# overview
My areas of research include:

1) Applications of algebraic geometry, differential geometry and representation theory to complexity theory,

2) Geometry of tensors and applications,

3) Exterior differential systems,

4) Geometry of projective varieties.

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
4903 Numerical And Computational Mathematics

# keywords[]
Classification Of Points Of Border Rank Three

# educationAndTraining[].organization.label
Brown University - (Providence, Rhode Island, United States)

# educationAndTraining[].organization.label
Duke University - (Durham, North Carolina, United States)

# educationAndTraining[].organization.label
Brown University - (Providence, Rhode Island, United States)

# publications[].label
On the geometry of border rank algorithms for matrix multiplication and other tensors with symmetry

# publications[].label
The complexity of matrix multiplication: developments since 2014.   Extended abstract of 2018 Oberwolfach Complexity meeting plenary lecture

# publications[].label
On minimal free resolutions of sub-permanents and other ideals arising in complexity theory

# publications[].label
Permanent v. determinant: An exponential lower bound assuming symmetry and a potential path towards Valiant's conjecture

# publications[].label
On a conjecture of Kontsevich and variants of Castelnuovo's lemma

# publications[].label
Rank and border rank of Kronecker powers of tensors and Strassen's laser   method

# publications[].label
Polynomials and the exponent of matrix multiplication

# publications[].label
The Geometry of Rank Decompositions of Matrix Multiplication I: 2  2 Matrices.

# publications[].label
Construction and classification of complex simple Lie algebras via projective geometry

# publications[].label
On the Identifiability and Stable Recovery of Deep/Multi-Layer Structured Matrix Factorization

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

# publications[].label
On the ideals of secant varieties of Segre varieties

# publications[].label
Towards finding hay in a haystack: explicit tensors of border rank greater than $2.02m$ in $C^motimes C^motimes C^m$

# publications[].label
ON 2ND FUNDAMENTAL FORMS OF PROJECTIVE VARIETIES

# publications[].label
An introduction to geometric complexity theory

# publications[].label
The complexity of matrix multiplication: developments since 2014. Extended abstract of 2018 Oberwolfach Complexity meeting plenary lecture

# publications[].label
Ranks of tensors and a generalization of secant varieties

# publications[].label
MINIMAL SUBMANIFOLDS DEFINED BY 1ST-ORDER SYSTEMS OF PDE

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Explicit Polynomial Sequences with Maximal Spaces of Partial Derivatives and a Question of K. Mulmuley.

# publications[].label
Secant varieties and the complexity of matrix multiplication

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Lines and osculating lines of hypersurfaces

# publications[].label
Complexity of Linear Circuits and Geometry

# publications[].label
On tangential varieties of rational homogeneous varieties

# publications[].label
Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture

# publications[].label
On the Ranks and Border Ranks of Symmetric Tensors

# publications[].label
Towards finding hay in a haystack: explicit tensors of border rank   greater than $2.02m$ in $C^motimes C^motimes C^m$

# publications[].label
On symmetric degeneracy loci, spaces of symmetric matrices of constant rank and dual varieties

# publications[].label
The method of shifted partial derivatives cannot separate the permanent from the determinant

# publications[].label
A 2n2 - log2(n)-1 Lower Bound for the Border Rank of Matrix Multiplication

# publications[].label
Matrix product states and the quantum max-flow/min-cut conjectures

# publications[].label
Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

# publications[].publisher
European Mathematical Society - EMS - Publishing House GmbH

# publications[].label
On the infinitesimal rigidity of homogeneous varieties

# publications[].label
GEOMETRY OF BACKFLOW TRANSFORMATION ANSATZE FOR QUANTUM MANY-BODY FERMIONIC WAVEFUNCTIONS

# publications[].label
Connections between conjectures of Alon-Tarsi, Hadamard-Howe, and integrals over the special unitary group

# publications[].label
On the Geometry of Matrix Multiplication

# publications[].label
Generalizations of Strassen's equations for secant varieties of Segre varieties

# publications[].label
Towards a Geometric Approach to Strassen's Asymptotic Rank Conjecture

# publications[].label
The sextonions and E 7 1/2

# publications[].label
On the complexity of the permanent in various computational models

# publications[].label
On the ideals and singularities of secant varieties of Segre varieties

# publications[].label
The Geometry of Rank Decompositions of Matrix Multiplication I: 2 x 2 Matrices

# publications[].label
Geometry of backflow transformation ansatz for quantum many-body fermionic wavefunctions

# publications[].label
THE METHOD OF SHIFTED PARTIAL DERIVATIVES CANNOT SEPARATE THE PERMANENT FROM THE DETERMINANT

# publications[].label
Triality, exceptional Lie algebras and Deligne dimension formulas

# publications[].label
Connections between conjectures of Alon-Tarsi, Hadamard-Howe, and integrals over the special unitary group

# publications[].label
Bad and good news for Strassen's laser method: Border rank of the 3x3 permanent and strict submultiplicativity

