Real Algebraic Geometry for Applications. 

   Real algebraic geometry is  a fundamental input for many
applications of algebraic geometry.   Its goals and methods
are  also distinct  from classical  algebraic geometry.   I
expect to cover topics such as real solutions to systems of
equations, including upper and lower bounds, positivity and
sums  of squares,  real toric  varieties, and  non-standard
real structures.

   This is  based on  parts of my  book "Real  solutions to
Equations from  Geometry", two chapters in  Theobald's book
"Real  Algebraic Geometry  and  Optimization",  as well  as
papers and other sources.

Grading is through class attendance and end-of-term presentations.

NOTE:   To sign up for this and for Landsberg's Algebraic Geometry II
      (both MATH 666), you need to write Briana Hague and ask for a waiver.

Course WWW: https://franksottile.github.io/teaching/25.1/RAGA.html

Course Outline:

Classical methods for real roots of equations
  Theorems of Sturm and Descartes
  Stability and Routh-Hurwicz
  Hermite trace form

Positive polynomials
  Positivity and sums of squares
  Hilbert's 17th problem
  Positivstellensatz
  Theorems of Polya, Handelman, Putinar, and Schmudgen

Bounds on numbers of real solutions
  Gale duality for upper bounds
  Lower bounds 
  Conditions for injectivity

Real toric varieties
  Real points of toric varieties
  Non-standard real toric varieties
  Irrational toric varieties