# overview
Algebraic combinatorics, real and computational algebraic geometry, applications of algebraic geometry, Mathematical Physics, Hopf algebras, discrete and computational geometry, tropical geometry.

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
Networking And Information Technology R&d

# educationAndTraining[].organization.label
University of Chicago - (Chicago, Illinois, United States)

# educationAndTraining[].organization.label
Michigan State University - (East Lansing, Michigan, United States)

# educationAndTraining[].organization.label
University of Chicago - (Chicago, Illinois, United States)

# educationAndTraining[].organization.label
University of Cambridge - (Cambridge, United Kingdom)

# awardsAndHonors[].label
Faculty Early Career Development (CAREER) Program

# awardsAndHonors[].label
Faculty Early Career Development (CAREER) Program

# publications[].label
Pieri's rule for flag manifolds and Schubert polynomials

# publications[].label
Betti number bounds for fewnomial hypersurfaces via stratified Morse theory

# publications[].label
Galois groups of Schubert problems via homotopy computation

# publications[].label
Grothendieck polynomials via permutation patterns and chains in the  Bruhat order

# publications[].label
Convex Hulls of Orbits and Orientations of a Moving Protein Domain

# publications[].label
Lines tangent to 2 n  2 2n-2 spheres in R n {mathbb R}^n

# publications[].label
An inequality of Kostka numbers and Galois groups of Schubert problems

# publications[].label
Two Murnaghan-Nakayama Rules in Schubert Calculus

# publications[].label
Skew Schubert functions and the Pieri formula for flag manifolds

# publications[].label
On the Sharpness of fewnomial bound and the number of components of a  fewnomial hypersurface

# publications[].label
A combinatorial proof that Schubert vs. Schur coefficients are  nonnegative

# publications[].label
Schubert Calculus, Schubert Cell, Schubert Cycle, and Schubert   Polynomials

# publications[].label
Structure of the Loday-Ronco Hopf algebra of trees

# publications[].label
Numerical Schubert Calculus in Macaulay2

# publications[].label
Lower Bounds for Real Solutions to Sparse Polynomial Systems

# publications[].label
Real k k -flats tangent to quadrics in R n mathbb {R}^n

# publications[].label
Elementary Transversality in the Schubert calculus in any Characteristic

# publications[].label
A new approach to Hilbert's theorem on ternary quartics

# publications[].label
Higher convexity for complements of tropical varieties

# publications[].label
The Horn recursion for Schur P- and Q- functions

# publications[].label
Irrational proofs for three theorems of Stanley

# publications[].label
Real Solutions to Equations from Geometry

# publications[].label
Fewnomial bounds for completely mixed polynomial systems

# publications[].label
The Equivariant Chow rings of quot schemes

# publications[].label
Certification for polynomial systems via square subsystems

# publications[].label
Galois Groups in Enumerative Geometry and Applications

# publications[].label
Cremona Convexity, Frame Convexity, and a Theorem of Santal

# publications[].label
Rational curves on Grassmannians: systems theory, reality, and   transversality

# publications[].label
Lower bounds in real algebraic geometry and orientability of real toric  varieties

# publications[].label
Skew Littlewood-Richardson Rules from Hopf Algebras

# publications[].label
The Horn recursion for Schur P- and Q- functions: Extended Abstract

# publications[].label
Pieri's formula for flag manifolds and Schubert polynomials

# publications[].label
Frontiers of Reality in Schubert Calculus

# publications[].label
Numerical Schubert Calculus via the Littlewood-Richardson Homotopy  Algorithm

# publications[].label
A sagbi basis for the quantum Grassmannian

# publications[].label
Some Remarks on Real and Complex Output Feedback

# publications[].label
A congruence modulo four in real Schubert calculus

# publications[].label
Enumerative geometry for the real Grassmannian of lines in projective space

# publications[].label
Double transitivity of Galois Groups in Schubert Calculus of   Grassmannians

# publications[].label
(Formulas of Brion, Lawrence, and Varchenko on rational generating functions for cones)

# publications[].label
Real Schubert Calculus: Polynomial systems and a conjecture of Shapiro   and Shapiro

# publications[].label
A numerical toolkit for multiprojective varieties

# publications[].label
Skew LittlewoodRichardson rules from Hopf algebras

# publications[].publisher
Centre pour la Communication Scientifique Directe (CCSD)

# publications[].publisher
Institute of Electrical and Electronics Engineers (IEEE)

# publications[].label
Real Schubert Calculus: Polynomial Systems and a Conjecture of Shapiro and Shapiro

# publications[].label
Irrational proofs for three theorems of Stanley

# publications[].label
Cohomological consequences of the pattern map

