## TORIC ARRANGEMENTS in BRISTOL

September 02..06, 2019

### Workshop's Venue

Room 1.58, **Queen's Building**, University of Bristol, University Walk, Bristol BS8 1TR (**Google maps**)

It is a good 20-minute walk from your hotel.

### Accommodation

**The Rodney Hotel**** (**Bristol BS8 4HY**) **has been booked for the participants (**Google maps**).

### Reaching The Rodney Hotel

From **Heathrow airport**: easiest is to take a National Express' coach to Bristol Bus & Coach Station, and a taxi to your hotel.

From **Bristol Airport**: taxi costs ~£35, **UBER** ~£25. Buses (Flyer Express or National Express) take you to the main train station.

Other **Airports in London: **use the public transport/taxi to **Paddington** train station to arrive at Bristol's main train station.

From **Bristol's main** **train station (Temple Meads)**: taxi costs < £10, a bus around £3 (you can buy tickets on-board).

### Taxi

Call VCars: 0117 925 2626, Arrow Cars: 01275 475000, or Book Arrow Cars Online , or use **UBER****. **

### Organisers' Contacts

Farhad Babaee (farhad.babaee@bristol.ac.uk) and Kevin Grace (kevin.grace@bristol.ac.uk)

### Timetable

### Titles and Abstracts

**Monday **

**9:15 – 9:30. **Introduction

**9:30 – 10:30**. Alex Fink, *Wonderful compactification of hyperplane arrangements and tropical compactifications.*

**11:00 – 12:00**. Giovanni Gaiffi, *Projective wonderful models for toric arrangements.*

Abstract. Given a toric arrangement A in torus T, it is possible to construct a projective wonderful model for the complement T-A. This is projective variety Y that contains T-A and such that the complement of T-A is a divisor D with normal crossings; the combinatorial properties of D are encoded by the initial combinatorial data. I will recall the main steps of this construction of Y providing some examples: one of the main ingredients is an algorithm that produces a fan whose associated toric variety is a suitable T- embedding. This is a recent work with Corrado De Concini (2018).

**2:00 – 3:00**. Diane Maclagan, *Tropicalisation of hyperplane arrangements, Bergman fans and toric ideals.*

**3:30 –4:30**. Yue Ren, *Zero-dimensional tropical varieties and tropical bases decertification.*

Abstract. In this talk, we discuss three approaches for computing zero-dimensional tropical varieties, each approach a non-Archimedean twist of a well-established counterpart in classical polynomial system solving. The first approach relates to root approximation, the second approach to root isolation, and the third approach to numerical linear algebra. We give a brief overview of the theory behind them and look at their strengths and weaknesses. Moreover, we highlight the difficulties of certifying whether a given generating set of a (not necessarily zero-dimensional) ideal forms a tropical basis, and discuss how the aforementioned techniques can be used to at least certify whenever a generating set fails to be a tropical basis. This talk contains joint works with Christian Eder and Tommy Hofmann (TU Kaiserslautern), Paul Goerlach and Avi Kulkarni (MPI MiS Leipzig), Jeff Sommars (formerly UI Chicago), and Leon Zhang (UC Berkeley).

**Tuesday**

**9:30-- 10:30.** Giovanni Gaiffi, *On the cohomology of projective wonderful models*.

Abstract. I will give a presentation by generators and relations of the cohomology ring of a projective wonderful model associated to a toric arrangement A in a torus T. This is a recent work with Corrado De Concini (2019). I will add some remarks on the dependence of the cohomology of T-A on the initial combinatorial data (work in progress with Corrado De Concini).

**11:00 – 12:00.** Emanuele Delucchi, *Matroidal polynomials for toric arrangements.*

Abstract. I will give an overview of the polynomial invariants of toric arrangements that appeared in the literature to date (including Tutte-like polynomials and their specializations). I will start with a mainly enumerative-combinatorial point of view and, time permitting, switch to a more structural frame of mind by illustrating some matroid-like objects that have been developed in order to give the aforementioned polynomials an abstract theoretical framework.

