journal="Discrete {\&} Computational Geometry",
year="2019",
month="Mar",
day="20",
issn="1432-0444",
doi="10.1007/s00454-019-00078-x",
url="https://doi.org/10.1007/s00454-019-00078-x"
}
(with Robert Davis and Dhagash Mehta)
"Counting equilibria of the Kuramoto model using birationally invariant intersection index"
SIAM Journal on Applied Algebra and Geometry, 2018, Vol. 2, No. 4 pp. 489-507
[ link ]
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{ChenDavisMehta2017,
author = {Chen, Tianran and Davis, Robert and Mehta, Dhagash},
doi = {10.1137/17M1145665},
issn = {2470-6566},
journal = {SIAM Journal on Applied Algebra and Geometry},
month = {jan},
number = {4},
pages = {489--507},
title = {{Counting equilibria of the Kuramoto model using birationally invariant intersection index}},
url = {https://epubs.siam.org/doi/10.1137/17M1145665},
volume = {2},
year = {2018}
}
"libtropicon: A Scalable Library for Computing Intersection Points of Generic Tropical Hyper-surfaces"
Mathematical Software -- ICMS 2018
[ link ]
[
abstract
]
[
bibtex
]
@InProceedings{Chen2018Libtropicon
author="Chen, Tianran",
editor="Davenport, James H. and Kauers, Manuel and Labahn, George and Urban, Josef",
title="libtropicon: A Scalable Library for Computing Intersection Points of Generic Tropical Hyper-surfaces",
booktitle="Mathematical Software -- ICMS 2018",
year="2018",
publisher="Springer International Publishing",
address="Cham",
pages="105--112",
isbn="978-3-319-96418-8"
}
(with Robert Davis)
"A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes"
Advances in Algebra: Research from the Southern Regional Algebra Conference 2017
[ link ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@InProceedings{ChenDavis2019Product,
author="Chen, Tianran and Davis, Robert",
editor="Feldvoss, J{\"o}rg and Grimley, Lauren and Lewis, Drew and Pavelescu, Andrei and Pillen, Cornelius",
title="A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes",
booktitle="Advances in Algebra",
year="2019",
publisher="Springer International Publishing",
address="Cham",
pages="111--119",
isbn="978-3-030-11521-0"
}
(with Christian Knoll, Dhagash Mehta, and Franz Pernkopf)
"Fixed points of belief propagation -- An analysis via polynomial homotopy continuation"
IEEE Transactions on Pattern Analysis and Machine Intelligence
[ link ]
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{knoll_fixed_2017,
author = {Knoll, Christian and Mehta, Dhagash and Chen, Tianran and Pernkopf, Franz},
doi = {10.1109/TPAMI.2017.2749575},
issn = {0162-8828},
journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
pages = {1--1},
title = {{Fixed Points of Belief Propagation - An Analysis via Polynomial Homotopy Continuation}},
url = {http://ieeexplore.ieee.org/document/8027142/},
year = {2017}
}
(with Dhagash Mehta)
"On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations"
IEEE Transactions on Power Systems (2018)
[ link ]
[ pdf ]
[
abstract
]
[
bibtex
]
@article{chen_network_2017,
author = {Chen, Tianran and Mehta, Dhagash},
doi = {10.1109/TPWRS.2017.2724030},
archivePrefix = {arXiv},
arxivId = {1512.04987},
eprint = {1512.04987},
issn = {08858950},
journal = {IEEE Transactions on Power Systems},
keywords = {Power systems,load flow,power system analysis computing,power system control},
title = {On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations},
url = {http://ieeexplore.ieee.org/document/7971956/},
number = {2},
pages = {1451--1460},
volume = {33},
year = {2018}
}
(with Tsung-Lin Lee and Tien-Yien Li)
"Mixed cell computation in Hom4PS-3"
Journal of Symbolic Computation (2017), pp. 516-534
[ link ]
[ pdf ]
[
abstract
]
[
bibtex
]
@article{chen_mixed_2017,
author = {Chen, Tianran and Lee, Tsung-Lin and Li, Tien-Yien},
doi = {10.1016/j.jsc.2016.07.017},
issn = {0747-7171},
