Jose Israel Rodriguez


Email

jo dot ro at berkeley dot edu

Office

743 Evans Hall


Research and Writings

Maximum likelihood geometry in the presence of data zeros.
we consider discrete algebraic statistical models and study the solutions to the likelihood equations when the data contain zeros. Focusing on sampling and model zeros, we show that, in these cases, the solutions to the likelihood equations are contained in a previously studied variety, the likelihood correspondence. The number of these solutions give a lower bound on the ML degree, and the problem of finding critical points to the likelihood function can be partitioned into computationally easier problems involving sampling and model zeros.

Bertini for Macaulay2. Submitted.
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.

Maximum Likelihood Duality for Determinantal Varieties. Int Math Res Notices (2013)
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.

Maximum Likelihood for Matrices with Rank Constraints. To appear in Journal of Algebraic Statistics.
We use numerical algebraic geometry to find maximum likelihood degrees of determinantal varieties and show these techniques can be useful for statisticians.

Combinatorial Excess Intersection. Submitted.
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.

A novel method for the solution of the forward displacement problem of spherical parallel manipulators. ZAMM Z. Angew. Math. Mech. 93 (2013), no. 1, 73–82.
This work uses algebraic geometry to find an analytical (solvable in radicals) solution to a well studied problem in kinematics.

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.

How to E-mail a Riemann Surface. An exposition for Bernd Sturmfels' class on algebraic curves.

Supplementary Materials

Interests

I am in my fourth year of graduate school at UC Berkeley. My advisor is Bernd Sturmfels, and my interests are in applied algebraic geometry. In particular, I am interested in solving systems of polynomial equations numerically.

Books I like

A Royal Road to Algebraic Geometry

The Numerical Solution of Systems of Polynomials Arising In Engineering and Science

Algebra Pure and Applied

Complex Variables and Applications

Teaching and Mentoring

Sections of Math 1B with Professor Serganova (Fall 2013): Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations.

Sections of Math 1B with Professor Christ (Spring 2013): Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations.

Berkeley Science Network

The UC Berkeley McNair Scholars Program

Software Resources

Lyx

M2 Table of Contents.

Bertini Documentation

Invited Talks and Seminars

- (August 2013) Symbolic Computation Seminar, North Carolina State University.

- (August 2013) Algebra Seminar, Georgia Institute of Technology, Atlanta.

- (August 2013) SIAM Conference of Applied Algebraic Geometry , Colorado State University.

- (July 2013) Computational Algebra Seminar, Max Planck Institute for Mathematics, Bonn.

- (July 2013) Algebra Seminar, Universitat des Saarlandes.

- (July 2013) Discrete Mathematics Seminar, Goethe-Universitat.

- (July 2013) Algebra Seminar, Universitat Konstanz.

- (June 2013) Effective Methods in Algebraic Geometry, Goethe-Universitat.

- (April 2013) Graduate Student Combinatorics Conference, University of Minnesota, Twin Cities.

- (March 2013) Math/ICES Center of Numerical Analysis Seminar, University of Texas at Austin.

- (March 2013) Colloquium, Sam Houston State University.

- (March 2013) AVID Program Seminar, Grand Prairie High School.

- (March 2013) SIAM Student Chapter Seminar, Texas Tech University.

- (March 2013) Colloquium, Southern Methodist University.

- (March 2013) Algebraic Statistics Workshop, University of California at Berkeley.

- (March 2013) AGC Seminar, San Francisco State University.

- (March 2013) Algebra and Discrete Mathematics, University of California at Davis.

- (February 2013) SIAM Student Chapter Seminar, University of California at Berkeley.

- (February 2013) FRAGMENT Seminar, Colorado State University.

- (February 2013) Colloquium, University of Colorado at Boulder.