# 3C in G Workshop on Computational Algebra

### Titles and Abstracts

**Winfried Bruns (Universität Osnabrück) - Convex Hull Computation in Normaliz**Normaliz is a versatile package for computations in discrete convex geometry: it computes lattice points in polyhedra, or, from a different perspective, solves linear diophantine systems of inequalities, equations and congruences.

Recently it has been included in benchmarks of convex hull computation (or vertex enumeration), and has often outperformed dedicated systems, especially on large problems. We will present Normaliz' approach to convex hulls. The main tool is pyramid decomposition whose development was sparked off by the more difficult task of computing triangulations. Pyramid decomposition allows "localization" and helps to break the often forbidding complexity of pure Fourier-Motzkin elimination. Furthermore it is parallelization friendly.

Based on: W. Bruns, B. Ichim and C. Söger, The power of pyramid decomposition in Normaliz. J. Symb. Comp. 74 (2016), 513--536.

**Daniel Cavey (University of Nottingham) - Classification of Minimal, Mutation Equivalent Polygons with Specified Singularity Content**- We first recall the definitions of minimality and mutations for Fano polygons. It is known that up to mutation equivalence, there are only finitely many Fano polygons of a given singularity content. We describe an algorithm to complete the classification of this problem. To illustrate we compute all Fano polygons with basket of singularities given by (i) B = {m
_{1 }× 1/3(1,1), m_{2}× 1/6(1,1)} and (ii) B = {m × 1/5(1,1)}. We are interested in the applications of our results in the classification of Fano varieties and this talk is part of an effort to classify orbifold del Pezzo surfaces X with h^{0}(X,−K_{X}) > 0 using mirror symmetry. **Wolfram Decker (Technische Universität Kaiserslautern) - What can be Computed in Algebraic Geometry? A Survey from Historical Roots to Future Plans**- We give an overview on computational algebraic geometry which may help algebraic geometers to decide whether today's or tomorrow's computational techniques and computer algebra systems provide useful tools for their own research.
**Paweł Dłotko (Inria France) - Computational Algebra, Topology and Applications**- In this talk I will present how various basic, and a bit more advanced, algorithms from computational algebra allow computational topology to solve real world problems. First we will revise classical simplicial complexes which are combinatorial representations of topological spaces. We will study persistent homology and see how one can use (and optimize) standard Gaussian elimination to compute it, and solve interesting practical problems. Later we will discuss ideas from discrete Morse theory and their relation with standard operations in group theory. We will conclude by providing an efficient partial algorithm to compute fundamental groups of spaces, and discuss some applications of it.
**Emilie Dufresne (University of Nottingham) - Mapping Toric Varieties into Small Dimensional Spaces**- A smooth d-dimensional projective variety X can always be embedded into (2d + 1)-dimensional space. In contrast, a singular variety may require an arbitrary large ambient space. If we relax our requirement and ask only that the map is injective, then any d-dimensional projective variety can be mapped injectively to (2d + 1)-dimensional projective space. A natural question then arises: what is the minimal m such that a projective variety can be mapped injectively to m-dimensional projective space? In this talk I discuss this question for normal toric varieties, with the most complete results being for Segre-Veronese varieties. (Joint work with Jack Jeffries).
**Anne Frühbis-Krüger (Leibniz Universität Hannover) - Determinantal Singularities - Questions, Algorithms and some Answers**Determinantal singularities form a class of singularities which seem at first glance more complicated than complete intersection singularities. However, this larger class also contains singularities which are from a certain point of view simpler than the simplest complete intersections - just think of the three coordinate axes in C

^{3}.Up to now the study of determinantal singularities has been impeded by the failure of many standard techniques, as soon as one left the realm of complete intersections. In this talk I shall give a brief overview on similarities and differences to the complete intersection case, on open questions and on tools to study the topology and deformation properties for some important subclasses of such singularities in detail.

**Simon Hettrick (Software Sustainability Institute) - Software Experts are Vital to Research**Most research would be impossible without software, and this reliance is forcing a rethink of the skills needed in a traditional research group. A survey of 15 Russell Group universities found that 92% of researchers used research software, 67% reported that it was fundamental to their research, and 56% said they developed their own software. With the emergence of software as the pre-eminent research tool used across all disciplines, comes the realisation that a significant majority of results are based, ultimately, on the skill of the experts who design and build it.

The work of software experts in academia is rarely recognised, and they suffer poor conditions of employment relative to other research roles. Since 2012, a community of these experts has grown around a campaign to raise awareness of the people who build the software used in research. These people have worked under many titles, but many now identify as Research Software Engineers.

By providing the expertise needed by modern research groups, we will promote the development of well-engineered software that will increase the scope, productivity and reliability of research.

