Jose Israel Rodriguez



University of Chicago,
Department of Statistics

CV (01/18) and Headshot

A talk about my thesis

Google Scholar

Advisors and Mentors

Postdoc Mentor: Lek-Heng Lim

Previous Mentor: Jonathan Hauenstein

PhD Advisor: Bernd Sturmfels

Undergraduate Advisor: Eric Katz


SIAM AN18 JSM18 SIAM AN16: Algebraic Statistics

AG17 AG15: Likelihood Geometry and Maximum Likelihood Degrees

JMM17: Numerical Algebraic Geometry

SIAM AN18: Numerical Differential Geometry


Galois Groups

I am a University of Chicago Provost's Career Enhancement Postdoc working with Lek-Heng Lim after transitioning from a NSF Postdoc. My previous postdoc was at the Simon's Institute and University of Notre Dame with Jonathan Hauenstein. In May 2014, I graduated from UC Berkeley with a PhD in mathematics. My thesis is titled Numerical algebraic geometry for maximum likelihood estimation, and my advisor was Bernd Sturmfels.

My interests are in applied algebraic geometry and algebraic methods for statistics. I have developed methods to compute Galois groups, braid groups, and Euler obstructions with my motivation coming from understanding problems in maximum likelihood estimation, kinematics, and enumerative geometry.

Here is a link to a talk at the Simons Institute about my thesis work: Link.

SIAM Journal in Applied Algebra and Geometry (SIAGA) in Seven Pictures and Tomorrow.




[24] Factoring tensor polynomials, graphs, and matrices.
(In preparation) We develop algorithms to factor a tensor polynomial, a tensor product of graphs, and Kronecker product of matrices.

[23] Numerical algebraic geometry for generalized method of moments.
(In preparation) We use homotopy continuation to solve GMM.

[22] Fiber product homotopies for multiparameter eigenvalue problems. (2018)
We introduce the fiber product homotopy to solve multiparameter eigenvalue problems. We show that our method is more accurate and faster for large dimensional problems.

[21] Numerical computation of braid groups. (2017)
We give a numerical algorithm to compute braid groups of curves, hyperplane arrangements, and parameterized system of polynomial equations using homotopy continuation.

[20] Accurate Solutions of Polynomial Eigenvalue Problems. (2017)
We use homotopy continuation to solve the polynomial eigenvalue problem. We show that this method produces substantially more accurate results and finds all eigenvalues with a certificate of correctness via Smale's alpha-theory.

[19] Computing Euler obstruction functions using maximum likelihood degrees. (2017)
Tools from algebraic topology have been used to solve problems in algebraic statistics. In this paper, we go the other way. We use the idea of maximum likelihood degree from algebraic statistics to compute values of the Euler obstruction function in algebraic topology.

[18] Solving parameterized polynomial system with decomposable projections. (2017)
We exploit the structure of decomposable projections in problems from statistics, kinematics, and benchmark problems in computational algebra. (Accepted for presentation at MEGA 2017 and Code).

[17] Multiprojective witness sets and a trace test. (2015)
We generalize regeneration, the trace test, and numerical irreducible decomposition to the multiprojective case. Applications include Alt's problem and tensor decomposition.

[16] Bertini for Macaulay2. (2015)
Numerical algebraic geometry is the field of computational mathematics concerning the numerical solution of polynomial systems of equations. Bertini, a popular software package for computational applications of this field, includes implementations of a variety of algorithms based on polynomial homotopy continuation. The Macaulay2 package Bertini.m2 provides an interface to Bertini.


[15] Trace test. Accepted to the Arnold Mathematical Journal. (2018)
We give a brief derivation of the trace test to verify completeness of a partial witness set of an irreducible variety in affine or projective space.

[14] The Maximum Likelihood Degree of Toric Varieties. Accepted to Journal of Symbolic Computation. (2018)
We study the maximum likelihood degree (ML degree) of toric varieties, known as discrete exponential models in statistics.

[13] The maximum likelihood degree of mixtures of independence models. SIAM J. Appl. Algebra Geometry, 1(1), 484-506. (23 pages)
We use Euler characteristics to prove an outstanding conjecture by Hauenstein, the first author, and Sturmfels by giving a recursion to determine ML degrees for mixtures of independence models. The recursion has been implemented here.

