Projects- Invariant theory
- Automorphisms of curves
- Field of moduli versus field of definition
- Theta functions and Jacobians
- Superelliptic curves
Theta FunctionsTheta functions are some of the most amazing objects in mathematics. They have been studied by Riemann, Picard, Kovalevski, Frobenious among many others.
We focus on theta functions of algebraic curves, and more explicitly on theta-nulls of superelliptic curves.
For more information click below
Genus 3 curves
Genus 3 curves are ternary quartics. If the curve is hyperelliptic then they are written as y^2=f(x) where f(x) is a polynomial of degree 8.
Genus 3 hyperelliptic curves are treated in the package of hyperelliptic curves.
Here we deal with non-hyperelliptic genus 3 curves. The most famous of them all is perhaps the Klein's quartic. AMS Special Session: March 2012, Tampa, Florida.
| Genus 2 Curves
A Maple package for genus 2 curves which computes the basic invariants of the curve: Igusa invariants, absolute invariants, field of moduli, field of definition, automorphism group, etc.
Conferences and ActivitiesAlgebraic Geometry Blog
- Theta Functions, Special Session, SIAM Conference, Raleigh, Oct. 6-8. 2011.
- Applied Algebraic Geometry, SIAM, Raleigh, Oct. 6-8. 2011.
- AMS sectional conference, Boise, Idaho, 2011
- AMS Annual Meeting, Boston, 2012
Databases for curvesI have seen every curve at least once ...
Genus: 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |Equation of curves over their field of definitionLet C be a curve defined over the complex numbers and F its field of definition. Finding a curve $X$, isomorphic to C over the complex numbers, is an old problem in algebraic geometry. Algorithms exist for genus 2 curves due to work of Clebsch, Mestre, Shaska, Cardona, et al. We focus on genus 3 curves and on superelliptic curves of any genus.
More information can be found here.
| Books
Algjebra Abstrakte (T. Shaska, L. Beshaj), AulonaPress, 2011, (Albanian). |
