Office: Blocker 601L
Email:
sottile"at"math.tamu.edu
WWW:
https://franksottile.github.io/
Office Hours: TBA.Seminars on Monday and Fridays Geometry (MF) and
Algebra and Combinatorics (F)
are a good venue for catching faculty members, including Frank.
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 would be 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. The expected background is a graduate course in algebra, possibly concurrent.
- Some geometrical aspects of control points for toric patches
Gheorghe Craciun, Luis Garcia, and Frank Sottile. - Sign conditions for injectivity of generalized polynomial
maps with
applications to chemical reaction networks and real algebraic geometry,
S Müller, E Feliu, G Regensburger, C Conradi, A Shiu, A Dickenstein
Foundations of Computational Mathematics 16 (1), 69-97 2016. -
Real Solutions to Systems of Equations in Macaulay2
Jordy Lopez-García, Kelly Maluccio, Frank Sottile, and Thomas Yahl.
- Sign Conditions for the Existence of at Least One Positive Solution of a Sparse Polynomial System, Frédéric Bihan, Alicia Dickenstein, and Magalí Giaroli.
- Descartes' Rule of Signs for Polynomial Systems Supported on Circuits, Frédéric Bihan and Alicia Dickenstein.
- Wronski Pairs of Honeycomb Curves, Laura Casabella, Michael Joswig, Rafael Mohr.
- A counterexample to the strong real Jacobian conjecture, S. Pinchuk, Math Z. 217 (1994) 1–4.
Hilbert's 1888 Theorem on positivity and sums of squares. John Ajayo and Marleigh Purgar-Mcdonald Lax's discriminant of a symmetric matrix is a sum of squares. Pooja Joshi and Zeytoon Kazemimoghaddam. "Newton-Okounkov bodies of chemical reaction systems" by Obatake and Walker. Jordy Lopez and Paul Dessauer.
Hilbert's 17th problem. Ozan Acikgoz
The Shapiro Conjecture and the Wronski map. C.J. Bott and Daniel Dale
A counterexample to the strong real Jacobian Conjecture. Kevin Le and Jonah Robinson.
Sums of squares and minimal varieties. Somak Dutta and Ruzho Sagayaraj.
Present Macaulay2 Package RealRoots.
Multivariate generalizations of stability of univariate polynomials.
Many of the Sources above would be fine for a presentation.