# publications[].label
Euclidean distance degree and mixed volume

# publications[].label
Irrational Toric Varieties and Secondary Polytopes

# publications[].label
Betti number bounds for fewnomial hypersurfaces via stratified Morse  theory

# publications[].label
A congruence modulo four in real Schubert calculus

# publications[].label
Degenerations of real irrational toric varieties

# publications[].label
Real solutions to systems of polynomial equations in Macaulay2

# publications[].label
Skew Schubert functions and the Pieri formula for flag manifolds

# publications[].label
Experimentation in the Schubert Calculus

# publications[].publisher
Mathematical Society of Japan (Project Euclid)

# publications[].publisher
Association for Computing Machinery (ACM)

# publications[].label
Structure of the Loday-Ronco Hopf algebra of trees

# publications[].label
Cocommutative Hopf algebras of permutations and trees

# publications[].label
KhovanskiiRolle Continuation for Real Solutions

# publications[].label
Fewnomial bounds for completely mixed polynomial systems

# publications[].label
Higher convexity for complements of tropical varieties

# publications[].label
Toric ideals, real toric varieties, and the algebraic moment map

# publications[].label
Some real and unreal enumerative geometry for flag manifolds

# publications[].label
General isotropic flags are general (for Grassmannian Schubert calculus)

# publications[].label
Complex static skew-symmetric output feedback control

# publications[].label
New fewnomial upper bounds from Gale dual polynomial systems

# publications[].label
Grothendieck polynomials via permutation patterns and chains in the Bruhat order

# publications[].label
New Fewnomial Upper Bounds from Gale Dual Polynomial Systems

# publications[].publisher
National Research University, Higher School of Economics (HSE)

# publications[].label
On the fundamental group of open Richardson varieties

# publications[].label
Numerical homotopies from Khovanskii bases

# publications[].label
Pieri-type formulas for maximal isotropic Grassmannians via triple   intersections

# publications[].label
Degenerations of real irrational toric varieties

# publications[].publisher
Society for Industrial & Applied Mathematics (SIAM)

# publications[].label
Tableau Switching: Algorithms and Applications

# publications[].label
The Secant Conjecture in the real Schubert calculus

# publications[].label
A Pieri-type formula for the K-theory of a flag manifold

# publications[].label
A lifted square formulation for certifiable Schubert calculus

# publications[].label
A geometric proof of an equivariant Pieri rule for flag manifolds

# publications[].label
Some remarks on real and complex output feedback

# publications[].label
The recursive nature of cominuscule Schubert calculus

# publications[].label
Line problems in nonlinear computational geometry

# publications[].label
Real solutions to equations from geometry

# publications[].label
Generic properties of dispersion relations for discrete periodic operators

# publications[].label
Shifted Quasi-Symmetric Functions and the Hopf algebra of peak functions

# publications[].label
Real enumerative geometry and effective algebraic equivalence

# publications[].label
Theorems of Brion, Lawrence, and Varchenko on rational generating   functions for cones

# publications[].label
Convex Hulls of Orbits and Orientations of a Moving Protein Domain

# publications[].label
Higher convexity of coamoeba complements

# publications[].publisher
Universidade de Sao Paulo, Agencia USP de Gestao da Informacao Academica (AGUIA)

# publications[].label
Maximally inflected real rational curves

# publications[].label
Common transversals and tangents to two lines and two quadrics in P^3

# publications[].label
A sagbi basis for the quantum Grassmannian

# publications[].label
Noncommutative Pieri Operators on Posets

# publications[].label
A numerical toolkit for multiprojective varieties

# publications[].label
Irrational toric varieties and secondary polytopes

# publications[].label
General witness sets for numerical algebraic geometry

# publications[].label
Combinatorial Hopf algebras and generalized Dehn-Sommerville relations

# publications[].label
A Pieri-type formula for the K {K} -theory of a flag manifold

# publications[].label
Nontrivial linear projections of the Grassmannian Gr_3(C^6)

# publications[].label
A Pieri-type formula for isotropic flag manifolds

# publications[].label
Lower bounds for real solutions to sparse polynomial systems

# publications[].label
Real Algebraic Geometry for Geometric Constraints

# publications[].publisher
Centre pour la Communication Scientifique Directe (CCSD)

# publications[].label
Certifiable Numerical Computations in Schubert Calculus

# publications[].label
Experimentation and conjectures in the real Schubert calculus for flag  manifolds (extended abstract)

