**Office: Blocker 601H**

Office hours: TBA

**E-mail: **jml@math.tamu.edu

TAMU math dept homepage

Supported by NSF grant DMS-1405368

Teaching Fall 2016:

Geometry and complexity theory Math 689

Meeting: MWF 10:20 - 11:10 Bloc 160

This will cover central problems in theoretical computer science from

a geometric perspective. Topics in computer science: the complexity of
matrix multiplication,

both upper and lower bounds, Valiant's conjecture on permanent v.

determinant and variants, the problem of explicitness: how to find

hay in a haystack. Geometry that will be covered: rank

and border rank of tensors, basic representation theory and algebraic

geometry.

I will follow these notes, which will

be rewritten in more polished form over the summer.

Background required: a strong background in linear algebra, e.g. Math 323

and co-organized (with P. Burgisser, K. Mulmuley and B. Sturmfels) of the semester long program Algorithms and Complexity in Algebraic Geometry During the semester I gave a course on Geometry and Complexity Theory, and have notes for the course.

- Honors combined ScB and MS, Mathematics, Brown University, 1986
(MS thesis director, K. Nomizu)

- PhD., Mathematics, Duke University, 1990 (director R.
Bryant)

- Habilitation., Mathematics, Universite Toulouse, 1997
(directeur, C. Simpson)

I organize and co-organize:My CV (last updated 1/16)

Geometry seminar, meeting Mondays 3-4 Blocker 220 and Fridays 4-5pm Blocker 117.

Working seminar for post-docs and graduate students meeting Thursdays, Blocker 624,

3-4pm

Everyone is welcome to the seminars, graduate students

are particularly encouraged to attend.

TAMU seminar calendar

My travel plans

I am on the editorial board of Foundation of Computational Mathematics, Differential Geometry and its Applications,

and Linear Algebra and its Applications.

Survey articles

PAPERS/PREPRINTS in past 5 years

An explicit description of the irreducible components of the set of matrix pencils with bounded normal rank (with F. De Teran and F. Dopico) On the geometry of border rank algorithms for matrix multiplication and other tensors with symmetry (with M. Michalek) On the geometry of border rank algorithms for n x 2 by 2 x 2 matrix multiplication (with N. Ryder, to appear in Exper. Math.)

Permanent vs determinant: an exponential lower bound assuming

symmetry and a potential path towards Valiant's conjecture (with N. Ressayre)On minimal free resolutions and the method of shifted partial derivatives in complexity theory (with K. Efremenko, H. Schenck and J. Weyman) Abelian Tensors (with Mateusz Michalek, to appear in JMPA) Connections between conjectures of Alon-Tarsi, Hadamard-Howe, and integrals over the special unitary group (with S. Kumar, Discrete Math. 2015) Complexity of linear circuits and geometry (with F. Gesmundo, G. Hauenstein and C. Ikenmeyer, FOCM 2016) Geometric Complexity Theory: an introduction for geometers (Ann. U. Ferrara 2015)

Computer aided methods for lower bounds on the border rank (with G. Hauenstein and C. Ikenmeyer, Exper. Math. 2013)

Explicit tensors of border rank at least 2n-2 (J. Pure. Appl. Alg. 2015)

New lower bounds for the rank of matrix multiplication (SICOMP, 2014)

Padded polynomials, their cousins, and geometric complexity theory (with H. Kadish, Communications in Algebra 2014) New lower bounds for the border rank of matrix multiplication (with G. Ottaviani, Theory of Computing 2015) On the third secant variety (with J. Buczynski, JAC 2014) On the geometry of Tensor Network States (with Y. Qi and K. Ye, QIC, 2012) Equations for secant varieties of Veronese and other varietites (with G. Ottaviani, Annali di Matematica Pura e Applicata, 2013) Fubini-Griffiths-Harris rigidity of homogeneous varieties (with C. Robles, IMRN 2013) Determinental equations for secant varieties and the Eisenbud-Koh-Stillman conjecture (with J. Buczynski and A. Ginesky, J.London Math. Soc. 2013)

Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program (with L. Manivel and N. Ressayre, CMH 2013) P versus NP and geometry ( J. Symb. Comp.2010, MEGA 2009 special issue)

Ranks of tensors and a generalization of secant varieties (with J. Buczynski, LAA special issue on tensors 2013) An overview of mathematical issues arising in the Geometric complexity theory approach to VP \neq VNP (with P. Buergisser, L. Manivel and J. Weyman, SIAM J. Comp. 2011) Holographic algorithms without matchgates (with Jason Morton and Serguei Norine, LAA special issue on tensors 2013) Fubini-Griffiths-Harris rigidity and Lie algebra cohomology (with C. Robles, (Asian Math. J. 2013)

all articles (since 2006)

Books:An introduction to Geometric Complexity Theory (Newsletter of the EMS 3/16)

Exterior differential systems, Lie algebra cohomology, and the rigidity of homogeneous varieties (2008) Differential geometry of submanifolds of projective space (2006) Exterior differential systems and billiards (2006) Representation theory and projective geometry (with L. Manivel), 2004

Tensors: Geometry and Applications.

AMS GSM 128. Click here to see table of contents and preface, and to order.

Click here for corrections and additionsCartan For Beginners: Differential geometry via moving frames and exterior differential systems (with T. Ivey)

AMS GSM 61 . To see the table of contents, preface, and selected pages click here

To order the book from the AMS, click here.

to see corrections to text, click here

Slides of recent talks:

Complexity theory and geometry (Berlin Mathematical School colloquium 2/15)

Perm v. det: an exponential lower bound assuming symmetry (Innovations in Theoretical Computer Science 1/16)

math reviews of all published papersLuke Oeding, PhD May 2009, Defining equations of the varietyof principal minors solved a conjecture of Holtz and Sturmfels.

PhD students:

Current students: Kashif Bari, Fulvio Gesmundo, and Yao Wang

Graduated students:

Cameron Farnsworth, August 2016, THE POLYNOMIAL WARING PROBLEM AND THE DETERMINANT

Yonghui Guan, August 2016, EQUATIONS FOR CHOW VARIETIES, THEIR SECANT VARIETIES AND

OTHER VARIETIES ARISING IN COMPLEXITY THEORY

Curtis Porter, August 2016 THE LOCAL EQUIVALENCE PROBLEM FOR 7-DIMENSIONAL, 2-NONDEGENERATE CR

MANIFOLDS WHOSE CUBIC FORM IS OF CONFORMAL UNITARY TYPE

Yang Qi, PhD August 2013 Geometry of Feasible Spaces of Tensors

determined defining equations for the third secant variety of a triple Segre product and closedness of tensor network states

Ke Ye, PhD August 2012, IMMANANTS, TENSOR NETWORK STATES AND THE GEOMETRIC

COMPLEXITY THEORY PROGRAM determined symmetry groups of immanents, and closedness of tensor network states

Ming Yang, PhD Sept. 2012, On partial and generic uniqueness of block term tensor decompositions in signal processing, solving questions originating in signal processing.

Frederic Holweck, PhD fall 04, Dual varieties, simple singularities and simple Lie algebras

Here is a summary of his results in English

E. Allaud, PhD spring 03, thesis: Nongenericity of variations of Hodge structure for hypersurfaces of high degree,

published in Duke. Math. J.

New to the area? Here are some Brazos Valley links

Now available:

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