Home /  Summer Graduate School /  Schedule

Schedule, Notes/Handouts & Videos

Geometry and Representation Theory of Tensors for Computer Science, Statistics, and other areas July 07, 2008 - July 18, 2008

Show All Collapse
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 David Garcia Puente (Colorado College)
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 (University of Chicago; Flatiron Institute)
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 (Auburn University)
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 (Jagiellonian 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 (Jagiellonian 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
Back to Summer Graduate School