Combinatorial Algebraic Geometry

ag.algebraic-geometry co.combinatorics
Start Date
2021-02-01 
End Date
2021-05-07 
Institution
ICERM 
City
Providence, RI 
Country
USA 
Meeting Type
thematic research program 
Homepage
https://icerm.brown.edu/programs/sp-s21/ 
Contact Name
 

Description

Combinatorial algebraic geometry comprises the parts of algebraic geometry where basic geometric phenomena can be described with combinatorial data, and where combinatorial methods are essential for further progress.

Research in combinatorial algebraic geometry utilizes combinatorial techniques to answer questions about geometry. Typical examples include predictions about singularities, construction of degenerations, and computation of geometric invariants such as Gromov-Witten invariants, Euler characteristics, the number of points in intersections, multiplicities, genera, and many more. The study of positivity properties of geometric invariants is one of the driving forces behind the interplay between geometry and combinatorics. Flag manifolds and Schubert calculus are particularly rich sources of invariants with positivity properties.

In the opposite direction, geometric methods provide powerful tools for studying combinatorial objects. For example, many deep properties of polytopes are consequences of geometric theorems applied to associated toric varieties. In other cases geometry is a source of inspiration. For instance, long-standing conjectures about matroids have recently been resolved by proving that associated algebraic structures behave as if they are cohomology rings of smooth algebraic varieties.

Much research in combinatorial algebraic geometry relies on mathematical software to explore and enumerate combinatorial structures and compute geometric invariants. Writing the required programs is a considerable part of many research projects. The development of new mathematics software is therefore prioritized in the program.

The program will bring together experts in both pure and applied parts of mathematics as well mathematical programmers, all working at the confluence of discrete mathematics and algebraic geometry, with the aim of creating an environment conducive to interdisciplinary collaboration. The semester will include four week-long workshops, briefly described as follows.

  • A 'boot-camp' aimed at introducing graduate students and early-career researchers to the main directions of research in the program.

  • A research workshop dedicated to geometry arising from flag manifolds, classical and quantum Schubert calculus, combinatorial Hodge theory, and geometric representation theory.

  • A research workshop dedicated to polyhedral spaces and tropical geometry, toric varieties, Newton-Okounkov bodies, cluster algebras and varieties, and moduli spaces and their tropicalizations.

  • A Sage/Oscar Days workshop dedicated to development of programs and software libraries useful for research in combinatorial algebraic geometry. This workshop will also feature a series of lectures by experts in polynomial computations.

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