[12] Numerical computation of Galois groups. Accepted to Foundations of Computational Mathematics (2017)
We use numerical homotopy continuation to compute Galois groups. Applications include formation shape control, the Alt-Burmester 4-Bar mechanism, and maximum likelihood estimation.

[11] A Probabilistic Algorithm for Computing Data-Discriminants of Likelihood Equations. Journal of Symbolic Computation Volume 83, November-December 2017, Pages 342-364
We develop a probabilistic algorithm with three different strategies for computing Data-Discriminants improving our previous version presented in ISSAC2015.

[10] The maximum likelihood data singular locus, Journal of Symbolic Computation Volume 79, Part 1, March-April 2017, Pages 99-107.
We describe the special locus of data for which the likelihood equations have a solution in the model's singular locus.

[9] Critical points via monodromy and local methods, Journal of Symbolic Computation. Volume 79, Part 3, March-April 2017, Pages 559-574
We use the numerical algebraic geometry tool of monodromy and local methods to compute critical points of the likelihood function and Euclidean distance function.

[8] Data discriminants of likelihood equations, ISSAC '15 Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pages 307-314
We develop a probabilistic algorithm for computing data discriminants that is experimentally more efficient than the standard elimination algorithm. Based on the computational results, we propose the real root classification conjecture for the 3 by 3 symmetric matrix model.

[7] Maximum likelihood for dual varieties. SNC '14 Proceedings of the 2014 Symposium on Symbolic-Numeric Computation (2014) Pages 43-49.
MLE for statistical models with discrete data is studied from an algebraic statistics viewpoint. A reformulation of the MLE problem in terms of dual varieties and conormal varieties is given.

[6] Maximum likelihood geometry in the presence of data zeros. ISSAC '14 Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (2014) Pages 232-239.
We consider discrete algebraic statistical models and study the solutions to the likelihood equations when the data contain zeros. We give new lower bounds for Maximum Likelihood degrees

[5] Maximum Likelihood Duality for Determinantal Varieties. International Mathematics Research Notices, Volume 2014, Issue 20, 1 January 2014, Pages 5648-5666.
We prove that the maximum-likelihood degree of the variety of rank-r matrices equals that of the variety of co-rank (r-1)-matrices; and also establish variants for symmetric and skew-symmetric matrices.

[4] Maximum Likelihood for Matrices with Rank Constraints. Journal of Algebraic Statistics, Volume 5, Number 1, Pages 18-38.
We use numerical algebraic geometry to find maximum likelihood degrees of determinantal varieties and show these techniques can be useful for statisticians.

[3] Combinatorial Excess Intersection. Journal of Symbolic Computation Volume 68, Part 2, May-June 2015, Pages 297-307
We provide formulas and algorithms for computing the excess numbers of certain ideals. The solution for monomial ideals is given by the mixed volumes of certain polytopes. These results enable us to design specific homotopies for numerical algebraic geometry.

[2] A novel method for the solution of the forward displacement problem of spherical parallel manipulators. ZAMM Z. Angew. Math. Mech. Volume 93, Issue 1, January 2013, Pages 73-82
This work uses algebraic geometry to find an analytical (solvable in radicals) solution to a well studied problem in kinematics.

[1] Bounding The Degree of Belyi Polynomials. J. Number Theory vol. 133 (2013), no. 9, 2892-2900.
This work began as an undergraduate research project advised by Eric Katz. A new combinatorial argument is given to bound the degree of a Belyi polynomial using the valuation of its roots.

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Travels, visitors, and advertisements

Latinx in the Mathematical Sciences Conference 2018: Discrete Mathematics: March 8 - 10, 2018 IPAM, UCLA.

Algebra, Geometry, and Combinatorics Seminar: March 14, 2018 San Francisco State University, San Francisco, CA.

Western Algebraic Geometry Symposium (WAGS): March 17 - 18, 2018 San Francisco State University, California.

AMS Eastern Sectional (Algebraic Statistics and Toric Geometry Sessions): April 21 - 22, 2018, Northeastern University, Boston, Massachusetts.

An NSF/CBMS Regional Conference in the Mathematical Sciences Applications of Polynomial Systems: June 4 - 8, 2018, Texas Christian University, Fort Worth, Texas.

The University of Chicago 2018 Summer Workshop: June 11 - 15, 2018, Chicago, Illinois.

