
Public Lecture:
Synopsis of the 5 lectures, in 10 min each

1.5 hr (Sat)
XXXXNeed to collect some pictures for the motivation
1.5 hr Sunday
XXXXNeed to decide what to say after the pictures

=====================
Lecture (1) and (2)

Motivation
-----------

CAD part and assembly design, 
  Teaching Geometry and automated theorem proving, 
  Robotics, 
  Graphics Animation

References
www.solidworks.com
www.dcubed.com
www.cise.ufl.edu/~sitharam - FRONTIER opensource software

Formalization
What are geometric constraint problems?
Roughly what is needed: solution space description, finding, charac?

References
--My surveys
www.cise.ufl.edu/~sitharam/drone.pdf
www.cise.ufl.edu/~sitharam/dimacs.pdf


--Class lecture notes (taken by students)
www.cise.ufl.edu/~sitharam/COURSES/GC/lec1.pdf
www.cise.ufl.edu/~sitharam/COURSES/GC/lec2.pdf
www.cise.ufl.edu/~sitharam/COURSES/GC/lec3.pdf
www.cise.ufl.edu/~sitharam/COURSES/GC/lec4.pdf
www.cise.ufl.edu/~sitharam/COURSES/GC/lec5.pdf
www.cise.ufl.edu/~sitharam/COURSES/GC/lec6.pdf

Easy-to-state formal Questions:      
-----------------------------
XXXspecific example questions for each- tie some of them later to the 
   ``what  is known'' part below

*optimal decomposition/ construction plan
         -given construction method
         -choosing construction method

*combinatorial detection of generic rigidity, unique realizability, 
                            algebraic dependence


*describing, sampling - points and  paths in solution space
  Collision avoiding paths from one realization to another

*inverse problem: designing low complexity constraint system for desired result

Mathematical directions/conjectures/known results:
---------------------------------------------

*Combinatorics: graphs, matroids and oriented matroids

(i)
2D Quadratic solvability - Owen's conjecture
http://www.maths.lth.se/matstat/ecmi/modw/reportgroup4/reportgroup4.pdf
what is known: characterization of ruler and compass constructibility

(ii)
Minimum dense subgraph problem
what is known: FA
http://www.cise.ufl.edu/~sitharam/drtwo.pdf

(iii)
Combinatorial characterization of rigidity
what is known:
Laman's theorem
Module rigidity
Seam graphs

http://www.math.cornell.edu/~connelly
http://www.math.cornell.edu/~connelly/BasicsI.BasicsII.pdf
http://biophysics.asu.edu/banf_files/jackson/banffprobs.pdf
http://www.cise.ufl.edu/~sitharam/module.pdf
http://www.cise.ufl.edu/~sitharam/overlap-new.pdf

(iv)
Global rigidity, Unique realizability
what is known: Hendrickson-Jackson-Jordan's theorem
http://www.math.cornell.edu/~connelly/global-6.pdf


*Algebraic geometry: varieties (Grassmanian, Cayley-Menger), 
                      semi-algebraic sets, (combinatorial)invariant theory

(i)
"Roadmapping"  varieties corresponding to classes of geometric 
                                             constraint systems
                      (finding an efficient projection/representation, then
                       characterizing connected components, 
                       Betti numbers etc.)
Why is this important: solution picking (chirality, variety 
                                         with grassmanian stratification), 
                       sampling 
                       singularities
                       motion

what is known: Heping's theorem
                              
what is known: Cayley-Menger conditions and properties (low-rank etc)
               of distance matrices
http://www.maths.lth.se/matstat/ecmi/modw/reportgroup4/reportgroup4.pdf
web.mit.edu/tfhavel/www/Public/dg-review.ps.gz
                and Cayley-Menger varieties
http://arxiv.org/PS_cache/math/pdf/0207/0207110.pdf


what is known: When panel and hinge structures collapse
http://maven.smith.edu/~streinu/Papers/megaProc.pdf

what is known: Carpenter's rule  
http://www.ams.org/featurecolumn/archive/links1.html
                             

(ii)
Is ``generic'' classification of all (isolated) solutions of a wellconstrained 
                     system any easier than
                     fully roadmapping the big variety corresponding to the 
                     given set of constraints with variable parameters? 

