Geometry Lab at Jyväskylä

The new Geometry Lab in the Department of Mathematics and Statistics offers master’s students the opportunity to gain hands-on experience with mathematical research and the activities of research groups. The goal of the initiative is to strengthen collaboration and communication skills within both national and international networks, lower the barrier between students and staff, and highlight the importance of scientific impact. The Geometry Lab is based on an internationally recognized model that has already been implemented at several universities in the United States.

How does the lab work in practice?

The main purpose of the projects is to investigate and illustrate a concept, phenomenon or theory emerging from mathematical research using technology, and thereby arrive at mathematical conclusions regarding the given topic.

The project lasts an entire semester (some can be extended to full academic year) and is carried out by a project group. The project group consists of one faculty member (professor or lecturer), one post-doc or doctoral student (from the professor’s/lecturer’s
research group), and at least three master’s students.
The project group is led by a professor or lecturer assigned to the group. They provide the project topic and explain
why researching it is important and how the topic could be approached (work time on average 1–2 hours per month depending on the situation
and willingness). The active leader of the project is a post-doc or doctoral student assigned to the group (work time
averaging 1–2 hours per week, depending on the situation).
Master’s students serve as active participants, developers, and idea generators for the project (work time averaging
1–2 hours per day on each weekday, depending on the situation). Students receive 5 credits/semester for the project work completed.
Good project work produces well-founded mathematical claims that appear to hold true based on simulations and modeling. The main goal is not to prove theorems but to provide evidence for plausible claims.
The final output of the project is a report and a poster on how these conclusions were reached. The project outputs are used for science outreach in elementary schools, high schools, and science pop-up events.

The Geometry Laboratory introduces master’s students to the world of mathematical research and collaboration at the University of Jyväskylä.

Goals of the Geometry Lab

matematiikan opiskelu — Students apply mathematics in practical situations and draw their own mathematical conclusions on a chosen topic.

The curriculum emphasizes the development of students’ teamwork skills, the societal significance of various academic disciplines, and the ability to apply knowledge. The Geometry Lab meets these requirements well because:

Students have the opportunity to participate in mathematical research. Projects involve collaboration among students and with the doctoral candidate and professor in the group. There is also the opportunity for collaborative projects with American students. Students’ professional skills will also develop.
Students will explore ways to illustrate the research topic so that anyone can understand its purpose and concept. The results of the projects and the reports produced from them are used for science communication to the general public, so students learn about the importance of scientific advocacy and its role in society.
Project work also gives doctoral students and professors the opportunity to practice teamwork skills, team leadership, and the implementation of research projects. Thus, the projects serve to build a sense of community at all levels.
The projects develop skills required in future professional life, including meeting and negotiation skills, presentation skills, interpersonal skills, teamwork skills, networking skills, and problem-solving abilities.

Available projects

Topics includes for example geometry and geometric analysis, metric geometry, fractal geometry, quasiconformal mappings, and interdisciplinary applications of geometry in mathematics and the sciences.

This project explores slowness surfaces, which play a role in physics, partial differential equations, and algebraic geometry. The main objective is to understand these surfaces from an algebraic perspective and to develop methods for visualizing them. During the project, participants will become familiar with various concepts of algebraic varieties and surfaces (real, complex, affine, projective, etc.) and the relationships between them. The emphasis is on visualization; from the algebraic viewpoint, it is sufficient to gain a solid understanding of the relevant definitions and concepts.

This project introduces hyperbolic buildings and their boundaries. Hyperbolic spaces are generalizations of Euclidean space in which the parallel postulate no longer holds, and triangles exhibit significantly different geometric behavior. Hyperbolic buildings are spaces constructed by gluing together a large number of hyperbolic spaces, called apartments, each of which is regularly tiled by chambers.

The project begins with an exploration of models of hyperbolic space and aims to visualize their regular tilings in various ways. The next goal is to develop visualizations of hyperbolic buildings themselves and, ultimately, to provide a description of the boundary of a building.

Prerequisites: Vector Analysis; knowledge of Metric Spaces and Differential and Riemannian Geometry is beneficial.

This project explores Kakeya sets, geometric sets that contain a unit line segment in every allowed direction. Although such sets might be expected to be large, a classical result of Besicovitch shows that they can have arbitrarily small area. The project investigates how this phenomenon changes when the allowed directions are restricted to a fractal subset of directions, such as a Cantor set. Participants will learn about geometric constructions of Kakeya sets, basic fractal geometry, and different notions of dimension (Hausdorff, Minkowski, and Assouad dimensions), with an emphasis on visualization and geometric intuition. The ultimate goal is to understand how the fractal structure of the direction set influences the existence of highly compressed Kakeya-type constructions, potentially extending these ideas to higher dimensions. No prior knowledge of Kakeya sets or fractal geometry is required.

If you are interested in any of the above topics, please contact Tuomo Äkkinen by email (tuomo.akkinen@jyu.fi) or stopping by at AgB321.6 to get your project going!