# publications[].label
AN OVERVIEW OF MATHEMATICAL ISSUES ARISING IN THE GEOMETRIC COMPLEXITY THEORY APPROACH TO VP  VNP

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

# publications[].label
Minimal submanifolds of E 2 n + 1 E^ {2n+1} arising from degenerate SO ( 3 ) 	extrm {SO} (3) orbits on the Grassmannian

# publications[].label
Geometric complexity theory: an introduction for geometers

# publications[].label
An explicit description of the irreducible components of the set of matrix pencils with bounded normal rank

# publications[].label
PADDED POLYNOMIALS, THEIR COUSINS, AND GEOMETRIC COMPLEXITY THEORY

# publications[].label
Exterior differential systems, lie algebra cohomology, and the rigidity of homogenous varieties

# publications[].label
New lower bounds for matrix multiplication and the 3x3 determinant

# publications[].label
Algebraic Geometry and Representation theory in the study of matrix   multiplication complexity and other problems in theoretical computer science

# publications[].label
Permanent v. determinant: an exponential lower bound assumingsymmetry and a potential path towards Valiant's conjecture

# publications[].label
Nontriviality of equations and explicit tensors in CmCmCm of border rank at least 2m2

# publications[].label
On the Geometry of Border Rank Algorithms for n  2 by 2  2 Matrix Multiplication.

# publications[].label
On Linear spaces of of matrices bounded rank

# publications[].label
Bad and Good News for Strassen's Laser Method: Border Rank of Perm3 and Strict Submultiplicativity

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ON THE GEOMETRY OF TENSOR NETWORK STATES

# publications[].label
On the complexity of the permanent in various computational models

# publications[].label
Griffiths-Harris rigidity of compact Hermitian symmetric spaces

# publications[].label
New lower bounds for matrix multiplication and the 3x3 determinant

# publications[].label
On degenerate secant and tangential varieties and local differential geometry

# publications[].label
Geometry and the complexity of matrix multiplication

# publications[].label
A universal dimension formula for complex simple Lie algebras

# publications[].label
Algebraic geometry and representation theory in the study of matrix multiplication complexity and other problems in theoretical computer science

# publications[].label
On secant varieties of compact Hermitian symmetric spaces

# publications[].label
The geometry of rank decompositions of matrix multiplication I: 2x2 matrices

# publications[].label
New lower bounds for matrix multiplication and $operatorname {det}_3$

# publications[].label
The border rank of the multiplication of 2x2 matrices is seven

# publications[].label
Kronecker Powers of Tensors and Strassen's Laser Method.

# publications[].publisher
Schloss Dagstuhl - Leibniz-Zentrum fr Informatik

# publications[].label
RANK AND BORDER RANK OF KRONECKER POWERS OF TENSORS AND STRASSEN'S LASER METHOD

# publications[].label
Equations for secant varieties of Veronese and other varieties

# publications[].label
On minimal free resolutions of sub-permanents and other ideals arising in complexity theory

# publications[].label
The projective geometry of Freudenthal's magic square

# publications[].label
On a conjecture of Kontsevich and variants of Castelnuovo's lemma (vol 115, pg 205, 1999)

# publications[].label
Permanent V. Determinant: An Exponential Lower Bound Assuming Symmetry

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

# publications[].label
Equations for Lower Bounds on Border Rank

# publications[].label
P versus NP and geometry

# publications[].label
The geometry of rank decompositions of matrix multiplication II: 3 x 3 matrices

# publications[].label
Equivariant spaces of matrices of constant rank

# publications[].label
On the Debarre-de Jong and Beheshti-Starr Conjectures on Hypersurfaces with Too Many Lines

# publications[].label
On space-time coding in the presence of spatio-temporal correlation

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

# publications[].label
On the geometry of border rank algorithms for n x 2 by 2 x 2 matrix multiplication

# publications[].label
Algebraic Geometry and Representation theory in the study of matrix multiplication complexity and other problems in theoretical computer science

# publications[].label
On the projective geometry of rational homogeneous varieties

# publications[].publisher
Research Institute for Mathematical Sciences, Kyoto University

# publications[].label
On the Geometry of Border Rank Algorithms for n x 2 by 2 x 2 Matrix Multiplication

# publications[].label
New lower bounds for the border rank of matrix multiplication

# publications[].label
Is a linear space contained in a submanifold? On the number of derivatives needed to tell

# publications[].label
On isometric and minimal isometric embeddings

# publications[].label
Cartan for Beginners Differential Geometry Via Moving Frames and Exterior Differential Systems

# publications[].label
Rank and border rank of Kronecker powers of tensors and Strassen's laser method

# publications[].label
FUBINI-GRIFFITHS-HARRIS RIGIDITY AND LIE ALGEBRA COHOMOLOGY

# publications[].label
Bad and good news for Strassen's laser method: Border rank of the 3x3   permanent and strict submultiplicativity

# researcherOn[].label
Texas Geometry and Topology Conference (TGTC)

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

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

# researcherOn[].label
AF: Small: Geometry and Complexity Theory

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

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

# researcherOn[].label
Texas Geometry and Topology Conference (TGTC)

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

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