# publications[].label
Pieri's Formula Via Explicit Rational Equivalence

# publications[].label
Galois Groups of Composed Schubert Problems

# publications[].label
Two Murnaghan-Nakayama rules in Schubert calculus

# publications[].label
Structure of the Malvenuto-Reutenauer Hopf algebra of permutations  (Extended Abstract)

# publications[].label
A Littlewood-Richardson rule for Grassmannian Permutations

# publications[].label
Dualit de Gale pour des intersections compltes

# publications[].label
Schubert polynomials, the Bruhat order, and the geometry of flag  manifolds

# publications[].label
Generic properties of dispersion relations for discrete periodic  operators

# publications[].label
The Secant Conjecture in the Real Schubert Calculus

# publications[].label
A primal-dual formulation for certifiable computations in Schubert  calculus

# publications[].publisher
Association for Computing Machinery (ACM)

# publications[].label
PieriS Formula Via Explicit Rational Equivalence

# publications[].label
Critical points of discrete periodic operators

# publications[].label
A geometric proof of an equivariant Pieri rule for flag manifolds

# publications[].label
Cofree compositions of coalgebras (extended abstract)

# publications[].label
The number of transversals to line segments in R^3

# publications[].label
Structure of the Malvenuto-Reutenauer Hopf algebra of permutations

# publications[].label
Galois groups of Schubert problems of lines are at least alternating

# publications[].label
The Envelope of Lines Meeting a Fixed Line and Tangent to Two Spheres

# publications[].label
Real enumerative geometry and effective algebraic equivalence

# publications[].label
A Littlewood-Richardson rule for Grassmannian permutations

# publications[].label
An excursion from enumerative goemetry to solving systems of polynomial  equations with Macaulay 2

