# Applied Algebra and Geometry Second meeting

## Date:

18-19 May 2017

## Location:

The seminar room, School of Mathematics, University of Edinburgh,

Room 5323, James Clerk Maxwell Building,

The King's Buildings,

Peter Guthrie Tait Road, Edinburgh, EH9 3FD

## Invited speakers:

- Sira Gratz, University of Glasgow
- Milena Hering, University of Edinburgh
- Fatemeh Mohammadi, University of Bristol
- Reidun Twarock, University of York
- Nelly Villamizar, Loughborough University
- Michael Wemyss, University of Glasgow
- Danyu Yang, University of Oxford

## Schedule:

### Thursday 18 May 2017

13:30 - 14:30 Fatemeh Mohammadi: *Combinatorial and Geometric View of the System Reliability Theory*

14:30-15:00 Coffee break

15:00-16:00 Nelly Villamizar: *Geometrically continuous spline spaces: construction and applications*

16:00-17:00 Danyu Yang: *Integration of rough paths*

Evening: informal dinner close to the train station

### Friday 19 May 2017

10:00-11:00 Reidun Twarock: Viruses and Geometry: *Group and Graph Theoretical Approaches for the Characterization of Viruses Structure and Function*

11:00-11:30 Coffee break

11:30-12:30 Michael Wemyss: *GV invariants, finite dimensional algebras and computer algebra*

12:30-14:00 Lunch break

14:00-15:00 Milena Hering: *Injectivity in Phase retrieval*

15:00-15:30 Coffee break

15:30-16:30 Sira Gratz: *Noncrossing partitions and thick subcategories*

## Local organizer:

Dimitra Kosta, University of Edinburgh.

## Conference photo:

## Abstracts

### Sira Gratz: Noncrossing partitions and thick subcategories

Noncrossing partitions are an interesting combinatorial object which play an important role in various topic of mathematics, such as in free probability. In this talk, we focus on their importance for the classification of thick subcategories in certain triangulated categories: Ingalls and Thomas have shown that the lattice of non-crossing partitions of a regular polygon with n+2 vertices is in one-to-one correspondence with the lattice of thick subcategories in the bounded derived category of representations of a Dynkin quiver of type A with n vertices. In joint work with Stevenson we provide an infinite version of this result by showing that the lattice of non-crossing partitions of the infinity-gon with a point at infinity is isomorphic to the lattice of thick subcategories in the bounded derived category of graded modules over the dual numbers.

### Milena Hering: Injectivity in Phase retrieval

In signal processing, often one cannot measure a complex vector directly, but instead one can measure the modulus of its inner product with a given spanning set. We study the question under what conditions on this spanning set the original vector can be uniquely obtained up to a global phase factor.

### Fatemeh Mohammadi: Combinatorial and Geometric View of the System Reliability Theory

Associated to every network/system there is a canonical ideal whose Hilbert series encodes the reliability of the system. We study various ideals arising in the theory of system reliability. Using ideas from the theory of orientations and graphic matroids we associate a polyhedral complex to our system so that the reliability formula can be read from this complex. Algebraically, this polyhedron resolves the minimal free resolution of these ideals. In each case, we give an explicit combinatorial description of terms in the reliability formula in terms of acyclic orientations of graph and the number of regions in the graphic hyperplane arrangement. This develops new connections between the theory of oriented matroids, the theory of divisors on graphs and the theory of system reliability.

### Reidun Twarock: Viruses and Geometry: Group and Graph Theoretical Approaches for the Characterization of Viruses Structure and Function

Viruses are remarkable examples of order at the nano-scale. Many viruses, including the common cold, package their genomes into protein containers that are organized according to icosahedral surface lattices. We present here group and graph theoretical approaches for the characterization of virus structure, and demonstrate that these techniques provide new insights into virus assembly and the structural transitions of viral capsid important for infection. We also show that these mathematical approaches can be exploited for the design of nanoparticle-based vaccines.

### Michael Wemyss: GV invariants, finite dimensional algebras and computer algebra

I will explain a counterexample, arising in joint work with Gavin Brown, which shows that Gopakumar--Vafa curve counting invariants (a tuple of integers) cannot distinguish between different types of surgeries called flops. Although the example is rigorous, and it boils down to comparing two 9-dimensional noncommutative rings, the construction of the counterexample is not obvious, and was only suggested after a serious of numerical computer algebra algorithms, most of which are still conjectural. I will also explain some of the numerical challenges, the motivation behind this question, and some other applications.

### Danyu Yang: Integration of rough paths

The theory of rough path develops a mathematical tool to model the evolution of controlled systems driven by highly oscillating signals, and the continuity theorem explains the convergence of controlled systems. There have been rapid developments in the field. This talk will focus on path integration. We build a connection between rough path theory and a noncommutative combinatorial Hopf algebra, and reduce the integration of rough path to a non-abelian version of the classical Young integration.