Geometry seminars

Previous seminars

2023

April 26, 2023

Pluripotential theory on Berkovich spaces

Speaker: Léonard Pille-Schneider, ENS Paris

Abstract: Berkovich spaces are an analog of complex manifolds/complex analytic spaces when the field of complex numbers is replaced by a non-archimedean field. They in particular enjoy nice topological properties, which contrasts with the ultrametric nature of the field. The goal of this talk is to explain how to perform (pluri)potential theory on those spaces, similar to the classical complex one. If time permits I will also present some applications to problems coming from differential geometry.

March 29, 2023

Large complex structure limits, polytopes and optimal transport

Speaker: Jakob Hultgren, Umeå University

Abstract: A cylindrical surface of soap film suspended between two circles will eventually snap if the circles are moved far enough apart. Similarly, if the data defining a complex algebraic manifold is adjusted in certain ways the manifold will break. Particularly severe cases of this are called large complex structure limits and these have proved to be very important both in theoretical physics (string theory, super symmetry) and algebraic geometry. A key insight of Yan Soibelman and Fields medalist Maxim Kontsevich is that large complex structure limits can be understood as the manifold collapsing onto the boundary of a polytope or, more generally, a simplicial complex. After giving a non-technical account covering some of this background I will explain that in order to get a finer understanding of it one is led to a partial differential equation on the boundary of a polytope and present the first general existence and uniqueness results for this equation (joint work with Mattias Jonsson, Enrica Mazzon and Nick McCleerey).

March 16, 2023

On the q-analogue of codes, matroids, simplicial complexes and their relations

Speaker: Tovohery Randrianarisoa, Umeå University

Abstract: For a code with the Hamming metric one can associate a matroid using its parity check matrix. The independent elements of this matroid form a simplicial complex. Simplicial complexes associated to a q-matroid were shown to be shellable and this property helps to relate some properties of the simplicial complex to the codes. Indeed it was shown that there is a relation between Betti Numbers associated to the simplicial complex and the generalized weights of the linear code.

In the first part of this talk, we will give a brief review of the relations between these objects. A very interesting question is to find whether similar relations exist when we replace the Hamming metric with the rank metric. As a partial answer to this question, a q-analogue of some of the previous results are presented in the second part of the talk.

March 15, 2023

Almost Hermitian Structures on Conformally Foliated Lie Groups

Speaker: Emma Andersdotter Svensson, Umeå University

Abstract: I will present my master’s thesis as well as give a brief introduction to Riemannian geometry. In the thesis, we let (G, g) be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation F with minimal leaves and adapt an almost Hermitian structure J on G to the foliation F. In 2014, S. Gudmundsson and M. Svensson showed that the corresponding Lie algebra of G must then belong to one of 20 families. In the thesis, we classify such structures J which are almost Kähler (AK), integrable (I) or Kähler (K). Hereby, we construct 16 multi-dimensional almost Kähler families, 18 integrable families and 11 Kähler families.

February 15, 2023

Determinantal point processes associated with Bergman kernels

Speaker: Thibaut Lemoine, Université de Lille

Abstract: Determinantal point processes (DPP) are point processes whose correlation functions are expressed as the determinant of an integral kernel on the ambient space. They often appear in models of particles with repulsive interactions, such as fermions, and have been studied extensively in quantum mechanics and random matrix theory. In this talk, I will describe a class of such processes on compact Kähler manifolds, first described by Berman a decade ago, whose correlation kernel is the Bergman kernel of a positive line bundle. In particular, I will show how the asymptotic expansion of the Bergman kernel, initiated by Tian, Catlin and Zelditch in the late 90's, translates into a universality phenomenon for these DPP.

January 26, 2023

Odd distances in colourings of the plane

Speaker: James Davies, Cambridge

Abstract: We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd integral distance from each other.

2022

October 20, 2022

Geometric Dominating Sets, after Aichholzer, Eppstein and  Hainzl

Speaker: Signe Lundqvist, Umeå University

Abstract: A geometric dominating set in an n by n grid is a set of points such that every grid point lies on a line defined by two points in the set. The geometric dominating set problem asks for the smallest geometric dominating set in an n by n-grid. We can also consider the same problem with the additional requirement that no three of the points in the set lie on a line. A geometric dominating set such that no three points lie on a line is said to be in general position.

This talk will follow a paper by Aichholzer, Eppstein and  Hainzl, where the authors show both lower and upper bounds on the size of minimal geometric dominating sets in an n by n grid. Furthermore, the authors provide optimal geometric dominating sets in general position for grids of size up to 12 by 12. They also consider the same problem on the discrete torus, where they prove upper bounds on the size of geometric dominating sets.

