GAPT Seminar
LMS Hardy Lectureship Talk 2025
Date: 27th June 2025
Time: 10:15am – 4:15pm
Location: Lecture Theatre 2.26, Abacws Building, Senghenndydd Road, CF24 4AG
If you would like to attend, please complete the registration form linked here by 20th June 2025.
The Cardiff University School of Mathematics is excited to be hosting Professor Emily Riehl (Johns Hopkins University) during her UK tour as the named LMS Hardy Lecturer 2025. The day will begin in the morning with an introductory talk aimed for students. There will then be two colloquium talks in the afternoon: the first by Prof. Grigory A. Garkusha and the second by Prof. Emily Riehl. Participants are invited to attend both the morning and the afternoon, or if they wish just the afternoon; please make this specification on the registration form above.
Schedule:
| 10:15 | Welcome for morning attendees |
| 10:30 | Overview of category theory (a lecture for students) |
| 11:30 | Lunch |
| 13:15 | Welcome for afternoon attendees |
| 13:30 | Prof. Grigory A. Garkusha |
| 14:30 | Break |
| 15:00 | Hardy Lectureship Talk, Prof. Emily Riehl |
| 16:15 | Reception |
The Hardy Lectureship Talk by Emily Riehl
Title: Homotopy types as homotopy types
Abstract: Classically, a “homotopy type” records the information in a topological space that is captured by its homotopy groups. The classical homotopy category of spaces is defined as a quotient of the ordinary category of topological spaces. Quillen famously proved that the homotopy category can also be understood as a quotient of the (better behaved) category of simplicial sets, with Kan complexes encoding homotopy types. Voevodsky extended this result to show that homotopy type theory (i.e., Martin-Löf’s dependent type theory plus the univalence axiom) can be modelled by the category of simplicial sets, again with the Kan complexes encoding homotopy types, now thought of more synthetically the primitive objects in this formal system. In this talk, I’ll highlight some constructions and proofs that are standard in homotopy type theory but less familiar in classical homotopy theory, focussing in particular on the principle of “path induction.”
Talk by Grigory A. Garkusha
Title: Motives, categories and stable homotopy types
Abstract: Algebraic geometry studies algebraic varieties which are of principal importance. They are relatively easy to understand since they are just defined by polynomial equations. Homotopy theory is a considerably newer area of mathematics, being an important branch of algebraic topology. It studies objects which are preserved under operations such as bending and stretching. Both areas appear naturally in quite a lot of subjects in theoretical physics, coding theory and computer sciences. Motivic homotopy theory is a blend of algebraic geometry and homotopy theory. Its primary object is to study algebraic varieties from a homotopy theoretic viewpoint. Besides quite spectacular applications such as the solution of the Milnor conjecture by Voevodsky and the Bloch-Kato conjecture by Rost and Voevodsky in algebraic geometry, motivic homotopy theory leads to explicit computations of various stable homotopy types in algebraic topology and algebraic K-theory by means of “motives and transfers”. These computations also lead to a variety of categories of motives which recover, in particular, classical stable homotopy theory of spaces. The resulting interplay of algebraic geometry, homotopy theory and enriched category theory provides a fascinating glimpse of the unity of mathematics.
What is the Hardy Lectureship award?
Named after former Society President and De Morgan Medallist, G. H. Hardy, the LMS Hardy Lectureship is awarded in odd-numbered years to a distinguished overseas mathematician; they are then invited to visit the UK for a period of about two weeks. This tour includes at least six lectures on varying topics at different venues in the UK. For more information, see their website here.
On 27th June, the 2025 LMS Hardy Lecturer, Professor Emily Riehl, will be visiting and delivering a lecture at Cardiff University.
Who is Prof. Emily Riehl?
Professor Emily Riehl works on higher category theory, homotopy type theory, and computer formalization at Johns Hopkins University, where she is the Kelly Miller Professor of Mathematics. Her many publications include the popular textbook Category Theory in Context. Emily was also a founding board member of Spectra, an association for LGBTQ+ mathematicians.