# Mathematical Sciences Research Institute

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Geometry and Representation Theory of Tensors for Computer Science, Statistics, and other areas July 07, 2008 - July 18, 2008
Location: Baker Board Room
Organizers J.M. Landsberg (Texas A&M), Lek-Heng Lim (UC Berkeley) and Jason Morton (UC Berkeley)
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Description Questions in computational complexity, statistical learning theory, signal processing, scientific data analysis, and other areas have recently been recast in terms of geometry and representation theory. Among them are: the complexity of matrix multiplication, Valiant's approach to P=NP, measures of entanglement in quantum information theory, matchgates in computer science, graphical models in statistical learning theory, the study of phylogenetic invariants, independent component analysis and other multilinear data analytic techniques in bioinformatics, signal processing, and spectroscopy. The geometric perspective allows one to understand the questions in a more general mathematical context. It explains known results in terms of standard theorems in geometry and helps to advance the relevant areas. The goals of this workshop are twofold: To introduce the relevant geometry and representation theory and to present and discuss open questions from the relevant areas that we believe could be resolved by workshop participants. We will introduce the problems that lead to varieties in spaces of tensors and cover the basic geometry and representation theory needed to study them. By the middle of the second week we expect to begin projects working on open questions. The preliminary schedule for the workshop is available at http://docs.google.com/View?docid=dcnmtggh_29px7qd3cw. For more advanced participants, there will be a follow-up research workshop at the American Institute of Mathematics (AIM) the week after the graduate workshop. Information on the research workshop is available at http://www.aimath.org/ARCC/workshops/repnsoftensors.html. If you are a student who has not been nominated for this workshop by one of our Academic Sponsors, please e-mail coord@msri.org for information about registration.

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Schedule
Jul 07, 2008
Monday
 09:15 AM - 10:15 AM Complexity of matrix multiplication, an overview of Ch. 2 including tensors, rank of tensors, and wiring diagrams. Joseph Landsberg (Texas A&M International University) 10:15 AM - 11:15 AM Algebraic varieties § 3.1, 3.2. Basic definitions from algebraic geometry: projective space, variety, ideal, Zariski topology. Segre, Veronese, and other examples of varieties. Graphical models and motivating examples in statistics and information t Jason Morton (Pennsylvania State University) 11:30 AM - 12:30 PM Tensor approximations Lek-Heng Lim (University of Chicago)
Jul 08, 2008
Tuesday
 09:00 AM - 10:00 AM Finish Ch. 2: skew-symmetric tensors, equations for rank at most r linear mappings, border rank, decomposing V^{\ot 3}., G-modules, isotypic components. § 4.1,2 Representations, Schur's Lemma, G-modules and decomposing spaces of tensors Jason Morton (Pennsylvania State University) 10:15 AM - 11:15 AM § 3.3,4,5,6 Tangent spaces to varieties, joins, cones, secant varieties, their dimension, Terracini's lemma. Joseph Landsberg (Texas A&M International University) 11:30 AM - 12:30 PM Notions of tensor ranks: rank, border rank, multilinear rank, nonnegative rank Vin de Silva 02:00 PM - 03:00 PM What is quantum information theory? David Gross
Jul 09, 2008
Wednesday