On the number of points in general position in the plane, after Balogh and Solymosi

Speaker: Joannes Vermant, Umeå University

Abstract: In this seminar I will go through the proof of a theorem from a paper of Balogh and Solymosi, in which some progress on the following question is made. Suppose we are given a set S of n points such that no four points are collinear. How small can the largest subset S’ of S which is in general position be? Balogh and Solymosi proved the upper bound of n^(5/6 + o(1)). The proof is based on the hypergraph container method.

September 15, 2022

Minimal hyperplane covers of finite spaces and applications

Speaker: Istvan Tomon, Umeå University

Abstract: At least how many hyperplanes are needed to cover the finite space $\mathbb{F}_p^{n}$ if we require that the normal vectors of the hyperplanes span the whole space, and none of the hyperplanes is redundant? This question is related to a number of long-standing conjectures in linear algebra and group theory, such as the Alon-Jaeger-Tarsi conjecture on non-vanishing linear maps, the Additive Basis conjecture, and a conjecture of Pyber about irredundant coset covers. I will talk about some progress on this question, which in particular leads to a solution of the first conjecture in a strong form, makes substantial progress on the second conjecture, and fully resolves the third conjecture. This is based on a joint work with Janos Nagy and Peter Pal Pach.

June 2, 2022

Inequalities on Projected Volumes

Speaker: Eero Räty, Umeå University

Abstract: Given 2^n-1 real numbers x_A indexed by the non-empty subsets A \subset {1,..,n}, is it possible to construct a body T \subset R^n such that x_A=|T_A| where |T_A| is the |A|-dimensional volume of the projection of T onto the subspace spanned by the axes in A? As it is more convenient to take logarithms we denote by \psi_n the set of all vectors x for which there is a body T such that x_A=\log |T_A| for all A. Bollobás and Thomason showed that \psi_n is contained in the polyhedral cone defined by the class of `uniform cover inequalities'. Tan and Zeng conjectured that the convex hull of \psi_{n} is equal to the cone given by the uniform cover inequalities. 

We prove that this conjecture is nearly right: the closure of the convex hull of \psi_n is equal to the cone given by the uniform cover inequalities.

Joint work with Imre Leader and Zarko Randelovic.

May 12, 2022

Rigidity, Combinatorics and Topology

Speaker: James (Jim) Cruickshank, NUI Galway, Ireland

Abstract: In 1776 Leonard Euler made the following startling and somewhat provocative claim: “A closed spacial figure allows not changes, as long as it is not ripped apart.”

This statement has inspired much investigation in the intervening centuries with many connections to various mathematical, scientific and engineering disciplines. I will survey some of the highlights in this story, focusing on the case of polyhedra and related structures, and on topics of current interest to the geometric rigidity community. I will also report on recent joint work with Shinichi Tanigawa and Bill Jackson which resolves a conjecture of Bob Connelly and also extends the well known lower bound theorem for simplicial complexes.

April 7, 2022

Minimal Ramsey graphs for cliques

Speaker: John Bamberg, University of Western Australia

Abstract: Burr, Erdős, and Lovász, in 1976, introduced the study of the smallest minimum degree s(r,k) of a graph G such that any r-colouring of the edges of G contains a monochromatic clique of size k, whereas no proper subgraph of G has this property. Burr, Erdős, and Lovász were able to show the rather surprising exact result, that if r=2, then s(2,k) = (k-1)^2. The behaviour of this function is still not so well understood for more than 2 colours. In 2016, Fox, Grinshpun, Liebenau, Person, and Szabó showed that for r>2, s(r,k) is at most 8(k-1)^6 r^3. The speaker, together with Anurag Bishnoi and Thomas Lesgourgues, have recently used finite geometry to improve this bound.

March 24, 2022

The geometry of extremal Cayley graphs

Speaker: Valentina Pepe, La Sapienza, University of Rome

Abstract: The geometric aspect of extremal Cayley graphs is highlighted, providing a different proof of known results and giving a new perspective on how to tackle such problems. Some new results about extremal pseudorandom triangle free graphs are also presented.

February 3, 2022

When is a planar rod configuration infinitesimally rigid?

Speaker: Signe Lundqvist, Umeå University

Abstract: A rod configuration is a realisation of an incidence geometry as points and lines in the Euclidean plane. In this talk, we will introduce notions of rigidity for rod configurations and discuss approaches for determining whether a given rod configuration is infinitesimally rigid. We will generalise a result due to Whiteley.

