Combinatorial Geometry and Number Theory
26 - 30 August 2024
First Conference in Lausanne
Combinatorial geometry is the study of extremal problems about finite arrangements of points, lines, circles, etc. Many questions have a strong intuitive appeal and can be explained to a layman. For instance, how many unit balls can be packed into a large box of a fixed volume? What is the maximum number of incidences between n points and n lines in the plane?
The works of László Fejes Tóth and C. Ambrose Rogers initiated new combinatorial approaches to some classical questions studied by Newton, Gauss, Minkowski, Hilbert, and Thue. They laid the foundations of the theory of packing and covering. At the same time, Paul Erdős continued bombarding the world with new questions of combinatorial geometry that even Euclid would appreciate.
Many of these problems turned out to be crucially important in coding theory, combinatorial optimization, computational geometry, robotics, computer graphics, etc. The explosive development of computer technology presented a powerful new source of inspiration for many areas of pure and applied mathematics. Combinatorial geometry is one of the fields that benefited most from this source.
Registration form
Deadline for registration: June 16, 2024.
Registration for the conference closes on June 16th 2024. If you are still interested in participating, please contact the organizer to ask for possible cancellations at the following email address: ilaria.viglino@epfl.ch
There is no registration fee for this Conference.
A one-week conference on aspects of combinatorial number theory and combinatorial geometry
The Conference will take place at the Bernoulli Center for Fundamental Studies, Located on the EPFL campus in Lausanne, Switzerland.
Address: Chemin des Alambics, Bâtiment GA, Station 5, 1015 Lausanne.
Geneva airport (GVA) is the nearest airport, around 50 minutes to Lausanne.
Zurich airport (ZRH) is located 2.5 hours by train from Lausanne.
Train timetables are available here. Detailed information on how to get to EPFL is available on the EPFL website (Coming to EPFL).
Below are some options for getting to the Bernoulli Center:
From Geneva airport:
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Take train to Morges, then bus line 701 from Morges Gare Ecublens VD, Champagne.
From Lausanne:
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Take metro line M2 from Lausanne-Gare station in the direction of Croisettes, get off at Flon and take M1. Travel on M1 Renens-Gare, Unil-Sorge walk 8 minutes to Bernoulli Center.
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Take bus line 1 from Lausanne-Gare station, travel in the direction EPFL/Colladon, get off at Ecublens VD, Champagne and walk 5 minutes to Bernoulli Center.
Facilities
Located on the 3rd floor of the building, the Center can be reached by elevator.The doors of the building are open from 07h00 to 19h00. Outside those hours, participants need an access card. The following spaces are available in the Center:
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One conference room with a maximal capacity of 75 attendees.
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10 working spaces in three separate offices.
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Three common rooms to be used for meetings, as break-out rooms, lunch and coffee breaks.
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A terrace for coffee breaks and lunches.
Free WIFI in the Bernoulli Center.
Accomodation
The EPFL campus has two hotels to welcome you:
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SwissTech Hotel, on EPFL campus.
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Starling Hotel, next to EPFL campus.
In addition, there are hotels and apartments next to EPFL campus and in
Lausanne to suit all budgets.
Hotels in Lausanne provide a free transit card to travel in and around the city for
the duration of your stay (max 15 days). You must ask for the card at the hotel
reception. Moreover, you can use public transport free of charge in Lausanne
(metro & bus) on the day of your arrival to reach your hotel; but you will need to
show your hotel booking and a proof of identity.
Schedule
Monday
8:15 - 9:00: Registration and Welcome session 🫶🏻
9:00 - 10:00: Igor Wigman (King's College London)
10:00 - 10:30: Coffee break ☕️
10:30 - 11:30: Marcelo Campos (University of Cambridge)
11:30 - 12:30: Claire Burrin (Universität Zürich)
12:30 - 14:00: Lunch break 🍽
14:00 - 14:30: Seungki Kim (University of Cincinnati)
14:30 - 15:00: Alessandro Gambini (Sapienza University of Rome)
15:00 - 15:30: Coffee break ☕️
15:30: Brainstorm session
Date
26/08
Date
27/08
Tuesday
9:00 - 10:00: Carlo Sanna (Polytechnic University of Turin)
10:00 - 10:30: Coffee break ☕️
10:30 - 11:30: Andreas Strömbergsson (Uppsala University)
11:30 - 12:30: Phong Nguyen (École normale supérieure de Paris)
12:30 - 14:00: Lunch break 🍽
14:00 - 14:30: Emil Rugaard Wieser (University of Copenhagen)
14:30 - 15:00: Henry Bambury (Ecole Normale Supérieure de Paris)
15:00 - 15:30: Coffee break ☕️
15:30: Brainstorm session
Wednesday
9:00 - 10:00: Anders Södergren (Chalmers University of Technology)
10:00 - 10:30: Coffee break ☕️
10:30 - 11:30: Mahbub Alam (Uppsala University)
11:30 - 12:30: Jiyoung Han (Korea Institute for Advanced Study)
12:30 - 14:00: Lunch break 🍽
19:30: Social dinner 🥂
Date
28/08
Date
29/08
Thursday
9:00 - 10:00: Rachel Newton (King's College London)
10:00 - 10:30: Coffee break ☕️
10:30 - 11:30: Matthew deCourcy-Ireland (Stockholm University)
11:30 - 12:30: Igor Shparlinski (The University of New South Wales)
12:30 - 14:00: Lunch break 🍽
14:00 - 14:30: Nathan Hughes (University of Exeter)
14:30 - 15:00: Sara Chari (St. Mary's College of Maryland)
15:00 - 15:30: Coffee break ☕️
15:30: Brainstorm session
Date
30/08
Friday
9:00 - 10:00: Maria-Romina Ivan (Magdalene College Cambridge)
10:00 - 10:30: Coffee break ☕️
10:30 - 11:30: Samantha Fairchild (Max Planck Institute)
11:30 - 12:30: Talk 15
12:30 - 14:00: Lunch and farewell 👏🏼
Speakers
Talks and abstracts
Invited speakers
Title: Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles
Name: Igor Wigman
Abstract
Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius-R disc by its area is O(R ). One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square-root cancellation of sums of these random variables. We examine this heuristic by studying how these lattice points interact with one another and prove that their autocorrelation is determined in terms of a random model. Additionally, it is shown that lattice points near the boundary which are "well separated" behave independently. We also formulate a conjecture concerning the distribution of pairs of these lattice points.
