Abstracts Leipzig - October 2017

Alexandru Constantinescu (Università di Genova)
Linear syzygies and hyperbolic Coxeter groups

This talk is based on a surprising connection between a question in commutative algebra regarding Castelnuovo-Mumford regularity, and a question of Gromov about hyperbolic Coxeter groups. We show that the virtual cohomological dimension of a Coxeter group is essentially the same as the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of its nerve. Using this, we modify a construction of Osajda in group theory to find for every positive integer r a monomial ideal generated in degree two, with linear syzygies, and regularity of the quotient equal to r. Previously known examples had regularity less than 5. For Gorenstein ideals we prove that the regularity of their quotients can not exceed four, thus showing that for d > 4 every triangulation of a d-manifold has a hollow square or simplex. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity is O(log(log(n)), where n is the number of variables, improving in this case a bound found by Dao, Huneke and Schweig. All results are in collaboration with with Thomas Kahle and Matteo Varbaro.

Lars Kastner (TU Berlin)
Parallel Enumeration of Triangulations

We report on the implementation of an algorithm for computing the set of all regular triangulations of finitely many points in Euclidean space. This algorithm, which we call down-flip reverse search, can be restricted, e.g., to computing full triangulations only; this case is particularly relevant for tropical geometry. Most importantly, down-flip reverse search allows for massive parallelization, i.e., it scales well even for many cores. Our implementation allows to compute the triangulations of much larger point sets than before. This is joint work with Charles Jordan and Michael Joswig.

Joseph Landsberg (Texas A&M University)
Symmetry and the complexity of matrix multiplication

In 1969 Strassen discovered the standard way of multiplying matrices is not the optimal one, which spurred an immense amount of research and the astounding conjecture that as n goes to infinity, it becomes almost as easy to multiply two nxn matrices as it is to add them. I will discuss the history of this problem, its reformulation in the language of tensors, and recent work attempting to better understand matrix multiplication in terms of its symmetries.

Diane Maclagan (University of Warwick)
Tropical ideals

Tropical geometry studies geometry over the tropical semiring, where multiplication is replaced by addition and addition by minimum. Over the last fifteen years there has been an explosion of work on varieties in this setting. Commutative algebra provides the "algebra" behind the "geometry" of varieties in algebraic geometry. From a commutative algebra perspective, however, the semiring of tropical polynomials is not as nice as the its standard counterpart. We no longer have unique factorization, cancellation, or the Noetherian property. I will discuss joint work with Felipe Rincon on a special class of "tropical" ideals in this semiring that is much better behaved, and on the geometric side allows us to expand from varieties to schemes. This uses the theory of valuated matroids.