First Meeting: 11 January 2024 at 10:00 AM
Venue: Online
Class timing: Meet once a week
Brief Description: Tiling problems have a rich and long history with strong connections to all major areas of mathematics. In this course we will concentrate and learn two specific aspects: the Combinatorics and the algbera. We will begin by studying the combinatorics of lozenge tilings and then dive into the combinatorial group theory arising from their study: Specifically we will look at the Conway-Lagarias-Thurston tiling groups. We will see how the tools arising from the combinatorial group theory will be useful in answering questions of the form: Local move connectedness of tilings, the completion problem for tilings among many others.
Course Evaluation: The quality and depth of the lectures will constitute 50 percent of the grade. The rest will come from a project report. The deadline for the project is two weeks before the respective institute deadline (in this case ICTS-TIFR-CAM and IIT Kanpur). The project will be based on an exploratory topic decided upon the first couple of weeks but can be chosen from the following list (I am open to other suggestions as well of course):
(1) Running simulations for various tiling problems and related models.
(2) Proving large deviations principle for some simple models where it is well-known but has not been written down.
(3) Exploring what happens in higher dimensions.
(4) The cohomology of SFTs following results by Schmidt and Einsiedler.
(5) Tilings of Rd and a study of finite local complexity.
(6) A study of some of the open problems listed on Rick Kenyon’s website.
These will be disjoint from the topics being covered during the lectures but are deeply related.
Prerequisites: The course does not have any serious pre-requisites except basic familiarity with graduate algebra and combinatorics. The specific projects will involve picking up serious programming skills, PDEs, topology, probability, geometry and analysis or any other area of mathematics, computer science or physics that one may deem useful.
References:
(1) Lectures on random lozenge tilings by Vadim Gorin.
(2) A Pedestrian Approach to a Method of Conway, or, A Tale of Two Cities by James Propp.
(3) Tiling with polyominoes and combinatorial group theory by Conway and Lagarias.
(4) Groups, tilings and finite state automata and Conway’s tilings groups by Thurston.
(5) Tiling a polygon with two kinds of rectangles by Bodini and R ́emila.
Expected course outcome
(1) Familiarity with counting problems in tilings and Conway-Lagarias-Thurston tiling groups
Venue: Online
Class timing: Meet once a week
Brief Description: Tiling problems have a rich and long history with strong connections to all major areas of mathematics. In this course we will concentrate and learn two specific aspects: the Combinatorics and the algbera. We will begin by studying the combinatorics of lozenge tilings and then dive into the combinatorial group theory arising from their study: Specifically we will look at the Conway-Lagarias-Thurston tiling groups. We will see how the tools arising from the combinatorial group theory will be useful in answering questions of the form: Local move connectedness of tilings, the completion problem for tilings among many others.
Course Evaluation: The quality and depth of the lectures will constitute 50 percent of the grade. The rest will come from a project report. The deadline for the project is two weeks before the respective institute deadline (in this case ICTS-TIFR-CAM and IIT Kanpur). The project will be based on an exploratory topic decided upon the first couple of weeks but can be chosen from the following list (I am open to other suggestions as well of course):
(1) Running simulations for various tiling problems and related models.
(2) Proving large deviations principle for some simple models where it is well-known but has not been written down.
(3) Exploring what happens in higher dimensions.
(4) The cohomology of SFTs following results by Schmidt and Einsiedler.
(5) Tilings of Rd and a study of finite local complexity.
(6) A study of some of the open problems listed on Rick Kenyon’s website.
These will be disjoint from the topics being covered during the lectures but are deeply related.
Prerequisites: The course does not have any serious pre-requisites except basic familiarity with graduate algebra and combinatorics. The specific projects will involve picking up serious programming skills, PDEs, topology, probability, geometry and analysis or any other area of mathematics, computer science or physics that one may deem useful.
References:
(1) Lectures on random lozenge tilings by Vadim Gorin.
(2) A Pedestrian Approach to a Method of Conway, or, A Tale of Two Cities by James Propp.
(3) Tiling with polyominoes and combinatorial group theory by Conway and Lagarias.
(4) Groups, tilings and finite state automata and Conway’s tilings groups by Thurston.
(5) Tiling a polygon with two kinds of rectangles by Bodini and R ́emila.
Expected course outcome
(1) Familiarity with counting problems in tilings and Conway-Lagarias-Thurston tiling groups
- Teacher: Nishant Chandgotia
Credit Score: 4