Jose Israel Rodriguez


Contact and Info

Jose@math.wisc.edu

Univ. of Wisconsin - Madison
Dept. of Mathematics

CV (10/19) Travel (Recent)
Publications (01/20)

Thesis talk Google Scholar

PhD Advisor

Bernd Sturmfels

Previous Postdoc Mentors

(U of Chicago) Lek-Heng Lim

(Notre Dame) Jonathan Hauenstein

Upcoming and Recently Organized

AMS20 (05/02-5/03) With Tingting Tang and Robert Krone.

Monodromy and Galois groups in Enumerative Geometry and Applications (08/31-09/02) With Frank Sottile and Alex Esterov.

Local Seminars and Events Organized

  • Applied Algebra (Ongoing)
  • WIMAW Meet and Greet (2019)
  • Matroids Day (2019)
  • Matroids (2019)
  • Algebraic Geometry (2016-18)
  • Minisymposia Organized

    Applications of Algebra: AMS19

    Algebraic statistics:
    AMS20 SIAM AG19 SIAM AN18 JSM18 SIAM AN16

    Likelihood Geometry and Maximum Likelihood Degrees SIAM AG17 SIAM AG15

    Numerical methods in algebraic geometry
    SIAM AG19 SIAM AN18 JMM17

    Software

    Decomposable Sparse Polynomial Systems

    Multiregeneration

    Algebraic Optimization

    Galois Groups

    ML Euler Obstruction



    In Fall 2020 I will start a tenure track position at UW Madison.


    Research Interests

    My interests are broadly in applied algebraic geometry and algebraic methods for statistics. In particular:
  • Nonlinear algebra and nonlinear eigenvalue problems
  • Algebraic Statistics and nearest point problems
  • Applications of monodromy and Galois groups
  • An exciting conference I attend is SIAM AG. The AG19 program has a wide range of topics. Here is a link to a talk at the Simons Institute about my thesis work.

    A list of collaborators.


    Teaching

    Spring 2020: CURL
    Spring 2020: Graduate Applied Algebra

    Fall 2019: Applied Linear Algebra

    Reading courses 2019:
    (1) Algebraic methods in statistics (Julia Lindberg)
    (2) Euler characteristics and optimization (Xiaxin Li)


    Undergraduate Research and Reading

    Collaborative Undergraduate Research Laboratory (CURL): This course runs Spring 2020, and applications were found here.

    I am available for directed reading courses on Ideal, Varieties, and Algorithms (introduction to computational algebraic geometry), Algebraic Statistics, solving polynomial systems, and Numerically solving polynomial systems.


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



    Preprints

    [37] Data loci in algebraic optimization
    We introduce methods for computing data loci (2020).

    [36] Decomposable Sparse Polynomial Systems
    This article surveys a Macaulay2 implementation for solving sparse decomposable systems. (2020)

    [35] A Morse theoretic approach to non-isolated singularities and applications to optimization
    We develop an approach to study positive dimensional critical sets with applications to SDP's and nearest point problems. (2020)

    [34] Solving Decomposable Sparse Systems
    We present a new recursive algorithm for solving sparse decomposable systems. (2020)

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

    [32] Defect of Euclidean distance degree. (2019)
    Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for nearest point problems of the algebraic variety. In this paper we compute the difference of these degrees by classical techniques in Singularity Theory, thereby deriving a new method for computing ED degrees of smooth complex projective varieties.

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

    [30] 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.

    [29] 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.

    [28] 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).

    [27] 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.

    Publications

    [26] A numerical approach for computing Euler characteristics of affine varieties (Lecture Notes in Computer Science, 2020)
    We develop a numerical nonlinear algebra approach for computing the Euler characteristic of an affine variety.

    [25] Algebraic Optimization Degree (To appear in the 2020 ISSAC Proceedings).
    This article surveys a Macaulay2 implementation for determining the algebraic degree of an optimization problem. (2020)

    [24] A numerical toolkit for multiprojective varieties. (2020, Accepted to mathematics of computation)
    We develop a toolkit for studying roots of polynomial systems in multiple variable groups.

    [23] Multiregeneration for polynomial system solving (To appear in the 2020 ISSAC Proceedings).
    We demonstrate our implementation of a continuation method for solving polynomials systems. (2020)

    [22] Euclidean distance degree of projective varieties. (2020, Accepted to IMRN)
    We give a positive answer to a conjecture of Aluffi-Harris on the computation of the Euclidean distance degree of a possibly singular projective variety.

    [21] The algebraic matroid of the funtf variety (Accepted 2020, Journal of Pure and Applied Alegebra)
    The affine funtf variety is the Zariski closure of the set of finite unit norm tight frames. This work characterizes the bases of the algebraic matroid underlying the affine funtf variety of funtfs in R^3, and partial results towards similar characterizations in higher dimensions.

    [20] Computing Euler obstruction functions using maximum likelihood degrees. (Accepted 2019, IMRN)
    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.

    [19] Euclidean distance degree of the multiview variety. (SIAM SIAGA)
    We solve the open problem in computer vision of determining the Euclidean distance degree of the affine multiview variety.

    [18] Multiprojective witness sets and a trace test. (Accepted 2019, Advances in geometry)
    We generalize regeneration, the trace test, and numerical irreducible decomposition to the multiprojective case. Applications include Alt's problem and tensor decomposition.

    [17] Homogenized funtf varieties and algebraic frame completion.
    (Extended abstract for ISSAC'18 Poster Session) We study the homogenized funtf variety and the degrees of its coordinate projections.

    [16] Solving the likelihood equations to compute Euler obstruction functions. Mathematical Software - ICMS 2018, Proceedings volume 10931 of Lecture Notes in Computer Science. Springer, pages 405--413, 2018.
    This paper describes a Macaulay2 package and algorithms to compute Euler obstructions using maximum likelihood degrees that are determined by solving likelihood equations.

    [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

    PACM IDeAS : Feb 26 - Feb 28, 2019 Princeton University, Princeton, NJ.

    Special Session on Applications of Algebraic Geometry : Mar 15-17, 2019 AMS Sectional, Auburn University, Auburn, AL.

    Junior Colloquium Series : Apr 12, 2019 University of Michigan, Ann Arbor, MI.

    SIAM AG:19 : July 9-13, 2019 University of Bern, Bern, Switzerland.

    ICIAM : July 15-19, 2019 Valencia, Spain.

    SIAM PNW : Oct 18 - Oct 20, 2019 Seattle, WA.

    Oberwolfach: Algebraic Statistics : Oberwolfach, Germany, March 15-21, 2020.

    Einstein Workshop: Nonlinear Algebra : TU Berlin, Berlin, Germany, March 26-27, 2020.

    AMS Special Session: Algebraic geometry in statistics and machine learning : Fresno, CA, May 2-3, 2020.

    Macaulay 2 Workshop (Virtual): Algebraic Statistics Group : Cleveland State University, Cleveland, Ohio, May 11-15, 2020.

    (Rescheduled COVID-19) HHXXI 2020 - Householder Symposium XXI : Hotel Sierra Silvana, Selva di Fasano (Br), Italy, 14-19 June 2020.

    Algebraic Statistics 2020 (Virtual) : University of Hawai'i at Manoa Honolulu, HI, June 22 - June 26, 2020

    ICMS (Virtual) : Braunschweig, 13-16 July 2020.

    ISSAC (Virtual) : Kalamata, Messinia, Greece, 20-23 July 2020.

    Algebraic Statistics 2021 : University of Hawai'i at Manoa Honolulu, HI, May 17 - May 21, 2021.

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    Fellow McNair Scholars

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


    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.

    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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