SIAM AN18: July 9 - 13, 2018, Oregon Convention Center, Portland, Oregon.

JSM 2018: July 28 - August 2, 2018 Vancouver Convention Center, Vancouver, British Columbia, Canada (Advertisement).

ICMS 2018 (Computational Algebraic Geometry Session): July 24 - 27, 2018 University of Notre Dame.

Joint CUNY Graduate Center-Courant Seminar in Symbolic-Numeric Computing : August 30, 2018 CUNY Graduate Center, New York.

ICERM Semester Program on Nonlinear Algebra : Sept 5 - Dec 7, 2018 ICERM, Brown University, Providence, RI.

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Fellow McNairScholars

Oscar D. Ayala Kyle Hale Martinque 'Marti' Jones Esmeralda Rodriguez


Carlos Amendola

Dan Bates

Nathan Bliss

Isaac Burke

Jan Draisma

Courtney Gibbons

Elizabeth Gross

Jonathan Hauenstein

Martin Helmer

Emil Horobet

Serkan Hosten

Anton Leykin

Lek-Heng Lim

Abraham Martin del Campo

Evan D. Nash

Maurizio Ruggiu

Daniel Smolkin

Frank Sottile

Bernd Sturmfels

Xiaoxian Tang

Botong Wang

Yiling You Go to top

Past Whereabouts

Mathematics Colloquium: December 1, 2014 UT Dallas.

Top Com Seminar: January 7, 2015 Freie Universitat Berlin.

Algebraic Geometry and Number Theory Seminar: February 3, 2015 Rice University, Houston.

Algebraic Geometry/Commutative Algebra Seminar: February 25, 2015 University of Notre Dame.

Algebraic Geometry Seminar: March 13, 2015 University of Wisconsin-Madison.

Algebra, Geometry, and Combinatorics Seminar: May 1, 2015 UIUC.

GOAL workshop: May 18-19, 2015 University of California, Berkeley.

UC Berkeley Commencement: May 22-23, 2015 University of California, Berkeley.

Tropical Doener Workshop: May 27-28, 2015 Freie Universität Berlin, Germany.

Workshop on Non-Linear Algebra: May 29-30, 2015 in Berlin, Germany.

GAeL XXIII: June 1-5, 2015 KU Leuven, Belgium.

Algebraic Statistics 2015: June 8-11, 2015 in Genoa, Italy.

MEGA 2015: Effective Methods in Algebraic Geometry: June 15-19, 2015 at The University of Trento, Povo (Trento), Italy.

Eindhoven Discrete Math Seminar : June 24, 2015 Technische Universiteit Eindhoven, Eindhoven, Netherlands.

ISSAC 2015: July 7-9, 2015 The University of Bath, Bath, UK.

SWAAG: July 13-17, 2015 NIMS, Daejeon, South Korea.

Applied Algebraic Geometry Workshop: July 30, 2015 Daegu Gyeongbuk Institute of Science and Technology, Daegu, South Korea.

SIAM AG15: August 03-07, 2015. I co-organized a mini-symposium Maximum Likelihood Degrees and Critical Points.

Dickson Day: October 2, 2015 University of Chicago, Chicago, IL.

AMS Sectional Meeting: Algebraic statistics and its interactions with combinatorics, computation, and network science session: October 3-4, 2015 Loyola University, Chicago, IL.

Geometry Seminar : November 1-3, 2015 Texas A&M, College Station, Texas.

Colloquium : November 16-18, 2015 Santa Clara University, Santa Clara, California.

California Alliance visit to Stanford : November 18-24, 2015 Stanford University, Stanford, California.

Algebraic Geometry Seminar : November 20, 2015 Stanford University, Stanford, California.

Workshop on Algebra, Geometry and Proofs in Symbolic: December 6-13, 2015 The Fields Institute-Toronto, Canada.

Algorithms and Complexity in Algebraic Geometry Reunion Workshop: December 14-17, 2015 Simons Institute, Berkeley, CA.

Joint Mathematics Meetings: January 6-9, 2016 Washington State Convention Center, Seattle, WA.

Applied Algebra Seminar: Apr 7, 2016 University of Wisconsin-Madison, Madison, Wisconsin.

Applied Algebra Days 3: Apr 29-May 1, 2016 University of Wisconsin-Madison, Madison, Wisconsin.