(iii)
Finding generators for a "hidden" ring of polynomial invariants
        (finding corresponding group of transformations)

Why is this important: choosing optimal solution method
Very little is known



*Geometric groups, representations and algebras

(i)
"Roadmapping" a variety corresponding to a classes of geometric
                                           constraint systems
                      (analyzing local(tangentspace) group structure) 

Why is this important: completely characterizing "motions," points and paths 
                       used a lot in robotics, elsewhere
Look at the first two books by Selig and Bayro-Corrochano et al
http://books.google.com/books?ie=UTF-8&q=geometric++groups++and++robotics&btnG=Search+all+books

(ii)
Detecting incidence geometry dependences
using Grassman-Cayley algebra
http://www.danroozemond.nl/site/archief/ProvingCinderella.pdf
http://www.math.ufl.edu/~white/antw2.ps
http://www.math.ufl.edu/~white/hbz.ps
http://www.math.ufl.edu/~white/tutorfig.ps


* Algorithms (algebraic-numeric) for solution of systems of polynomial 
  equations and inequalities 

---------------------
Lecture (3) (i)
============
--Sensor/Wireless network Localization

Easy-to-state formal Questions:
-----------------
XXXspecific example questions for each- tie some of them later to the 
   ``what  is known'' part below

*Given (partial/inaccurate) distance matrix for some dimension - 
--complete it
--find "optimal" realization in a particular dimension

Mathematical directions/conjectures/known results:
------------------


*All of the directions from Lecture 1/2 that apply 
    to purely distance constraint systems 

*Especial interest in  complete distance graphs - 
    Isometric Embeddability of finite metric spaces 
          in fixed (Euclidean) dimension.

What is (additionally) known: SUV of complete distance matrix
http://www.maths.lth.se/matstat/ecmi/modw/reportgroup4/reportgroup4.pdf
web.mit.edu/tfhavel/www/Public/dg-review.ps.gz 

* In addition (for the partial/inaccurate case) 
           convex analysis and convex programming

What is known: semidefinite program for low rank completions              
http://www-personal.engin.umich.edu/~alfakih/
http://www.stanford.edu/~yyye/local-mpb.pdf

----------------------
Lecture 3 (ii):
===========
--Modeling materials, Nanoscale macromolecular self assembly

Easy-to-state formal Questions:
--------------
XXXspecific example questions for each- tie some of them later to the 
   ``what  is known'' part below

*Adapt known results from Lectures 1/2 to leverage symmetry and 
  regularity in constraint systems 

*No overall constraint graph, just local constraint structure
  What possible composites does the local constraint structure result in?
  With what probability?
  Designing "robust and low complexity" constraint systems for a desired result

Mathematical directions/conjectures:
-------------------
In addition to the list for Lecture 1:

*
Enumerative combinatorics and algorithms

(i)Optimization and sampling using graphs and markov chains 
   over discrete structures
(ii)discrete probabilistic methods and randomized algorithms

What is known: enumeration of posets, isomorphic, nonisomorphic
               probability that two are isomorphic, etc.
What is known: basis exchange graphs, enumeration, random walks 
                    for optimization, sampling
http://citeseer.ist.psu.edu/avis93reverse.html
http://www-static.cc.gatech.edu/~mihail/www-papers/stoc92.pdf

*Coxeter groups, tilings, automorphism groups of polyhedral graphs
   and their representations

What is known:Coxeter matroid polytopes  and  tilings
http://citeseer.ist.psu.edu/486847.html
citeseer.ist.psu.edu/article/borovik97coxeter.html
http://citeseer.ist.psu.edu/469343.html

What is known: Mani's theorem
               Babai-Imrich 

What is known: quasi-equivalence theory, later theories
http://viperdb.scripps.edu/icos_server.php?icspage=paradigm
http://arxiv.org/find/grp_q-bio,grp_cs,grp_physics,grp_math,grp_nlin/1/all:+twarock/0/1/0/all/0/1                    



*Finite geometries, designs, finite groups and representations

(i)Designs, Pappus like (special) constraint systems
                        and configurations, realizability, dependences
What is known:  Since Hilbert
http://mathworld.wolfram.com/Configuration.html

(ii) symmetry-adapted combinatorial rigidity analysis
What is known: analysis of local symmetry group structure of realization-space variety
http://www2.eng.cam.ac.uk/~sdg/preprint/tetrahedra.pdf
http://dx.doi.org; doi:10.1016/S0020-7683(02)00350-5  

Lecture 4:
==========
--Quantum information processing, quantum algorithms
  Quantum and classical:
  Illuminating the  quantum-classical gap

Easy-to-state formal Questions:
--------------------
XXXspecific example questions for each- tie some of them later to the 
   ``what  is known'' part below

