Numerical_Algebraic_Geometry_(IMA_2011)

title: Numerical Algebraic Geometry (IMA 2011) permalink: wiki/Numerical_Algebraic_Geometry_(IMA_2011)/ layout: wiki

Interfaces: Some functions in NumericalAlgebraicGeometry already have ability to easily substitute one homotopy solver with another. The idea is to have all interfaces to Numerical AG software as separate packages that would integrate seemlessly and use the same data types.
- Dan(Bates) will look at the interface for Bertini.
- Jan, Sonja, Elizabeth are in charge of the interface for PHCpack: work on functions for numerical irreducible decomposition: e.g., make cascade return witness sets, export monodromyBreakup, etc. ''' Make sure any new methods are documented. '''

Data types: NAGtypes.m2 collects the common data types such as Point and WitnessSet.
- Discuss what the common types should be. (Everybody in the group should participate.)
- Go over the issue of interdependencies between the packages. Dan(Grayson) can help.

Schubert calculus: Abraham is in charge of NumericalSchubertCalculus
- Check that the current functionality works with all available homotopy trackers.
- Jan has worked before on Littlewood-Richardson homotopies. Abraham, Jan, and Anton should discuss their implementation in M2.

Dual bases and Hilbert functions: NumericalHilbert is developed by Robert.
- All functions should be able to take both exact (in QQ[...]) and approximate (in CC[...]) input.
- Termination criterion in dual basis computation for a positive-dimensional ideal.
- Flesh out DualSpace data type.

Tropical G meets Numerical AG: Compute tropical curves using heuristics and numerics.
- Henry, Josephine, Anders (and Anton) use NumericalAlgebraicGeometry to slice amoebas and extract integer vectors from their tentacles.
- Henry thinks on how to go about arbitrary precision linear algebra in M2.

Symbolic-Numeric Toolbox: Build tools that are useful for implementing existing and future symbolic-numeric algorithms (and implement some!).
- Dan is suggesting this (see comments below), though we (all or some appropriate subset) will need to think about how this fits into other existing packages, as well as the current numerical linear algebra capabilities within Macaulay2.
- The main idea is to have available several operations that are common in hybrid symbolic-numeric operations, then implement several (though this latter part will likely stretch beyond this workshop).

More on the toolbox (from Dan): There are now algorithms for computing the geometric genus of a curve (Bates-Peterson-Sommese-Wampler), chern classes (Di Rocco-Eklund-Peterson-Sommese, Bates-Eklund-Peterson, Eklund-Jost-Peterson), low-degree generators of irreducible components of algebraic sets (Bates-Hauenstein-McCoy-Peterson-Sommese - the "LLL method"), real curves and surfaces within higher-dimensional complex components (Lu-Bates-Sommese-Wampler, Besana-Di Rocco-Hauenstein-Sommese-Wampler), etc., all with numerical algebraic geometry plus a bit of symbolic manipulation. Most of these have barely been implemented anywhere - maybe a few ad hoc, partial implementations using Maple - and my goal is to build an environment (using some of the existing work in NAG4M2) where these sorts of algorithms can be (a) implemented efficiently and (b) found and used (!). I won't put all the details here, but the idea is to finish the basic Bertini/M2 interface (as indicated by Anton above), then build some M2 functions to build certain sorts of polynomial systems and/or homotopies that can be plugged in to the algorithms above. For example, for the genus algorithm, we need one polynomial system to find all singular points in a certain (generic) projection, then a bunch of other parameter homotopies to do "monodromy loops" around the ramification points. (The current work and plans for NAG4M2 are a great start in this direction!)

NAGtypes discussion and to-do list (7/26/11):
- New/modified fields to Point:
  - ErrorBoundEstimate (should be absolute)
  - Multiplicity (either "path"-multiplicity or the actual multiplicity)
  - MaxPrecision (max precision used during the homotopy tracking)
  - WindingNumber
  - DeflationNumber (number of first-order deflations)
- Service functions:
  - convertCoordsToFloats
  - residual(System,Point,Norm=>...)
  - relativeError(Point)
  - areEqual -- move it from NAG
  - isReal(Point, Tolerance=>...)
  - classify(Point) -- heuristic that says whether the point is Regular, Singular, etc.
- Additional WitnessSet fields
  - type? -- irreducible, reducible, unclassified
- Service functions
  - degree
  - dim
  - union? (binary operator)
- Prospective new type: something that represents a variety numerically (produced by numerical irreducible decomposition), what is a good name?
  - Decomposition?
  - NumericalVariety?
  - GeneralWitnessSet?
- ... good structure?
  - a hashtable with dimensions for keys and lists of (irreducible) WitnessSets as entries?
  - a list of WitnessSets?
- ... service functions?
  - dim
  - isIrreducible?
  - isReduced?