**Tommy Hofmann (Technische Universität Kaiserslautern) - Computing Tropical Points and Links**- Computing tropical varieties is an algorithmically challenging task, requiring sophisticated techniques from computer algebra and convex geometry. Based on triangular decompositions and Newton polygons, we describe a new approach for computing a tropical starting point as well as tropical links. We apply this to the investigation of tropical Grassmannians. This is joint work with Yue Ren.
**Nathan Ilten (Simon Fraser University) - Semi-Canonical Embeddings for Rational Complexity-One T-Varieties**- A normal affine toric variety X with no torus factors has a canonical equivariant embedding in affine space, determined by the Hilbert basis of the weight cone of X. This embedding has many nice properties: the tropicalization Trop(X) with respect to this embedding is a linear subspace of R
^{n}, every initial ideal corresponding to a point in Trop(X) is simply the ideal of X, and the generators for the coordinate ring of X form a Khovanskii basis with respect to any full-rank homogeneous valuation. In this talk, I will report on joint work with Chris Manon in which we generalize this situation to normal rational affine varieties equipped with the action of a codimension-one torus. We produce explicit affine embeddings of such varieties, and show that these embeddings enjoy properties similar to those found in the toric situation. **Anders Jensen (Aarhus University) - Finding Binomials in Polynomial Ideals**Deciding if a polynomial ideal contains monomials is a problem which can be solved by usual Groebner basis techniques. Deciding if a polynomial ideal contains binomials is more complicated. We show how the general case can be reduced to the case of zero-dimensional ideals using projections and stable intersections in tropical geometry. In the case of rational coefficients the zero-dimensional problem can then be solved via Ge's algorithm using methods by Babai, Beals, Cai, Ivanyos and Luks. In case binomials exist, one can be computed.

This is joint work with Thomas Kahle and Lukas Katthän.

**Michael Joswig (Technische Universität Berlin) - Smooth Fano Polytopes and their Triangulations**- The classification of the d-dimensional simplicial, terminal, and reflexive polytopes with at least 3d-2 vertices is presented. It turns out that all of them are smooth Fano polytopes. This builds on and improves previous results of Casagrande (2006) and Øbro (2008). The second half of the presentation is concerned with triangulations of such polytopes. Joint work with Benjamin Assarf, Andreas Paffenholz, and Julian Pfeifle.
**Sara Lamboglia (University of Warwick) - Computing Toric Degenerations of Flag Varieties**- Toric degenerations of flag varieties have been studied quite intensively in the last few decades and several methods have been applied. In this talk I am going to show results on the toric degenerations arising from the tropicalization of the full flag varieties of C
^{4}and C^{5}. These are obtained as Gröbner degenerations associated to the initial ideals corresponding to the maximal cones of the tropicalization of the flag variety. I will then compare these degenerations with the ones arising in representation theory from string polytopes and FFLV polytopes. Moreover I will present a general procedure to find toric degenerations in the cases where the initial ideal arising from a cone of the tropicalization of a variety is not prime. This is joint work with L. Bossinger, K. Mincheva and F. Mohammadi. **Diane Maclagan (University of Warwick) - Computational Algebra Lessons from Tropical Geometry**- Tropical geometry has introduced several computational algebra challenges, the solution to which should have applications outside tropical geometry. One of the first of these is a generalization of the theory of Gröbner bases to polynomial rings where the coefficients have a nontrivial valuation. Potential application of this outside tropical geometry comes from the fact that these Gröbner bases can be significantly smaller than usual ones. Another challenge comes from considering Gröbner bases of ideals in the semiring of tropical polynomials. With some additional combinatorial constraints we get a notion of Gröbner complex/Gröbner fan in this setting, and a Nullstellensatz, which sheds some light on the corresponding notions for usual polynomial rings.
**Vidit Nanda (Alan Turing Institute) - Discrete Morse Theory for Computing Homology**- Recent applications of algebraic-topological methods to scientific and engineering contexts have produced intriguing results: from the discovery of a new type of breast cancer, to the detection of coverage in mobile sensor networks and accurate prediction of protein compressibility directly from crystallography data. This talk will survey the underlying methodology of topological data analysis, describe its primary computational challenges, and show how a discrete version of Morse theory renders enormous computations tractable. Nothing is asked of the audience beyond a basic understanding of linear algebra and a burning desire to chase gradient trajectories between critical points.
**Michele Nicolussi (Universität Tübingen) - Terminal Fano Threefolds with a Torus Action of Complexity One**- In toric geometry, Fano varieties correspond to certain lattice polytopes that detect the type of singularities of the variety. In this spirit, we approach the broader family of varieties with a torus action of complexity one. We associate to any such variety a polyhedral complex with the same detecting properties. Moreover, we present the classification results obtained up to this point and highlight the role of computational methods.
**Andreas Paffenholz (Technische Universität Darmstadt) - Polyhedral Adjunction and the Q-Codegree Spectrum**Polyhedral adjunction theory allows to study questions of classical adjunction theory for polarised toric varieties from a convex-geometric viewpoint using the associated polyhedral fans and lattice polytopes. Important algebraic invariants like the (unnormalised) spectral value or the nef-value can be defined and studied purely in terms of these polytopes.