journal = {Journal of Symbolic Computation},
month = {mar},
pages = {516--534},
series = {SI: Numerical Algebraic Geometry},
title = {{Mixed cell computation in Hom4PS-3}},
url = {http://www.sciencedirect.com/science/article/pii/S0747717116300542},
volume = {79, Part 3},
year = {2017}
}
(with Dhagash Mehta)
"Parallel degree computation for binomial systems"
Journal of Symbolic Computation (2017)
[ link ]
[ pdf ]
[
abstract
]
[
bibtex
]
@article{chen_parallel_2017,
author = {Chen, Tianran and Mehta, Dhagash},
doi = {10.1016/j.jsc.2016.07.018},
issn = {0747-7171},
journal = {Journal of Symbolic Computation},
keywords = {Algebraic Geometry,BKK root-count,Binomial systems,Folder - Unmixing,GPU computing,Supersymmetric gauge theories,homotopy continuation},
month = {mar},
pages = {535--558},
series = {SI: Numerical Algebraic Geometry},
title = {{Parallel degree computation for binomial systems}},
url = {http://www.sciencedirect.com/science/article/pii/S0747717116300554},
volume = {79, Part 3},
year = {2017}
}
(with Dhagash Mehta, John Morgan, and David Wales)
"Response to "Comment on 'Exploring the potential energy landscape of the Thomson problem via Newton homotopies'""
The Journal of Chemical Physics 143, 247102
[ link ]
[
abstract
]
[
bibtex
]
@article{mehta_response_2015,
author = {Mehta, Dhagash and Chen, Tianran and Morgan, John W. R. and Wales, David J.},
doi = {10.1063/1.4939011},
issn = {0021-9606, 1089-7690},
journal = {The Journal of Chemical Physics},
keywords = {Bifurcations,Difference equations,Flow instabilities,Flow simulations,Folder - LogEval,Folder - Unmixing,Singularity theory},
month = {dec},
number = {24},
pages = {247102},
title = {{Response to “Comment on ‘Exploring the potential energy landscape of the Thomson problem via Newton homotopies”' [J. Chem. Phys. 143, 247101 (2015)]}},
url = {http://scitation.aip.org/content/aip/journal/jcp/143/24/10.1063/1.4939011 http://scitation.aip.org/deliver/fulltext/aip/journal/jcp/143/24/1.4939011.pdf;jsessionid=23ev3700uj07.x-aip-live-03?itemId=/content/aip/journal/jcp/143/24/10.1063/1.4939011{\&}mimeTyp},
volume = {143},
year = {2015}
}
(with Tien-Yien Li)
"Homotopy continuation method for solving systems of nonlinear and polynomial equations"
Communications in Information and Systems 15(2):119--307 (2015)
[ link ]
[
abstract
]
[
bibtex
]
@article{chen_homotopy_2015,
author = {Chen, Tianran and Li, Tien-Yien},
doi = {10.4310/CIS.2015.v15.n2.a1},
issn = {15267555, 21634548},
journal = {Commun. Inf. Syst.},
number = {2},
pages = {119--307},
title = {Homotopy continuation method for solving systems of nonlinear and polynomial equations},
volume = {15},
year = {2015}
}
(with Dhagash Mehta, John Morgan, and David Wales)
"Exploring the potential energy landscape of the Thomson problem via Newton homotopies"
The Journal of Chemical Physics 142 194113 (2015)
[ link ]
[ pdf ]
[
abstract
]
[
bibtex
]
@article{mehta_exploring_2015,
author = {Mehta, Dhagash and Chen, Tianran and Morgan, John W. R. and Wales, David J.},
doi = {10.1063/1.4921163},
issn = {0021-9606, 1089-7690},
journal = {The Journal of Chemical Physics},
keywords = {Eigenvalues,Folder - Finished - NewtonHomotopyReply,Folder - Finished - Solving survey,Folder - LogEval,Folder - Power flow,Folder - Unmixing,Newton Raphson method,Nonlinear dynamics,Numerical solutions,Potential energy surfaces},
month = {may},
number = {19},
pages = {194113},
title = {{Exploring the potential energy landscape of the Thomson problem via Newton homotopies}},
url = {http://scitation.aip.org/content/aip/journal/jcp/142/19/10.1063/1.4921163},
volume = {142},
year = {2015}
}
(with Tien-Yien Li and Xiaoshen Wang)
"Theoretical aspects of mixed volume computation via mixed subdivision"
Communications in Information and Systems (2014)
[ link ]
[ pdf ]
[
abstract
]
[