-The (approximate) MUB problem
-The general extremal line set problem (1 or 2 angles)
-Hidden nonabelian subgroup problem
-Problem of approximation and nonapproximability from (a collection of) MUBs

Mathematical areas and  big questions/conjectures
--------------------------

*
Finite Groups and Fields and codes:
(i) Extremal Euclidean  Line sets
http://www.uoregon.edu/~kantor/PAPERS/Z4.pdf

What is known: 4^n

(ii)Orthogonal decompositons of Lie algebras
http://arxiv.org/abs/quant-ph/0506089

What is known: complex d+1
http://arxiv.org/pdf/quant-ph/0103162
*
Combinatorics: Designs 
Mutually orthogonal latin squares
 
What is known: latin mubs
http://arxiv.org/abs/quant-ph/0407081

* 
Asymptotic geometry of  finite dimensional spaces

(i)isometric embedding of finite point sets
What is known:    
Johnson-Lindenstrauss
http://citeseer.ist.psu.edu/dasgupta99elementary.html
Bourgain
http://ftp.cs.toronto.edu/~avner/teaching/2414/LN2.pdf

                   
                        
Lecture 5:
===============
--      (classical  and  quantum) 
        Communication and circuit complexity (upper and general lower) bounds; 
        pseudorandom generators and cryptographic protocols, 
        learning,sampling,approximation algorithms, 
        Boolean function spectral analysis
        data mining

        approximation algorithms for NP-hard problems
        locally sensitive hashing
        nearest neighbor searches
http://research.microsoft.com/research/theory/naor/homepage%20files/lsh-new.pdf
http://research.microsoft.com/research/theory/naor/homepage%20files/lowdim-journal.pdf
            

        design and decoding of error correcting codes
        illuminating the quantum/classical gap
          

Easy-to-state formal Questions:
--------------
XXXspecific example questions for each- tie some of them later to the 
   ``what  is known'' part below

*Generalization of questions from Lecture 4, for 
     more general (still complete) 
       constraint systems than the MUB system, 
     approximation from more general bases chosen from more general 
        basis families.

Examples:
-infty norm approximation (2 equivalent questions)
-2 norm (1 question,for datamining app)                   
-low spread approximation
-revisiting the hidden subgroup problem

*Isometric (low distortion) (non)embeddability in given(arb) dimension.
of one finite dimensional space into another
including embedding  finite metric spaces into another or into L_p spaces

*Embedding by sections, projections, etc

*Relation between embeddability and approximability questions,
 specifically,  approximation from an (unknown) low dimensional
  subspace spanned by a specific basis family (whose structure is
 typically based on groups or fields)

*Relation between embeddability and nonembeddability questions
   (is there approximation type duality?) 


Mathematical directions/conjectures:
----------------------------------
*(local/asymptotic) geometry 
   of banach spaces - 

What is known: Dvoretzky's theorem
http://research.microsoft.com/research/theory/naor/homepage%20files/Ramsey.pdf
What is known:  Forster's theorem
www.cise.ufl.edu/~sitharam/forster.pdf
What is known: finite metricspace embeddings

Indyk-Matousek survey
http://citeseer.ist.psu.edu/657835.html
Linial survey
http://citeseer.ist.psu.edu/linial02finite.html

Nonembeddability
http://research.microsoft.com/research/theory/naor/homepage%20files/nonembed-final-new.pdf

*Approximation from finite dimensional spaces, convex geometry
     Duality theorems, sampling, spectral analysis 
www.cise.ufl.edu/~sitharam/CIRCUIT/PAPER3/linapprox.pdf
www.cise.ufl.edu/~sitharam/CIRCUIT/TIM/aaecc-396.pdf

     Boolean functions - spectral analysis
http://research.microsoft.com/research/theory/naor/homepage%20files/social.pdf
www.ma.huji.ac.il/~kalai/bki.ps
http://www.ma.huji.ac.il/~ehudf/
http://citeseer.ist.psu.edu/kindler03noiseresistant.html
http://citeseer.ist.psu.edu/496380.html
Bourgain's Boolean paper
citeseer.ifi.unizh.ch/context/2277782/0
Israel Journal of Math 131, 2002
Khot's Course
http://www-static.cc.gatech.edu/~khot/Fourier.htm

*Algebraic geometry - 
    using Grassmanians (for exhibiting special  constructions, 
                                         non-realizability),
    Nullstellensatz
    (For non-realizability) 
http://www.danroozemond.nl/site/archief/ProvingCinderella.pdf
Bokowski
http://portal.acm.org/citation.cfm?id=86263.86266