# publications[].label
Shifted quasi-symmetric functions and the Hopf algebra of peak functions

# publications[].label
Bounds on the Number of Real Solutions to Polynomial Equations

# publications[].label
Galois groups of Schubert problems of lines are at least alternating

# publications[].label
Algorithm 921: alphaCertified: Certifying Solutions to Polynomial Systems

# publications[].publisher
Association for Computing Machinery (ACM)

# publications[].label
Galois groups of Schubert problems via homotopy computation

# publications[].label
Computing monodromy via parallel homotopy continuation

# publications[].publisher
Association for Computing Machinery (ACM)

# publications[].label
Transversals to Line Segments in Three-Dimensional Space

# publications[].label
Some problems at the interface of approximation theory and algebraic   geometry

# publications[].label
Equivariant cohomology theories and the pattern map

# publications[].label
The envelope of lines meeting a fixed line that are tangent to two  spheres

# publications[].label
Experimentation and Conjectures in the Real Schubert Calculus for Flag Manifolds

# publications[].label
Skew Littlewood-Richardson rules from Hopf algebras

# publications[].label
Numerical Schubert calculus via the Littlewood-Richardson homotopy algorithm

# publications[].label
Frontiers of reality in Schubert calculus

# publications[].label
alphaCertified: certifying solutions to polynomial systems

# publications[].label
A Monoid for the Universal K-Bruhat Order

# publications[].label
Lines Tangent to Four Triangles in Three-Dimensional Space

# publications[].label
Combinatorial Hopf algebras and generalized Dehn-Sommerville relations

# publications[].label
Linear precision for toric surface patches

# publications[].label
Nontrivial linear projections on the Grassmannian Gr3(C6)

# publications[].label
Elementary transversality in the Schubert calculus in any characteristic

# publications[].label
Explicit Enumerative Geometry for the Real Grassmannian of Lines in   Projective Space

# publications[].label
Certification for Polynomial Systems via Square Subsystems

# publications[].label
New Hopf Structures on Binary Trees (Extended Abstract)

# publications[].label
Higher convexity of coamoeba complements

# publications[].label
Cohomological consequences of the pattern map

# publications[].label
Euclidean Distance Degree and Mixed Volume

# publications[].label
Experimentation and conjectures in the real Schubert calculus for flag  manifolds

# publications[].label
Experimentation in the Schubert Calculus

# publications[].label
Some geometrical aspects of control points for toric patches

# publications[].label
Common Transversals and Tangents to Two Lines and Two Quadrics in P

# publications[].label
Complex Static Skew-Symmetric Output Feedback Control

# publications[].publisher
Society for Industrial & Applied Mathematics (SIAM)

# publications[].label
General witness sets for numerical algebraic geometry

# publications[].publisher
Association for Computing Machinery (ACM)

# publications[].label
Some real and unreal enumerative geometry for flag manifolds

# publications[].label
A New Proof of Hilbert's Theorem on Ternary Quartics

# publications[].label
Cocommutative Hopf algebras of permutations and trees

# publications[].label
Bounds on the number of real solutions to polynomial equations

# publications[].label
The Monotone Secant Conjecture in the Real Schubert Calculus

# publications[].label
A Pieri-type formula for isotropic flag manifolds

# publications[].label
Experimentation at the frontiers of reality in Schubert calculus

# publications[].label
Classification of Schubert Galois Groups in Gr(4,9)

# publications[].label
Some Geometrical Aspects of Control Points for Toric Patches

# publications[].label
Schubert polynomials, the Bruhat order, and the geometry of flag manifolds

# publications[].label
The monotone secant conjecture in the real Schubert calculus

# publications[].label
Structure of the Malvenuto-Reutenauer Hopf algebra of permutations

# publications[].label
A Congruence Modulo Four for Real Schubert Calculus with Isotropic Flags

# publications[].label
Lower Bounds in Real Algebraic Geometry and Orientability of Real Toric Varieties

# publications[].label
Skew Littlewood-Richardson Rules from Hopf Algebras

# publications[].label
A congruence modulo four in real Schubert calculus with isotropic flags

# publications[].label
Quiver coefficients are Schubert structure constants

# publications[].label
Classification of Schubert Galois groups in Gr(4,9)

# publications[].label
Line problems in nonlinear computational geometry

# publications[].label
Discrete and Computational Geometry: Guest editors' foreword

# publications[].label
Cremona convexity, frame convexity and a theorem of Santal

# publications[].label
Identities of Structure Constants for Schubert polynomials and Orders on  S_n

# publications[].label
Software for the Gale transform of fewnomial systems and a Descartes  rule for fewnomials

# publications[].label
Linear Precision for Toric Surface Patches

# publications[].label
A Primal-Dual Formulation for Certifiable Computations in Schubert Calculus

# publications[].label
A Monoid for the Grassmannian Bruhat Order

# publications[].label
General isotropic flags are general (for Grassmannian Schubert calculus)

# publications[].label
Line tangents to four triangles in three-dimensional space

# publications[].label
A lifted square formulation for certifiable Schubert calculus

# publications[].label
Software for the Gale transform of fewnomial systems and a Descartes rule for fewnomials

# publications[].label
On the sharpness of fewnomial bounds and the number of components of fewnomial hypersurfaces

# publications[].label
Non-commutative Pieri operators on posets

# publications[].label
Quiver coefficients are Schubert structure constants

# publications[].publisher
Centre pour la Communication Scientifique Directe (CCSD)

# publications[].label
Experimentation at the Frontiers of Reality in Schubert Calculus

# publications[].label
An inequality of Kostka numbers and Galois groups of Schubert problems

# publications[].label
The recursive nature of cominuscule Schubert calculus

# publications[].label
THE EQUIVARIANT CHOW RINGS OF QUOT SCHEMES

# researcherOn[].label
Texas Algebraic Geometry Symposium: TAGS 2015

# researcherOn[].awardedBy.label
National Science Foundation - (Arlington, Virginia, United States)

# researcherOn[].label
Combinatorial and Real Algebraic Geometry

# researcherOn[].awardedBy.label
National Science Foundation - (Arlington, Virginia, United States)

# researcherOn[].label
Texas Algebraic Geometry Symposium 2018-2020

# researcherOn[].awardedBy.label
National Science Foundation - (Arlington, Virginia, United States)

# streetAddress
Texas A&M University; Mathematics; 3368 TAMU

# etdChairOf[].label
Ruffo, James Vincent (2009-05). A straightening law for the Drinfel'd Lagrangian Grassmannian. Doctoral Dissertation.

# etdChairOf[].label
Buczynska, Weronika J. (2010-10). Phylogenetic Toric Varieties on Graphs. Doctoral Dissertation.

# etdChairOf[].label
Martin Del Campo Sanchez, Abraham (2012-10). Galois Groups of Schubert Problems. Doctoral Dissertation.

# etdChairOf[].label
Hein, Nickolas Jason (2013-06). Reality and Computation in Schubert Calculus. Doctoral Dissertation.

# etdChairOf[].label
Williams, Robert Lee (2017-08). Restrictions on Galois Groups of Schubert Problems. Doctoral Dissertation.

# etdChairOf[].label
Irving, Corey 1977- (2012-11). Wachspress Varieties. Doctoral Dissertation.

# etdChairOf[].label
Pir, Ata Firat (2018-08). Irrational Toric Varieties. Doctoral Dissertation.