Algebraic Geometry

Gudrun spent an afternoon at the Max Planck Institute for
Mathematics in the Sciences (MPI MSI) in Leipzig. There she met
the Colombian mathematician Eliana Maria Duarte Gelvez. Eliana is
a PostDoc at the MPI MSI in the Research group in Nonlinear
Algebra. Its head is Bernd Sturmfels.


They started the conversation with the question: What is
algebraic geometry? It is a generalisation of what one learns in
linear algebra insofar as it studies properties of polynomials
such as its roots. But it considers systems of polynomial
equations in several variables so-called multivariate
polynomials. There are diverse applications in engineering,
biology, statistics and topological data analysis. Among them
Eliana is mostly interested in questions from computer graphics
and statistics.


In any animated movie or computer game all objects have to be
represented by the computer. Often the surface of the geometric
objects is parametrized by polynomials. The image of the
parametrization can as well be defined by an equation. For
calculating interactions it can be necessary to know what is the
corresponding equation in the three usual space variables. One
example, which comes up in school and in the introductory courses
at university is the circle. Its representation in different
coordinate systems or as a parametrized curve lends itself to
interesting problems to solve for the students.


Even more interesting and often difficult to answer is the simple
question after the curve of the intersection of surfaces in the
computer representation if these are parametrized objects.
Moreover real time graphics for computer games need fast and
reliable algorithms for that question. Specialists in computer
graphics experience that not all curves and surfaces can be
parametrized. It was a puzzling question until they talked to
people working in algebraic geometry. They knew that the genus of
the curve tells you about the possible vs. impossible
parametrization.


For the practical work symbolic algebra packages help. They are
based on the concept of the Gröbner basis. Gröbner basis help to
translate between representations of surfaces and curves as
parametrized objects and graphs of functions. Nevertheless, often
very long polynomials with many terms (like 500) are the result
and not so straightforward to analyse.


A second research topic of Eliana is algebraic statistics. It is
a very recent field and evolved only in the last 20-30 years. In
the typical problems one studies discrete or polynomial equations
using symbolic computations with combinatorics on top. Often
numerical algebraic tools are necessary. It is algebraic in the
sense that many popular statistical models are parametrized by
polynomials. The points in the image of the parameterization are
the probability distributions in the statistical model. The
interest of the research is to study properties of statistical
models using algebraic geometry, for instance describe the
implicit equations of the model.


Eliana already liked mathematics at school but was not always
very good in it. When she decided to take a Bachelor course in
mathematics she liked the very friendly environment at her
faculty in the Universidad de los Andes, Bogotá. She was
introduced to her research field through a course in
Combinatorial commutative algebra there. She was encouraged to
apply for a Master's program in the US and to work on elliptic
curves at Binghamton University (State University of New York)
After her Master in 2011 she stayed in the US to better
understand syzygies within her work on a PhD at the University of
Illinois at Urbana-Champaign. Since 2018 she has been a postdoc
at the MPI MSI in Leipzig and likes the very applied focus
especially on algebraic statistics.


In her experience Mathematics is a good topic to work on in
different places and it is important to have role models in your
field.

References

E. Duarte, Ch. Görgen: Equations defining probability tree
models

E. Duarte: Implicitization of tensor product surface in the
presence of a generic set of basepoints. 2016. Journal of Algebra
and Applications(to appear).

Rigidity of Quasicrystal Frameworks - webpage

E. M. Duarte, G. K. Francis: Stability of Quasicrystal
Frameworks in 2D and 3D Proceedings of the First Conference
Transformables 2013.In the Honor of Emilio Perez Piñero 18th-20th
September 2013, Seville, Spain

Portraits of people working in Nonlinear Algebra


Podcasts

P. Schwer: Metrische Geometrie, Gespräch mit G. Thäter im
Modellansatz Podcast, Folge 102, Fakultät für Mathematik,
Karlsruher Institut für Technologie (KIT), 2016.