**2:00 – 3:00.** Jeffrey Giansiracusa, *Tropical modifications.*

**3:30 – 3:45. **Roberto Pagaria, *Arithmat: Sage implementation of arithmetic matroids and toric arrangements* (joint work with Giovanni Paolini).

**Wednesday**

**9:30 – 10:30.** Omid Amini, *Log-concavity of characteristic polynomials and the Bergman fan of matroids.*

Abstract. In this talk, we will go over the details of Huh—Katz’s proof of the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0.

** 11:00 – 12:00.** Emanuele Delucchi, *Stanley-Reisner rings for toric arrangements.*

Abstract. Recently, several authors have suggested generalizations of Stanley-Reisner rings from hyperplane arrangements (more generally, matroids) to toric arrangements (and, at times, more general arrangements or matroidal structures). I will explain these constructions showing that - in the case of toric arrangements - they are essentially equivalent, and I will summarize what structural properties of these rings are known. This is a joint work with Alessio D'Alì.

**2:00 – 3:00. **Karim Adiprasito, *Face rings, intersection numbers and Hodge theory.*

Abstract. I will discuss two central ideas to the Rota conjecture via face rings/Stanley-Reisner rings. 1. Intersection numbers of combinatorial classes can occasionally be computed using combinatorial means. 2. Why the Hodge-Riemann relations imply log-concavity of intersection numbers in certain cases.

**Thursday**

**9:30 – 10:30.** Roberto Pagaria, *Orlik-Solomon-type presentations for the cohomology algebra of toric arrangements.*

Abstract. I will talk about my joint work with Filippo Callegaro, Michele D'Adderio, Emanuele Delucchi, Luca Migliorini, where we give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias "toric arrangement"). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes, and builds on De Concini and Procesi's work on the rational cohomology of unimodular toric arrangements. As a byproduct we extend Dupont’s rational formality result to formality over Z. The data needed in order to state the presentation of the rational cohomology ring is fully encoded in the poset of connected components of intersections of the arrangement.

**11:00 – 12:00.** Alex Fink,* Z-matroids, Grothendieck groups and motivic definition of Tutte polynomials.*

**4:00 – 5:00. **Zeev Rudnick, *Quantum chaos, eigenvalue statistics and the Fibonacci sequence* (Room 1.15, Registration).

Abstract*. *One of the outstanding insights in the field of “Quantum Chaos” is a conjectural description of local statistics of the energy levels of simple quantum systems according to crude properties of the dynamics of classical limit, such as integrability, where one expects Poisson statistics, versus chaotic dynamics, where one expects Random Matrix Theory statistics. These insights were obtained by physicists in the last quarter of the 20th century (much of it in Bristol!). However, mathematicians are far behind in understanding the scope and validity of this theory. The first part of the lecture will be dedicated to an introduction to these conjectures, which I believe deserve to be better known in the mathematics community. In the second part, I will describe more recent work on statistics of the minimal gap between the first N eigenvalues for one such simple integrable system, a rectangular billiard having irrational squared aspect ratio. When the aspect ratio is the “golden ratio”, the problem involves some curious and entertaining properties of the Fibonacci sequence.

**Friday**

**9:30 – 10:00.** Neil Gillespie, *Equiangular Lines and Incoherent Sets.*

Abstract. A classical question in Euclidean geometry is to determine the maximum number of equiangular lines that can fit in any given dimension **d**. An absolute upper bound for this is **d(d+1)/2**; however, this bound is only known to be saturate in dimensions **d**=2,3,7 and 23, where the constructions are related to highly symmetrical objects (the regular hexagon, icosahedron, E8 lattice and Leech lattice respectively). Furthermore, it is a longstanding open question whether the absolute bound is saturated in any other dimension. By investigating *incoherence sets of equiangular lines* and the *incoherence bound, *I will prove that a set of lines saturates the absolute bound and the incoherence bound if and only if **d**=2,3,7 or 23.

In addition to the absolute bound, there exists an upper bound on the number of equiangular lines in dimension **d** with a specified angle **a**, known as the relative bound**. **I will also investigate sets of lines that saturate the relative bound and the incoherence bound, and show that such lines exist if and only if certain combinatorial designs exist. I will then characterise the known sets of lines that saturate the relative and incoherence bounds.