Rod configurations generalise frameworks of graphs. Rigidity of graphs is well-studied. There is a combinatorial characterisation of the minimally rigid graphs, due to Pollaczek-Geiringer (1927) and later Laman (1970).

January 27, 2022

MacWilliams Identities for q-Polymatroids and Applications

Speaker: Eimear Byrn, University College Dublin (UCD)

Abstract: A q-polymatroid consists of a lattice of subspaces of a vector space endowed with a rank function that is both increasing and submodular. They were introduced independently by Gorla et al (2020) and Shiromoto (2019) as q-analogues of polymatroids and in reference to matrix codes. A number of invariants of codes are in fact matroid invariants, including the MacWilliams duality theorem. MacWilliams identities for classical matroids have been studied by a number of authors (e.g. Brylawski, Oxley, Britz, Shiromoto).  In this talk we will consider duality of q-polymatroids and will give a version of a MacWilliams theorem for q-polymatroids, using the characteristic polynomial. As as application of this result, we will state an Assmus-Mattson-like theorem that establishes criteria for the existence of weighted subspace designs arising from a q-polymatroid. This talk is based on joint work with Michela Ceria, Relinde Jurrius, and Sorina Ionica.

2021

December 21, 2021

Finite Semifields and their Invariants

Speaker: John Sheekey, University College Dublin

Abstract: Finite Semifields are  division algebras in which multiplication is not assumed to be associative. They have been studied in many contexts over the years; as algebraic objects, as the coordinatisation of projective planes with certain symmetries, and more recently as rank-metric codes. Determining the equivalence of two semifields is difficult, and so various equivalence invariants have been proposed and studied.

In this talk we will discuss some of these invariants, with particular emphasis on recent work with Michel Lavrauw on the tensor rank. The tensor rank is an invariant naturally arising from multilinear algebra, and can be viewed as a measure of multiplicative complexity. We present the first known examples of finite semifields of lower tensor rank than the finite field of the same size.

December 2, 2021

Hermitian self-orthogonal codes

Speaker: Simeon Ball, Universitat Politécnica Catalunya, Barcelona, Spain

Abstract: Let C be a [n,k]_{q^2} linear code, i.e. a k-dimensional subspace of the n-dimensional vector space over the finite field with q^2 elements F_{q^2}. The code C is linearly equivalent to a Hermitian self-orthogonal code if and only if there are non-zero a_i in the finite field with q elements F_{q} such that a_1 u_1 (v_1)^q+…+a_n u_n (v_n)^q=0 for all u and v in C. For any linear code C of length n over the finite field with q^2 elements, Rains defined the puncture code P(C) to be

P(C)={a=(a_1,…,a_n) in (F_q)^n : a_1 u_1 (v_1)^q+…+a_n u_n (v_n)^q=0 for all u and v in C }.

There is a truncation of a linear code C over F_{q^2} of length n to a linear over F_{q^2} of length r <= n which is linearly equivalent to a Hermitian self-orthogonal code if and only if there is an element of P(C) of weight r. Rains was motivated to look for Hermitian self-orthogonal codes, since there is a simple way to construct a [[ n,n-2k]]_q quantum code, given a Hermitian self-orthogonal code. This construction is due to Ketkar et al, generalising the F_4-construction of Calderbank et al.

In this talk, I will detail an effective way to calculate the puncture code. I will outline how to prove various results about when a linear code has a truncation which is linearly equivalent to a Hermitian self-orthogonal linear code and how to extend it to one that does in the case that it has no such truncation. In the case that the code is a Reed-Solomon code, it turns out that the existence of such a truncation of length r is equivalent to the existence of a polynomial g(X) in F_{q^2}[X] of degree at most (q-k)q-1 with the property that g(X)+g(X)^q has q^2-r distinct zeros in F_{q^2}.

(Joint work with Ricard Vilar)

November 25, 2021

"Graphical" introduction to parapolar spaces

Speaker: Anneleen De Schepper, Ghent University, Belgium

Abstract: In this talk I want to give a gentle introduction to parapolar spaces. These are are axiomatically defined point-line geometries,  introduced by Cooperstein in the 70ies to capture the behaviour of the Grassmannians of spherical buildings. Actually, all interesting known examples arise from buildings (not necessarily spherical). I will talk about some results on how to recognise a parapolar space given local information, starting from some graph related examples, and ending with an application in the Freudenthal-Tits magic square.