Geometry

Geometry is the branch of mathematics concerned with properties of space. Our research covers a wide range of areas in geometry. More information can be found on the personal pages of our group members and further down on this page.

We are generally very open to collaboration, so if you want to do a research visit, write a thesis project or attend one of our seminars, do not hesitate to get in touch!

Our research

Combinatorial geometry

Combinatorial Geometry is the branch of geometry which studies combinatorial properties of discrete geometric objects. This involves the study of arrangements of points, lines, balls, polygons, etc. Some of the major problems in this area are:

  1. What is the most efficient way to pack unit balls in the d-dimensional space? (Kepler's problem, 17th century)

  2. At most how many unit distances can be determined by n points in the plane? (Erdős 1946)

  3. How can we arrange n points in the unit square without creating a triangle with small area? (Heilbronn 1940's)
  
In our group, one of the topics studied is the Ramsey theoretic properties of families of simple shapes. That is, given a collection of n geometric objects (e.g. disks, boxes, convex sets, curves), how many pairwise disjoint elements can we find, assuming the collection does not contain a large family of pairwise intersecting elements.

Complex geometry

Complex geometry is a branch of geometry studying complex manifolds. These manifolds often arise as the zero sets of polynomials and tools from differential geometry, partial differential equations and algebraic topology are applied to study their properties.  The field is partly driven by its interaction to other fields of mathematics (algebraic geometry, number theory) and theoretical physics, but it also have several central open problems of its own. Three of these are

1. The Hodge Conjecture, stating that all cohomology classes of certain types come from complex sub manifolds (this is one of the Clay Millennium Problems).

2. The Yau-Tian-Donaldson Conjecture, asking under what conditions certain Riemannian metrics on complex manifolds (Kähler metrics) can be deformed into metrics satisfying special curvature properties, for example Einstein metrics.

3. The Strominger-Yau-Zaslov Conjecture, regarding special Lagrangian Torus fibrations in Calabi-Yau manifolds.

The latter two are closely related to a partial differential equation: the Monge-Ampere equation, which is a central topic of research at Umeå University.

Geometry and symmetries of differential equations

Differential equations are equations that impose relations between an unknown function and its derivatives; the solution of a differential equation is a family of functions. Differential equations are ubiquitous in pure mathematics and a large variety of applications spanning from physics and biology to computer science. In many applications the functions that constitute the solutions take values on some algebraic variety or satisfy a given symmetry. We explore how knowledge about the geometry and the symmetries of the differential equations can be used to characterise and construct solutions to differential equations, and how it features in applications, e.g., conservation laws in physics, identifiability and stability of dynamical systems, and construction of explicit solutions in mathematical biology.

Geometric deep learning

In modern machine learning research, geometry is playing an increasingly prominent role. In many applications the data has an intrinsic geometric structure or is invariant under some symmetry transformations. Important examples of geometric structures include graphs, which can be used to describe, e.g., molecular structures or computer networks; groups, which represent data on spaces with global symmetries, e.g., climate data or the cosmic microwave background which are defined on the surface of a sphere; and manifolds, which represent more general spaces, e.g., space-times in General Relativity. Geometric deep learning is the research field concerned with the construction and analysis of machine learning models which incorporate various geometric structures as inductive biases to improve performance and stability.

Incidence geometry, finite geometry and coding theory

Incidence geometry is the branch of geometry concerned with the underlying combinatorial incidence relation behind a geometric structure, The geometric structure can be embedded in a geometric space or be a geometric space itself. Geometry is often interpreted in a very wide sense (in the spirit of Klein and Tits); as a combinatorial incidence structure with a flag-transitive action of the symmetry group. ​

Finite geometry, or Galois geometry is geometry when studied over a finite field. In finite geometry classical topics such as projective and axiomatic geometry are still important and there are connections to algebraic geometry, combinatorics, linear algebra, and coding theory.​

Algebraic coding theory is concerned with the construction and analysis of good codes for digital communication. Examples of applications of such codes are error-correction over noisy channels, network coding, storage codes and post-quantum cryptography.

Structural rigidity and kinematics

Structural rigidity and kinematics is a branch of geometry studying the motions of articulated structures - rigid bodies connected in joints or hinges - in geometric spaces. Combinatorics, algebra and group theory are combined to solve geometric problems, motivated by applications in physics, chemistry, robotics and architecture. The theory was initiated by J.C. Maxwell in the 1860's. A lot of progress have been made since, but the mathematics describing the motions of articulated structures is not yet as well-developed as the mathematics describing the rigid motions. For example, a long-standing open question is the characterization of the rigid bar and joint frameworks in 3-dimensional Euclidean space. Some of the major results in the area are:​

1. Geiringer-Laman's theorem, giving a necessary and sufficient combinatorial criteria for when a bar and joint structure is rigid in the Euclidean plane;​

2. Cauchy's theorem stating that a three-dimensional convex polyhedron constructed with rigid plates for its faces, connected by hinges along its edges, forms a rigid structure;​

3. Kempe's universality theorem saying that given an algebraic curve in the plane it is always possible to find a mechanism consisting of bars and joints that traces the curve. In other words, there is a bar and joint linkage that writes your name!​

Seminars