Tropical Geometry

Monday, we looked at tropical arithmetic and algebra but that is only half of it.  The other half is tropical geometry which takes questions on Algebraic varieties and transforms these questions so they apply to polyhedral complexes.  Tropical geometry developed as an offshoot to Tropical Algebra due to questions that arose in computer science.  In addition, this topic is often referred to as tropical algebraic geometry.

There are three approaches to tropical geometry.  First is the synthetic approach, and is considered algebraic geometry applied to a tropical semifield.  It led to finding tropical theorems equivalent to those applied to algebraic curves and their associated geometry.  The second is the valuation theoretical approach which looks at tropical versions as shadows for the algebraic ones.  The final approach is the degeneration theoretical approach which is related to the degeneration theoretical approach for algebraic varieties.

Tropical geometry looks at things differently such as when polynomials which may be continuous in algebra are turned into piecewise functions. In fact, tropical geometry makes it easier to do polyhedral geometry because algebraic geometry doesn’t explain polyhedrals as well. It has been found that many of the invariant varieties are preserved when using tropical geometry.

Furthermore, tropical geometry works well with enumerative geometry, especially Mikhalken’s work.  Tropical geometry has developed its own formulas to determine the number of rational curves based on its degree. Furthermore, tropical geometry looks at solution spaces.  

If you look at this site, you can read four lectures by Dianne Maclagan given back in 2008/2009 to students taking a class in this topic.  The four lectures cover an introduction, fundamental theorems, more examples and explanations, finishing off with enumerative geometry.  Although it is only 67 pages long, it provides some fascinating information.

If you notice, tropical math encompasses both algebra and geometry to work together to form a new way of explaining things.  Let me know what you think, I’d love to hear.  Have a great day.