bibtex
]
@article{chen_theoretical_2014,
author = {Chen, Tianran and Li, Tien-Yien and Wang, Xiaoshen},
doi = {10.4310/CIS.2014.v14.n4.a1},
issn = {15267555, 21634548},
journal = {Communications in Information and Systems},
number = {4},
pages = {213--242},
title = {Theoretical aspects of mixed volume computation via mixed subdivision},
url = {http://www.intlpress.com/site/pub/pages/journals/items/cis/content/vols/0014/0004/a001/},
volume = {14},
year = {2014}
}
(with Dhagash Mehta, Jonathan Hauenstein, and David Wales)
"Newton homotopies for sampling stationary points of potential energy landscapes"
The Journal of Chemical Physics 141 (12), 121104 (2014)
[ link ]
[ pdf ]
[ arXiv ]
[
abstract
]
[
bibtex
]
@article{mehta_newton_2014,
author = {Mehta, Dhagash and Chen, Tianran and Hauenstein, Jonathan D and Wales, David J},
doi = {10.1063/1.4896657},
issn = {0021-9606, 1089-7690},
journal = {The Journal of Chemical Physics},
keywords = {Atomic and molecular clusters,Biomolecular structure,Nonlinear differential equations,Numerical modeling,Numerical solutions,real homotopy},
month = {sep},
number = {12},
pages = {121104},
shorttitle = {Communication},
title = {Newton homotopies for sampling stationary points of potential energy landscapes},
url = {http://scitation.aip.org/content/aip/journal/jcp/141/12/10.1063/1.4896657},
volume = {141},
year = {2014}
}
(with Tien-Yien Li)
"Solutions to systems of binomial equations"
Annales Mathematicae Silesianae 28:7–34 (2014)
[ link ]
[ pdf ]
[
abstract
]
[
bibtex
]
@article{chen_solutions_2014,
author = {Chen, Tianran and Li, Tien-Yien},
journal = {Annales Mathematicae Silesianae},
pages = {7--34},
title = {{Solutions to systems of binomial equations}},
volume = {28},
year = {2014}
}
(with Tsung-Lin Lee, and Tien-Yien Li)
"Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods (extended abstract)"
Mathematical Software -- ICMS 2014 -- 4th International Congress, Seoul, South Korea, August 5-9, 2014. Proceedings 8592:183–190
[ link ]
[
abstract
]
[
bibtex
]
@incollection{chen_hom4ps-3,
author = {Chen, Tianran and Lee, Tsung-Lin and Li, Tien-Yien},
booktitle = {Mathematical Software – ICMS 2014},
editor = {Hong, Hoon and Yap, Chee},
isbn = {978-3-662-44198-5 978-3-662-44199-2},
keywords = {Algorithm Analysis and Problem Complexity,Discrete Mathematics in Computer Science,Math Applications in Computer Science,Numeric Computing,Software Engineering/Programming and Operating Sy,Theory of Computation,binomial system,homotopy continuation,polyhedral homotopy,polynomial systems},
month = {jan},
number = {8592},
pages = {183--190},
publisher = {Springer Berlin Heidelberg},
series = {Lecture Notes in Computer Science},
shorttitle = {Hom4PS-3},
title = {{Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods}},
url = {http://link.springer.com/chapter/10.1007/978-3-662-44199-2{\_}30},
year = {2014}
}
(with Tsung-Lin Lee and Tien-Yien Li)
"Mixed volume computation in parallel"
Taiwanese Journal of Mathematics 18(1):93–114 (2014)
[ link ]
[ pdf ]
[
abstract
]
[
bibtex
]
@article{chen_mixed_2014,
author = {Chen, Tianran and Lee, Tsung-Lin and Li, Tien-Yien},
journal = {Taiwanese Journal of Mathematics},
keywords = {bkk,mixed cells,mixed volume},
number = {1},
pages = {93--114},
title = {Mixed volume computation in parallel},
volume = {18},
year = {2014}
}
(with Tien-Yien Li)
"Spherical projective path tracking for homotopy continuation methods"
Communications in Information and Systems 12(3):195-220 (2012)
[ link ]
[ pdf ]
[
abstract
]
[
bibtex
]
@article{chen_spherical_2012,
author = {Chen, Tianran and Li, Tien-Yien},
journal = {Communication in Information and Systems},
number = {3},
pages = {195--220},
title = {Spherical projective path tracking for homotopy continuation methods},
volume = {12},
year = {2012}
}