 09:00 AM - 10:00 AM Finish Chap 3 - Terracini's lemma cont'd and applications to computing the dimension of secant varieties. The geometric definition of border rank, projective second fundamental form. Joseph Landsberg (Texas A&M International University) 10:15 AM - 11:15 AM § 4.3,4,5 - Representations of the symmetric group, Young diagrams, Young symmetrizers and wiring diagrams. Using these tools to decompose V^{\otimes d} as a GL(V) module. Schur-Weyl Duality. Jason Morton (Pennsylvania State University) 11:30 AM - 12:30 PM Conditioning, computations, applications Lek-Heng Lim (University of Chicago)
Jul 10, 2008
Thursday
 09:00 AM - 10:00 AM Toric varieties, toric ideals, moment map, exponential families. Jason Morton (Pennsylvania State University) 10:15 AM - 11:15 AM § 4.6,7,8 Highest weight vectors, bases of highest weight space. Ideals of Segre, Veronese varieties and homogeneous varieties in general, decomposing S^d(A_1\otimes \cdots \otimes A_n), characters. Joseph Landsberg (Texas A&M International University) 11:30 AM - 12:30 PM Constructibility of the set of tensors of a given rank Vin de Silva 02:00 PM - 03:00 PM Phylogenetic algebraic geometry Luis Garcia-Puente
Jul 11, 2008
Friday
 09:00 AM - 10:00 AM finish Ch 4 (Littlewood-Richardson rule and other handy formulas, more decompositions of spaces of tensors) Jason Morton (Pennsylvania State University) 10:15 AM - 11:15 AM § 5.1-5.3 Equations for secant varieties I: special Segre varieties, subspace varieties, flattenings Joseph Landsberg (Texas A&M International University) 11:30 AM - 12:30 PM Hyperdeterminants and optimal approximability Vin de Silva 02:00 PM - 03:00 PM What are graph states? David Gross
Jul 14, 2008
Monday
 09:00 AM - 10:00 AM § 5.4, 5.5 Equations II: inheritance, and prolongation Jason Morton (Pennsylvania State University) 10:15 AM - 11:15 AM § 5.6 Equations III: Strassen's equations and variants Joseph Landsberg (Texas A&M International University) 11:30 AM - 12:30 PM Uniqueness of tensor decomposition, direct sum conjecture Vin de Silva 02:00 PM - 03:00 PM Non-commutative harmonic analysis in machine learning Risi Kondor
Jul 15, 2008
Tuesday
 09:00 AM - 10:00 AM § 6.1,6.2,6.6,6.7 The Alexander-Hirshowitz theorem and dimensions of secant varieties of Segre varieties Joseph Landsberg (Texas A&M International University) 10:15 AM - 11:15 AM Ch 7. An algorithm for explicitly writing down polynomials in a given submodule of the space of polynomials. Further combinatorics of Young tableaux. Working with tensors in factored vs. expanded form. Jason Morton (Pennsylvania State University) 11:30 AM - 12:30 PM Ch 8: Rank vs border rank of tensors and symmetric tensors Joseph Landsberg (Texas A&M International University) 02:00 PM - 03:00 PM The variety of principal minors of symmetric matrices Luke Oeding (University of California, Berkeley)
Jul 16, 2008
Wednesday
 09:00 AM - 10:00 AM (a) general statements on linear mixtures of random variables, (b)cumulants, (c) tensors Pierre Comon 10:15 AM - 11:15 AM What do the words "ACM", "Gorenstein", and " rational singularites" mean and why are these properties useful? Jerzy Weyman (Northeastern University) 11:30 AM - 12:30 PM Nonnegative hypermatrices, symmetric tensors Lek-Heng Lim (University of Chicago)
Jul 17, 2008
Thursday
 09:00 AM - 10:00 AM (d) the invertible case: Independent Component Analysis - optimization criteria and some numerical algorithms Pierre Comon 10:15 AM - 11:15 AM Introduction to the study of G-varieties via desingularizations by homogeneous vector bundles Jerzy Weyman (Northeastern University) 11:30 AM - 12:30 PM Ch 9: Spaces of tensors admitting normal forms Joseph Landsberg (Texas A&M International University)
Jul 18, 2008
Friday
 09:00 AM - 10:00 AM (e) the UDM case: some selected statistical blind identification approaches, all involving tensors. Local identifiability and numerical algorithms (including BIOME and FOOBI). Pierre Comon 10:15 AM - 11:15 AM Induction for the rank of tensors Giorgio Ottaviani 11:30 AM - 12:30 PM Student Lecture 02:00 PM - 03:00 PM The Alexander-Hirschowitz theorem Giorgio Ottaviani