Special session for applied algebraic geometry: (Depart May 13) May 14-May 22, 2016 KSIAM Daejeon, Korea.

Bertini Workshop: May 23-25, 2016 Notre Dame University, Notre Dame, Indiana.

Spring Research Conference 2016: May 25-27, 2016 Illinois institute of Technology, Chicago, Illinois.

MRC: Algebraic Statistics: June 12-18, 2016 Snowbird Resort, Utah.

Applications of Algebraic Methods to Statistics: (Depart June 18) June 19-25, 2016 Research Institute of Mathematical Science of Kyoto University, Kyoto, Japan.

SIAM Annual Meeting: July 11-14, 2016 Westin Boston Waterfront, Boston, Massachusetts.

Algebraic Statistics Session: July 11, 2016 Westin Boston Waterfront, Boston, Massachusetts.

XXI Coloquio Latinoamericano de Álgebra : (Depart July 26) July 27-29, 2016 Buenos Aires, Argentina.

Computational Algebra, Algebraic Geometry and Applications : August 1-3, 2016 Buenos Aires, Argentina.

Research Seminar: (Bay area Sept 16-21) September 19, 2016 San Jose State University, San Jose, CA.

Algebra, Geometry, and Combinatorics Seminar: September 21, 2016 San Francisco State University, San Francisco, CA.

Fall Western Sectional Meeting: (Depart Oct 7) October 8-12, 2016 University of Denver, Denver, CO.

AMS special on Applied Algebraic Geometry: November 12-13, 2016 North Carolina State University in Raleigh, North Carolina.

JMM17: Num AG Special Session: Thursday January 5 (Conference January 4-7), 2017 Hyatt Regency Atlanta and Marriott Atlanta Marquis, Atlanta, GA.

JMM17: MRC Alg Stat Special Session: Thursday January 5 (Conference January 4-7), 2017 Hyatt Regency Atlanta and Marriott Atlanta Marquis, Atlanta, GA.

Scientific and statistical computing seminar: February 2, 2017 University of Chicago, Chicago IL.

Colloquium: March 10, 2017 University of Chicago, Chicago, Illinois.

Algebraic Geometry Seminar: April 4, 2017 (Return April 5) The Ohio State University, University in Columbus, Ohio.

Oberwolfach: Algebraic Statistics : (Depart Apr 15, Return Apr 24) April 16 - April 22, 2017 Oberwolfach, Germany.

Polynomials, Kinematics, and Robotics : June 5-7, 2017 University of Notre Dame, Indiana.

String Phenomenology Conference (StringPheno 2017) : July 3-7, 2017 Virginia Tech, Blacksburg, Virginia.

Foundations of Computational Mathematics Conference 2017 : (Depart July 8) July 10 -July 19, 2017 Barcelona, Spain.

Applied Macaulay2 tutorials : July 27-29, 2017 Georgia Institute of Technology, Atlanta, Georgia.

SIAM Conference on Applied Algebraic Geometry (AG17) : July 31-August 4, 2017 Georgia Institute of Technology, Atlanta, Georgia.

AMS Special Session on Applicable and Computational Algebraic Geometry : September 9-10, 2017 University of North Texas, Denton TX.

SILO Seminar Series: October 3-5, 2017 University of Wisconsin-Madison.

Geometry Seminar: October 16-October 18, 2017 Massachusetts Institute of Technology, Cambridge, MA.

Algebra meets numerics: condition and complexity : November 5-8, 2017 TU Berlin, Berlin, Germany.

Non-Linear Algebra : November 8-12, 2017 Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany.

Seminar: November 12-15, 2017 Sapientia University, Targu Mures, Romania.

Seminar: November 15-18, 2017 KU Leuven, Belgium.

Random Structures Seminar: November 29, 2017 University of Texas, Austin, Texas.

Geometry Seminar: November 30-December 1, 2017 Texas A&M, College Station, Texas.

Seminar: December 6-8, United States Naval Academy, College in Annapolis, Maryland.

2017 Korea-China International Conference on Matrix Theory with Applications: (Arrive in Seoul Dec 11 and depart Dec 22) December 14-December 17, 2017 AORC-IRCTMT Joint Meeting Sungkyunkwan University, Suwon, South Korea.

Applied analysis and algebra seminar: December 18, Yonsei University, Seoul, South Korea.

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