Info: The AGDM seminar is a joint seminar of the University of Vienna and TU Wien. In winter semesters, we meet on Tuesdays from 15:00 to 16:30 at the University of Vienna. In summer semesters, we meet on Tuesdays from 15:15 to 16:45 at TU Wien.
Since 2025, we use a mailing-list to advertise the seminar. You can register here.
Place: University of Vienna, Oskar-Morgenstern-Platz 1, BZ 09 (first room on the right on the 9th floor)
Time:Tuesday, 15:00-16:30
| Date: | Tuesday 02.12.2025 |
| Title: | The $1/4$-phenomenon of placement probabilities of tilings in the Aztec diamond |
| Speaker: | Marcus Schönfelder (University of Vienna) |
| Abstract: | Let $R$ be a region in the plane consisting of a union of unit squares which align with the square grid $Z^2$. A domino tiling of $R$ is a covering of the region with $1\times 2$- or $2\times 1$-rectangles called domino tiles. Given $R$, enumerative combinatorics asks for the number of domino tilings. Naturally, This number heavily depends on the underlying region $R$. We consider domino tilings of the Aztec diamond, a well studied model region in this discipline. Using the "Domino Shuffling" algorithm introduced by Elkies, Kuperberg, Larsen, and Propp (1992), we are able to generate domino tilings uniformly at random. In this talk, we investigate the probability of finding a domino at a specific position in such a random tiling. We prove that this placement probability is always equal to $1/4$ plus a rational function, whose shape depends on the location of the domino, multiplied by a position-independent factor that involves only the size of the diamond. This result leads to significantly more compact explicit counting formulas compared to previous findings. As a direct application, we derive explicit counting formulas for the domino tilings of Aztec diamonds with $2\times 2$-square holes at arbitrary positions. |
Maintaining a respectful environment is essential to fostering meaningful dialogue and intellectual growth. Participants are expected to refrain from any form of disrespectful or inappropriate behaviour, including offensive comments, harassment, or disruptive conduct. Questions and contributions should be constructive, relevant to the topic, and posed in a professional manner that encourages healthy academic exchange. Harassment of any kind—including verbal, moral or physical—will not be tolerated, and all attendees are urged to uphold these principles to ensure a safe and welcoming atmosphere for everyone.
| 09.12.2025 | Alin Bostan and Anastasia Matveeva | |
| 16.12.2025 | Atsuro Yoshida | |
| 13.01.2026 | Abdulhafeez Abdulsalam | |
| 20.01.2026 | Markus Reibnegger | |
| 27.01.2026 | Mona Gatzweiler |
| 25.11.2025 | Joshua Jeishing Wen (University of Vienna) | Tesler identities for wreath Macdonald polynomials |
| 18.11.2025 | Fabián Levicán (University of Vienna) | Embeddings of weighted projective spaces |
| 11.11.2025 | Nicolas Allen Smoot (University of Vienna) | Some New Examples of Modular Congruence Multiplicities |
| 04.11.2025 | Eva-Maria Hainzl (TU Wien) | Functional equations with catalytic variable 101 |
| 28.10.2025 | Shane Chern (University of Vienna) | The Koutschan-Krattenthaler-Schlosser determinants and their combinatorics |
| 21.10.2025 | Sergio Alejandro Fernandez de Soto Guerrero (TU Graz) | MathMagic: A positroidal action over a deck of cards |
| 14.10.2025 | Christian Krattenthaler (University of Vienna) | Two Topics, Four Lessons |
| 07.10.2025 | Matija Bucic (University of Vienna) | Equiangular lines via improved eigenvalue multiplicity |