1/2+o(1)
Title: New Lower Bounds for Sphere Packing
Name: Marcelo Campos
Abstract
In this talk I'll show the existence of a packing of identical spheres in ℝ with density
as d → ∞. This improves the best known asymptotic lower bounds for sphere packing density. The proof uses a connection with graph theory and a new result about independent sets in graphs which is proved probabilistically.
This is joint work with Matthew Jenssen, Marcus Michelen and Julian Sahasrabudhe.
(1-o(1))
dlogd
2
d+1
d
Title: TBA
Name: Alessandro Gambini
Abstract
Title: TBA
Name: Rachel Newton
Abstract
Title: TBA
Name: Andreas Strömbergsson
Abstract
Title: TBA
Name: Phong Nguyen
Abstract
Title: TBA
Name: Anders Södergren
Abstract
Title: TBA
Name: Matthew deCourcy-Ireland
Abstract
Title: TBA
Name: Carlo Sanna
Abstract
Title: TBA
Name: Jiyoung Han
Abstract
Title: Pairs of saddle connections in translation surfaces
Name: Samantha Fairchild
Abstract
A translation surface is a collection of polygons in the plane with parallel sides identified by translation to form a Riemann surface with a singular Euclidean structure. A saddleconnection is a special type of geodesic, and the set of saddleconnections form discrete subsets of the Euclidean plane. Studying the set of saddle connections is a long standing problem in the field of translation surfaces. I will discuss problems and results related to counting pairs of saddle connections. This talk will include some computer experiments, number theory, dynamics, and geometry.
Title: TBA
Name: Maria-Romina Ivan
Abstract
Title: Geometric properties of points on modular hyperbolas
Name: Igor Shparlinski
Abstract
It is well know that counting integer points under a hyperbola xy ≤ N, x,y ≥ 1, is equivalent to the Dirichlet divisor problems. In this talk we will discuss various questions concerning geometric properties of the set of points on a modular hyperbola xy = a mod q, 1 ≤ x,y ≤ q. In fact, this is closely related to the question about the sum of the divisor function over an arithmetic progression.
These questions include: equidistribution and the covering radius, the number of distinct distances they define, and the number of vertices in their convex hull. Some of these questions have been answered reasonably satisfactory, some are still widely open and even heuristically are poorly understood.
Title: Rational points on spheres
Name: Claire Burrin
Abstract
I will discuss recent results on the distribution of rational points on n-dimensional spheres that build on the theory of modular forms.
Contributed speakers
n-1
Title: Equidissections of the cross-polytope
Name: Emil Rugaard Wieser
Abstract
Monsky's theorem states that a square may only be dissected into an even number of triangles with equal area. The proof relies on a coloring of the real plane by a 2-adic valuation. Work by Mead and Kasimatis generalize this to many classes of polytopes (in particular regular polygons and n-cubes) by colorings of spaces by p-adic valuations, often employing useful affine transformations to get results. It is shown that the n-dimensional cross-polytope may only be dissected into a multiple of 2 simplices of equal n-volume generalizing the square case by Monsky and the tetrahedron by Kasimatis and Stein, with the proof strategy being largely an extension of the latters. With the work by Mead and Kasimatis this leaves only equidissections of regular polytopes in dimensions 3 and 4. Partial results on these (due again to Kasimatis in dimension 3) and a potential proof strategy for a full result employing 2-adic valuations extended to the quaternions will be discussed.
Title: Effective Counting and Spiralling of Diophantine Approximates in Number Fields
Name: Nathan Hughes
Abstract
Using techniques from ergodic theory and the geometry of numbers, we will give error terms for the asymptotic count of rational approximates to almost all linear forms over the Minkowski space of any given number field, and describe their distribution. The proof relies on a maximal inequality of a class of stochastic processes and a variant of Rogers' mean value formula. We will also indicate a possible proof direction for the case of 1x1 linear forms over number fields with complex places.
Title: TBA
Name: Henry Bambury
Abstract
Title: Mathematical Origami Constructions
Name: Sara Chari
Abstract
Origami is the art of folding paper into various patterns without cutting or tearing the paper. By viewing the paper as a complex plane, we record all intersection points to construct mathematical origami sets, and one may ask whether the resulting set is a lattice or a dense set. This idea is generalized to construct lattices in higher dimensions. We will also discuss the symmetry group of the resulting set together with the lines through each point at the allowed angles.
Financial support
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