NZMRI Summer Workshop 2027
Photo by Amos Haring on Unsplash
Overview
The theme of the 2027 NZMRI Summer Workshop is Combinatorics, Probability and Symmetry.
This workshop features leading researchers giving lecture series on selected topics. The broad theme of the meeting is the interplay of combinatorics, probability, and symmetry. Topics covered include problems of geometric flavour with roots in analysis, probability and combinatorics, probabilistic and additive combinatorics, enumerative combinatorics, and more.
Location
The workshop will be held at the Function Centre at Tahunanui Beach, Nelson.
Lecture Series
The workshop features six outstanding researchers, each giving a series of three lectures. Click a title to read the abstract.
-
This course will explore techniques for the asymptotic enumeration of regular graphs, which have numerous applications beyond counting, including generating graphs with specified degree sequences, aiding in network design and testing, enhancing our understanding of network dynamics, and modeling real-world systems. Existing techniques can be roughly categorized into three methods: the configuration model and switchings, the use of generating functions combined with saddle point methods, and a relatively recent approach utilising degree switchings to derive contracting formulae. In this course, we will give an overview of all three methods.
-
Combinatorial enumeration is the art of counting discrete objects, either by listing them or by discovering a mathematical way to count them without listing. Sometimes the approximate (asymptotic) number can be found when the exact count is too hard. Methods can be classed as generative (listing the objects), combinatorial (bijections, decompositions, recurrences, etc), or analytic (generating functions, complex analysis, etc). We will provide a limited view of each of the three methods by way of examples that include recent research.
-
An important class of problems in combinatorial number theory asks how dense a subset of the integers must be to guarantee that it contains a fixed arithmetically defined pattern. One example of such a problem is to prove the best possible bounds in Szemerédi’s 1975 theorem on arithmetic progressions, i.e., to determine the size of the largest subset of {1, …, N} containing no k-term arithmetic progressions x, x + y, …, x + (k − 1)y with y nonzero. In the late 1990s/early 2000s, Gowers initiated the study of higher-order Fourier analysis and proved the first reasonable upper bounds in this problem for arbitrary k. He then posed the problem of doing the same for the multidimensional and polynomial generalizations of Szemerédi’s theorem (proven in 1978 by Furstenberg and Katznelson and in 1996 by Bergelson and Leibman, respectively). This has turned out to be very difficult, with little progress made until recently. In this minicourse, I will give a gentle introduction to higher-order Fourier analysis and some of the new tools that have gone into obtaining bounds for the sizes of sets lacking certain multidimensional and polynomial patterns.
-
In these lectures I will give a gentle introduction to some exciting recent developments in the study of lower bounds for the Ramsey numbers. The Ramsey numbers are fundamental quantities in combinatorics and, while they a priori have nothing to do with either probability or symmetry, the study of these numbers has uncovered a rich connection between these concepts.
-
A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of covering systems was initiated by Erdős in 1950 and over the following decades many beautiful questions and conjectures were posed regarding their properties, but until recently little was known. One particularly notorious question, due to Erdős, asks whether there exist covering systems with distinct moduli whose minimum modulus is arbitrarily large. This problem was resolved in 2015 by Hough, who showed that in any such system the minimum modulus is bounded. Another notorious question, due to Erdős, asks whether there exist covering systems with distinct odd moduli. This question is still open. The purpose of these lectures is to review important problems on covering systems and to give a gentle exposition of a modern technique that was recently used to resolve several of these problems. We hope that this technique will have further applications in other combinatorial settings.
-
I will give three lectures at the interface of knot theory, representation theory, combinatorics and AI. The first two lectures will centre around the Jones polynomial. This seems particularly fitting, as these lectures are named in Jones’ honour. I’ll discuss the fascinating open question of whether the Jones polynomial detects the unknot, and report on recent developments. There are fascinating connections to braid groups, representation theory and AI methods, which I’ll try to touch on. In the last lecture, I’ll change tack somewhat and discuss some interesting combinatorial problems that arise from moduli spaces in algebraic geometry.
Schedule
Arrival: by the evening of Saturday, 9 January
Departure: lunchtime Saturday, 16 January
Talks will start at 9am Sunday 10 January, and finish by lunchtime Saturday 16 January. Lectures will run each morning. Wednesday is a free day.
Lecture Notes
These will be available later.
Accommodation
Accommodation is reserved for NZ students at the nearby Tahuna Beach Holiday Park.
All other participants must arrange their own accommodation bookings. Some nearby options:
Meals
Dinner will be provided at the Function Centre for participants and their families on Sunday and Friday. Lunch will be provided on Monday, Tuesday, and Thursday. Wednesday is a free day.
Financial Support
NZ-based students
We will cover the cost of accommodation and catering for all NZ-based student participants.
Students at NZ universities are encouraged to apply for a Kalman Summer Scholarship to assist their participation at this workshop. The Kalman Summer Scholarships were established from a generous bequest in support of New Zealand mathematics from the late John Kalman, a former professor at the University of Auckland.
Four or five scholarships will be awarded to help cover costs of travel and incidental expenses (on top of accommodation and catered meals).
Applicants must:
- Be studying for an Honours, Masters, or PhD degree at a NZ university
- Submit an academic transcript and brief CV
- Provide a statement from their academic supervisor or mentor explaining how participation will benefit their development
Applications should be sent by email to the NZMRI Secretary, Prof. Rod Gover (r.gover@auckland.ac.nz), by 15th November 2026.
Scholarship recipients will be expected to submit a short report by mid-February 2027 describing what they gained from the workshop.
NZ-based academics
We will provide some support for NZ-based academics to attend. Cost of catering is covered for participants and families. We will make a contribution of at most $500 to accommodation costs. You must provide a GST receipt for your accommodation charges.
Participants outside New Zealand
You are most welcome to attend. However, the NZMRI's funding for the workshop does not permit support for participants based outside of New Zealand. If you need assistance, please email us.
Registration
Please complete the form below to register for the NZMRI Summer Workshop 2027. The organisers will be in touch closer to the workshop.
Travel Options
Nelson Airport is located 2 kilometres from the workshop venue. There are flights between Nelson and Auckland, Wellington and Christchurch. By car, Nelson is a pleasant 2-hour drive (110 km) from Picton, where there are car and passenger ferries connecting with Wellington.
Organisers
Additional Information
This page will be updated regularly. To obtain further information, email us.