**10:00 – 10:30.** Ollie Clarke, *Combinatorics and geometry of matroid varieties.*

Abstract. A particularly interesting family of varieties are given by configurations of points on line arrangements. These varieties are examples of matroid varieties and we can use techniques from combinatorics and commutative algebra to answer questions which are typically very difficult. In this talk I will define point/line configurations, their associated matroid varieties and give some techniques which can be used to study them.

**10:50 – 11:30.** Leonid Monin, *Cohomology ring of toric bundles.*

Abstract. Using Bernstein--Kushnirenko theorem one can describe the cohomology ring of a smooth projective toric variety via volume polynomial on the space of polytopes. One can extend this description to the case of equivariant compactifications of a torus principal bundle. In my talk I will formulate a version of Bernstein Kushnirenko theorem for toric bundles, and will explain how it leads to a computation of their cohomology rings.

**11:30 – 12:00.** Kevin Grace, *Templates for representable matroids.*

Abstract. The matroid structure theory of Geelen, Gerards, and Whittle has led to an announced result that the highly connected members of certain minor-closed class of matroids representable over a finite field are mild modifications (known as perturbations) of frame matroids, which generalise graphic matroids. They introduced the notion of a template to describe these perturbations in more detail. In this talk, we will describe templates and discuss how templates are related to each other. We use templates to obtain results about representability, extremal functions, and excluded minors for various minor-closed classes of matroids.

### List of Participants

Karim Adiprasito, Hebrew University of Jerusalem

Omid Amini, École Polytechnique of Paris

Farhad Babaee, University of Bristol

Ollie Clarke, University of Bristol

Emanuele Delucchi, University of Fribourg

Alex Fink, Queen Mary University of London

Giovanni Gaiffi, University of Pisa

Jeffrey Giansiracusa, Swansea University

Kevin Grace, University of Bristol

Kevin Hughes, University of Bristol

Diane Maclagan, University of Warwick

Fatemeh Mohammadi, University of Bristol

Leonid Monin, University of Bristol

Roberto Pagaria, Scuola Normale Superiore

Yue Ren, MPI MIS Leipzig

### More info, things to do and restaurants

**Work Spaces**

- Breakout rooms: We can use Room 1.18 from 15.00 - 17.30 and Room 0.12 all day
- Maths library is in the Queen's Building accessible from the ground floor (Summer opening times: 8:45am - 4:45pm)

**Things to do and restaurants**

- Buffet lunch will be delivered for us in front of the seminar room
- Wednesday's dinner is at the Lido Restaurant
- Check out the Royal Fort Gardens in the University Campus where you can see a familiar arrangement!
- Nice restaurants near your Hotel: Indian (Nutmeg), French (Côte Brasserie), more options
- Other nice restaurants: Persian (Kuch , farther but more authentic Mithra), Caribbean (Caribbean Croft), more options
- Spots not to miss: Suspension Bridge, Cabot Tower, Brunel's SS Great Britain
- Visit Bristol Website!

### Group photo: The ultimate arrangement

### Reimbursement

You will be given a form (or you can download UK Banks/Non-UK Banks, Per diem) upon arrival to fill out, and you'll be required to send the forms with the receipts to the Heilbronn's address below, or just send a **readable scanned copy** of the forms and receipts to the following email address. For each dinner receipt we'll reimburse precisely £15 if you have filled out the per diem form. Don't forget to **sign** the documents please!

**Heilbronn's coordinators email**: heilbronn-coordinator@bristol.ac.uk

**Address**: Heilbronn Inst. for Mathematical Research, School of Maths, Uni. of Bristol, Howard House, Queen's Avenue, Clifton, Bristol BS8 1SN, UK

**Tel.** 0117 33 15260.

**Website**: https://heilbronn.ac.uk/contact-us/

A big thank you to the Heilbronn's coordinators, Jasmine Truman, Francoise Blake and Chloe Biddle, and indeed the Heilbronn Institute!