# Applied Algebra & Geometry Workshop

Thursday 30th-Friday 31st May, 2019 Maths & Stats Building

#### Description

Matthew England, Coventry University

Jessica Enright, University of Edinburgh

Jeffrey Giansiracusa, Swansea University

Delaram Kahrobaei, University of York

Elana Kalashnikov, Imperial College London

Piotr Zwiernik, Universitat Pompeu Fabra, Barcelona

30-31 May 2019

School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ

**30th May 2018**

10:30-11:00 Coffee & welcome

11:00-12:00 Delaram Kahrobaei

12:00-13:30 Lunch & discussions

13:30-14:30 Jeff Giansiracussa

14:30-14:45 Coffee break

14:45-15:45 Elana Kalashnikov

18:00 Dinner

**31 May 2018**

10:00-11:00 Jessica Enright

11:00-12:00 Piotr Zwiernik

12:00-12:30 Coffee break

12:30-13:30 Matthew England

13:30-14:30 Lunch

If you are planning to attend this workshop, please register by sending an e-mail to Dimitra Kosta (Dimitra.Kosta@glasgow.ac.uk).

In this talk, I explore how group theory playing a crucial role in data science and artificial intelligence as well as cyber security and quantum computation. At the same time, how computer science for example machine learning algorithms and computational complexity could help group theorists so tackle their open problems.

Symmetry is present in all forms in the natural and biological structures as well as man-made environments. Computational symmetry applies group-theory to create algorithms that model and analyze symmetry in real data set. The use of symmetry groups in optimizing the formulation of signal processing and machine learning algorithms can greatly enhance the impact of these algorithms in many fields of science and engineering where highly complex symmetries exist.

At the same time, Machine Learning techniques could help with solving long standing group theoretic problems. ...more

One of the key properties of 1-parameter persistent homology is that its output can entirely encoded in a purely combinatorial way via persistence diagrams or barcodes. However, many applications of topological data analysis naturally present themselves with more than 1 parameter. Multiparameter persistence suggests itself as the natural invariant to use, but the problem here is that the moduli space of multiparameter persistence diagrams has a much more complicated structure and we lack a combinatorial diagrammatic description. An alternative approach was suggested by work of Giansiracusa-Moon-Lazar, where they investigated calculating a series of 1-parameter persistence diagrams as the other parameter is varied. In this talk I will discuss work in progress to produce a refinement of their perspective, making use the Algebraic Stability Theorem for persistent homology and work of Bauer-Lesnick on induced matchings.

Fano varieties are only classified in dimension up to 3. However, all these (dimension up to 3) Fano varieties can be constructed as either toric complete intersections or as quiver flag zero loci. While not all Fano varieties are of these two types, it might be hoped that most small dimensional ones are. Coates, Kasprzyk and Prince completed the search for all Fano toric complete intersections in codimension at most four. In this talk, I discuss the same search for Fano quiver flag zero loci (joint work with Coates and Kasprzyk), as well as the theoretical basis behind it. I will also explain the underlyng mirror symmetry motivation.

British farms have a variety of different types of potentially infection-spreading contacts that can be interpreted as edges in graphs with different (and often very useful) structure. Some of this structure is geographically geometric: e.g. farms are arranged in a landscape on the surface of the earth, and so a graph that records physical fence-line adjacency between farms will be planar with edge-length and density constraints. Other structure is more related to trading behaviour, e.g. the important hub role of markets. I’ll describe some of this structure, and outline approaches from parameterised algorithmics that we’ve used to solve problems related to the spread of infectious disease over these graphs. This will include both graph modification problems, in particular edge deletion to limit disease spread, as well as subgraph counting problems.

Felsenstein's classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of inverse covariance matrices. We present an exact semialgebraic characterization of this model. The main part of the talk will be based on a paper written jointly with Bernd Sturmfels and Caroline Uhler (arXiv:1902.09905). In the second part of the talk I will present recent results on the maximum likelihood estimation problem for this model class.

We will start with a gentle introduction to the problem of Quantifier Elimination; and the idea of Cylindrical Algebraic Decomposition (CAD) which can solve this problem, and others in the field of non-linear real arithmetic. These algorithms are usually implemented within Computer Algebra systems, although they have recently found their way into Satisfiability Modulo Theory (SMT) Solvers. We will illustrate with recent examples the author has worked on from economics and bio-chemical network analysis. Time permitting, the author will describe some of his recent work: adaptations to the core CAD algorithm to take advantage of the logical structure in problems; and machine learned heuristics for optimizing CAD algorithm settings.

Funding has been secured by the London Mathematical Society, Edinburgh Mathematical Society and Glasgow Mathematical Journal Trust to support the travel costs of PhD students/early career researchers who wish to attend the meeting in York. Please email Dimitra Kosta Dimitra.Kosta@glasgow.ac.uk (Scottish based students) or Emilie Dufresne emilie.dufresne@york.ac.uk (rest of UK) if you wish to attend the meeting in York and require financial support with the travel costs.

- How to get to the University of Glasgow
- Campus Map - We are located at C3
- How to get to the Mathematics & Statistics Building

Dimitra Kosta, University of Glasgow

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