## 6th meeting, 10th September 2018,

## Swansea University

## Date:

10 September 2018

## Location:

Department of Mathematics, Swansea University, Singleton Campus, Faraday Tower room: 314

## Local organiser

Nelly Villamizar, Swansea University

## Schedule

13:00 - 13:40 Hamza Fawzi (University of Cambridge)

13:40 - 14:20 Dimitra Kosta (University of Glasgow)

14:20 - 14:40 Coffee break

14:40 - 15:20 Heidi Dahl (SINTEF, Norway)

15:20 - 16:00 Dante Kalise (Imperial College)

16:00 - 16:40 Samet Sarioglan (Hacettepe University)

## Abstracts

**Heidi Dahl: ****From Big Data to Smart Data: Geometric modelling for Big Data and Artificial Intelligence**

Abstract: In ANALYST, a new research project funded by The Norwegian Research Council, we are combining geometric modelling with artificial intelligence to visualize and analyse big datasets from medical imaging and geospatial modelling. In this presentation we will show some early results, using volumetric Locally Refined (LR-) splines to model data from Magnetic Resonance Imaging (MRI) scans. Evaluating the LR-model at each level of the approximation, we investigate how the fit of the model evolves. In the next stage of the project, such LR-models will be the basis for advanced analytics such as segmentation and feature detection.

**Hamza Fawzi****: A lower bound on the positive semidefinite rank of convex bodies**

The positive semidefinite rank of a convex body C is the size of its smallest positive semidefinite formulation. We show that the positive semidefinite rank of any convex body C is at least sqrt(log d) where d is the smallest degree of a polynomial that vanishes on the boundary of the polar of C. This improves on the existing bound which relies on results from quantifier elimination. The proof relies on the B\'ezout bound applied to the Karush-Kuhn-Tucker conditions of optimality. We discuss the connection with the algebraic degree of semidefinite programming and show that the bound is tight (up to constant factor) for random spectrahedra of suitable dimension. Joint work with Mohab Safey El Din.

* Dante Kalise*:

**On Polynomial Systems Arising in Model-based Optimization and Control**

Abstract: in this talk we review some classical problems arising in optimal control theory which require the solution of polynomial systems. These include the characterisation of equilibrium points in nonlinear dynamics, the pole assignment problem, the solution of Algebraic Riccati Equations in linear-quadratic control, and the identification of switching structures in minimum time control. While there is a vast amount of literature on the efficient numerical solution of such systems by means of iterative methods, this approach does not preserve the structural properties of the control system. In this talk, we discuss the application of Groebner basis for the solution of polynomial systems arising in minimum time control, and discuss the advantages of this technique as opposed to standard discretization-based methods.

**Dimitra Kosta: ****Log canonical thresholds in birational geometry and applications**

Abstract: There has been a lot of current interest in log canonical thresholds in algebraic statistics due to the work of Watanabe connecting them to singular statistical models and model selection. In this talk I will talk about how one can compute log canonical thresholds for del Pezzo surfaces with Du Val singular points. Log canonical thresholds are also connected with the existence of Kahler-Einstein metrics on Fano varieties. As a consequence I will show that del Pezzo surfaces of degree 1 with Du Val singular points of type An with n less than or equal to 6 admit a Kahler-Einstein metric answering a question initiated by Tian and Yau.

**Samet Sarıoğlan: **** Algebraic Structure of Generalized Splines**

Abstract: Splines are piecewise polynomial functions over polyhedral complexes that agree up to a smoothness degree at the intersection of faces. They increase the power to control of the shape of a surface. Splines are used in several areas related with industry, computer based animations and geometric design. While splines are defined on polyhedral complexes, generalized splines are defined on edge labeled graphs with a base ring. The set of generalized splines on an edge labeled graph has ring and module (over the base ring) structure algebraically. We study the module structure of generalized splines.

In this talk, the motivation and basic definitions of generalized spline theory will be given. We also introduce some results related to the freeness and bases of generalized spline modules on arbitrary graphs and base rings. These results are obtained from joint work with Selma Altınok Bhupal.

## Visitor information

There are directions to the university here http://www.swansea.ac.uk/the-university/location/directions/

The talks will be in room Faraday Tower 314 (building 8.2 on the campus map that you can find here).

If you arrive before, the Maths Department is located on the second floor of the Talbot Building (number 8.3 on the campus map). The best way to access the Talbot Building is via the Faraday Building. Note that the main entrance of the Faraday building is actually already on the first floor, so you only need to ascend one set of stairs.

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## Sponsors:

We are grateful for the financial support from the **Glasgow Mathematical Journal Learning** and Research Support Fund, from the **Edinburgh Mathematical Society, ** the **London Mathematical Society**.