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. A PDF version of the schedule is also available.
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. |
- Jason Morton, Toric varieties, toric ideals, moment map, exponential families. July 10,2008, 09:00 AM to 10:00 AM
- Joseph Landsberg, § 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. July 10,2008, 10:15 AM to 11:15 AM
- Vin de Silva, Constructibility of the set of tensors of a given rank July 10,2008, 11:30 AM to 12:30 PM
- Luis Garcia-Puente, Phylogenetic algebraic geometry July 10,2008, 02:00 PM to 03:00 PM
- Jason Morton, finish Ch 4 (Littlewood-Richardson rule and other handy formulas, more decompositions of spaces of tensors) July 11,2008, 09:00 AM to 10:00 AM
- Joseph Landsberg, § 5.1-5.3 Equations for secant varieties I: special Segre varieties, subspace varieties, flattenings July 11,2008, 10:15 AM to 11:15 AM
- Vin de Silva, Hyperdeterminants and optimal approximability July 11,2008, 11:30 AM to 12:30 PM
- David Gross, What are graph states? July 11,2008, 02:00 PM to 03:00 PM
- Jason Morton, § 5.4, 5.5 Equations II: inheritance, and prolongation July 14,2008, 09:00 AM to 10:00 AM
- Joseph Landsberg, § 5.6 Equations III: Strassen's equations and variants July 14,2008, 10:15 AM to 11:15 AM
- Vin de Silva, Uniqueness of tensor decomposition, direct sum conjecture July 14,2008, 11:30 AM to 12:30 PM
- Risi Kondor, Non-commutative harmonic analysis in machine learning July 14,2008, 02:00 PM to 03:00 PM
- Joseph Landsberg, § 6.1,6.2,6.6,6.7 The Alexander-Hirshowitz theorem and dimensions of secant varieties of Segre varieties July 15,2008, 09:00 AM to 10:00 AM
- Jason Morton, 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. July 15,2008, 10:15 AM to 11:15 AM
- Joseph Landsberg, Ch 8: Rank vs border rank of tensors and symmetric tensors July 15,2008, 11:30 AM to 12:30 PM
- Luke Oeding, The variety of principal minors of symmetric matrices July 15,2008, 02:00 PM to 03:00 PM
- Pierre Comon, (a) general statements on linear mixtures of random variables, (b)cumulants, (c) tensors July 16,2008, 09:00 AM to 10:00 AM
- Jerzy Weyman, What do the words "ACM", "Gorenstein", and " rational singularites" mean and why are these properties useful? July 16,2008, 10:15 AM to 11:15 AM
- Lek-Heng Lim, Nonnegative hypermatrices, symmetric tensors July 16,2008, 11:30 AM to 12:30 PM
- Pierre Comon, (d) the invertible case: Independent Component Analysis - optimization criteria and some numerical algorithms July 17,2008, 09:00 AM to 10:00 AM
- Jerzy Weyman, Introduction to the study of G-varieties via desingularizations by homogeneous vector bundles July 17,2008, 10:15 AM to 11:15 AM
- Joseph Landsberg, Ch 9: Spaces of tensors admitting normal forms July 17,2008, 11:30 AM to 12:30 PM
- Pierre Comon, (e) the UDM case: some selected statistical blind identification approaches, all involving tensors. Local identifiability and numerical algorithms (including BIOME and FOOBI). July 18,2008, 09:00 AM to 10:00 AM
- Giorgio Ottaviani, Induction for the rank of tensors July 18,2008, 10:15 AM to 11:15 AM
- Giorgio Ottaviani, The Alexander-Hirschowitz theorem July 18,2008, 02:00 PM to 03:00 PM
- Joseph Landsberg, Complexity of matrix multiplication, an overview of Ch. 2 including tensors, rank of tensors, and wiring diagrams. July 7,2008, 09:15 AM to 10:15 AM
- Jason Morton, 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 July 7,2008, 10:15 AM to 11:15 AM
- Lek-Heng Lim, Tensor approximations July 7,2008, 11:30 AM to 12:30 PM
- Jason Morton, 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 July 8,2008, 09:00 AM to 10:00 AM
- Joseph Landsberg, § 3.3,4,5,6 Tangent spaces to varieties, joins, cones, secant varieties, their dimension, Terracini's lemma. July 8,2008, 10:15 AM to 11:15 AM
- Vin de Silva, Notions of tensor ranks: rank, border rank, multilinear rank, nonnegative rank July 8,2008, 11:30 AM to 12:30 PM
- David Gross, What is quantum information theory? July 8,2008, 02:00 PM to 03:00 PM
- Joseph Landsberg, 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. July 9,2008, 09:00 AM to 10:00 AM
- Jason Morton, § 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. July 9,2008, 10:15 AM to 11:15 AM
- Lek-Heng Lim, Conditioning, computations, applications July 9,2008, 11:30 AM to 12:30 PM
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