Using this approach we obtain structural results for lattice polytopes and their toric varieties. In particular we can show that a lattice polytope with sufficiently large spectral value has a Cayley structure, i.e. it allows a non-trivial lattice projection onto a lattice simplex. We can also affirmatively answer a conjecture of Fujita on the spectrum of the spectral value in the toric case.

This is partially based on joint work with Benjamin Nill, Christian Haase, and Sandra Di Rocco.

**Michael Sagraloff (Max-Planck-Institut für Informatik) - On the Complexity of Solving Polynomial Equations**The computation of the roots of a univariate polynomial is one of the best studied problems in the areas of computer algebra and numerical analysis. Nevertheless, until recently, there has still been a large discrepancy between methods that are considered to be efficient in practice and those that achieve good theoretical bounds. In the first part of this talk, I will review some recently developed algorithms for real and complex root finding, which are simple and practical on the one hand and whose worst-case bit complexity matches that of the best methods in theory on the other hand. Combining our techniques with a classical approach that considers so-called fractional derivatives, we can further derive a polynomial-time algorithm to compute approximations of all real roots of a sparse polynomial with arbitrary real coefficients. For very sparse polynomials with integer coefficients, the method performs near-optimal.

In the second part of my talk, I will focus on the more general problem of computing the solutions of a given zero-dimensional polynomial system in d variables. A common approach for computing the set S of solutions is to first project S into one dimension (e.g. using resultant computation and univariate root finding) and then to recover the solutions from the projections. The latter step turns out to be difficult if the projection direction is not known to be separating for the solutions. I will review a simple and efficient algorithm for the certified computation of S, which only needs to compute projections of the solutions but no further algebraic manipulations. It computes a separating direction of small bit-size as well as isolating boxes for all solutions for the cost of O(d) projections of the solutions. Finally, I will sketch a symbolic-numeric algorithm to count the number of solutions within a local region, which is based on a "local variant" of the rather global method from above. Assuming the existence of an oracle that provides arbitrary good approximations of the solutions of the system, our method can be used to verify correctness of these approximations. In contrast to global approaches, the complexity of our method mainly depends on local parameters such as the number of roots within the given region or the size of this region.

**Harald Skarke (Technische Universität Wien) - Classification of Cones for Calabi-Yaus**- Calabi-Yau hypersurfaces in toric varieties can be described via reflexive polytopes. A generalized construction involving reflexive Gorenstein cones affords descriptions of Calabi-Yau manifolds of higher codimension as well as Landau-Ginzburg models. After reviewing some of the relevant concepts and the algorithm that led to the classification of reflexive polytopes in three and four dimensions, I will indicate how this algorithm needs to be be modified so that it can be applied to reflexive Gorenstein cones. I will also point out open classification problems that may well lie within the range of contemporary computer power.
**Isabel Stenger (Technische Universität Kaiserslautern) - Computing Numerical Godeaux Surfaces**- Numerical Godeaux surfaces provide the first case in the study of minimal surfaces of general type. It is known that the torsion group of such a surface is cyclic of order m ≤ 5 and a full classification has been given for m = 3, 4, 5 by Reid, Miyaoka. In my talk I will discuss a homological approach to construct a numerical Godeaux surface X based on a former project of Frank-Olaf Schreyer. The main idea is to study the syzygies of the canonical ring R(X) considered as a module over some (weighted) graded polynomial ring. We focus on the case Tors(X) = 0 and pay particular emphasis to the genus-4 fibration given by the bicanonical system. The general fibre of this fibration is known to be non-hyperelliptic, and the number h of the hyperelliptic fibres is bounded by 3, provided that the bicanonical system has no fixed part and four base points. Our construction yields an 8-dimensional family of numerical Godeaux surfaces with h = 0 and rediscovers the Barlow surface which satisfies h = 2. Furthermore, for the first time, we find surfaces with h = 1.
**Michael Stillman (Cornell University) - Computational Algebraic Geometry meets String Theory: the Search for Rigid Divisors and Computing Sheaf Cohomology on Calabi-Yau Hypersurfaces of Toric 4-folds**Calabi-Yau 3-folds play a large role in string theory. Cohomology of sheaves on such varieties has many uses in string theory, including counting the number of particles or fields in a theory, as well as to help identify terms in the superpotential that determines the equations of motion of the corresponding string theory, and many other uses as well. As a computational algebraic geometer, string theory provide a rich source of new computational problems to solve.

In this talk, we focus on the search for rigid divisors on these Calabi-Yau hypersurfaces of toric varieties. We have had methods to compute sheaf cohomology on these varieties for many years now (Eisenbud-Mustata-Stillman, around 2000), but these methods fail for many of the examples of interest, in that they take a very long time, or the software (wisely) refuses to try!

We provide techniques and formulas for the sheaf cohomology of certain divisors of interest in string theory, that other current methods cannot handle. Along the way, we describe a Macaulay2 package for computing with these objects, and show its use on examples.

This is joint work with Andreas Braun, Cody Long, Liam McAllister, and Benjamin Sung.

This page is maintained by Alan Thompson and was last updated on 15/04/17. Please email comments and corrections to verily(at)alanthompson.rocks.