We are currently inviting applications for a Postdoctoral Research Associate (PDRA) to work with Professor Marc Lackenby at the Mathematical Institute, University of Oxford. This is a three-year, fixed-term position, funded by a research grant from the Engineering and Physical Sciences Research Council (EPSRC). We anticipate the starting date of this position to be 01 October 2024.
We are currently inviting applications for a Postdoctoral Research Associate (PDRA) to work with Professor Marc Lackenby at the Mathematical Institute, University of Oxford. This is a three-year, fixed-term position, funded by a research grant from the Engineering and Physical Sciences Research Council (EPSRC). We anticipate the starting date of this position to be 01 October 2024.
We will survey work of Birman-Craggs, Johnson, and Sato on the abelianization of the level 2 congruence group of the mapping class group of a surface, and of the corresponding Torelli group. We will then describe recent work of Lewis providing a common framework for both abelianizations, with applications including a partial answer to a question of Johnson.
Let C,D be classes of finitely presented groups. The epimorphism problem from C to D is the following decision problem:
Input: Finite descriptions (presentation, multiplication table, other) for groups G in C and H in D
Question: Is there an epimorphism from G to H?
I will discuss some cases where it is decidable and where it is NP-complete. Spoiler alert: it is undecidable for C=D=the class of 2-step nilpotent groups (Remeslennikov).
This is joint work with Jerry Shen (UTS) and Armin Weiss (Stuttgart).
Turning thought in to form is a mysterious process with which artists, scientists, philosophers and, indeed, all of us engage. But though the outcomes, mathematical, artistic, philosophical, may be different, might there be much that is common to all?
In the last lecture of the four-part series organised as part of Conrad Shawcross' 'Cascading Principles' exhibition in Oxford Mathematics, we bring together a panel comprising Jon Keating, Sedleian Professor of Natural Philosophy at the University of Oxford, Emma Ridgway, Director of the Foundling Museum, Maya B Kronicg, philosopher and Director of Urbanomic, and Conrad himself. The discussion will be chaired by Fatos Ustek, curator of the 'Cascading Principles' exhibition.
There will be an opportunity to view the exhibition with the curator at 4pm on the day of the lecture.
Please email @email to register for the in-person event.The lecture will be broadcast on the Oxford Mathematics YouTube Channel at a later date.
Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.
We show that given $m$ disjoint chains in the Boolean lattice, we can create $m$ disjoint skipless chains that cover the same elements (where we call a chain skipless if any two consecutive elements differ in size by exactly one). By using this result we are able to answer two conjectures about the asymptotics of induced saturation numbers for the antichain, which are defined as follows. For positive integers $k$ and $n$, a family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is $k$-antichain saturated if it does not contain an antichain of size $k$ (as induced subposet), but adding any set to $\mathcal{F}$ creates an antichain of size $k$. We use $\textrm{sat}^{\ast}(n,k)$ to denote the smallest size of such a family. With more work we pinpoint the exact value of $\textrm{sat}^{\ast}(n,k)$, for all $k$ and sufficiently large $n$. Previously, exact values for $\textrm{sat}^{\ast}(n,k)$ were only known for $k$ up to 6. We also show that for any poset $\mathcal{P}$, its induced saturation number (which is defined similar as for the antichain) grows at most polynomially: $\textrm{sat}^{\ast}(n, \mathcal{P})=O(n^c)$, where $c \leq |\mathcal{P}|^2/4+1$. This is based on joint works with Carla Groenland, Maria-Romina Ivan, Hugo Jacob and Tom Johnston.
Consider the following "controlled" random graph process: edges of the complete graph are revealed one by one in random order to an online algorithm, which immediately decides whether to retain each observed edge. The algorithm's objective is to construct a graph property within specified constraints on the total number of observed edges ("time") and the total number of retained edges ("budget").
During this talk, I will present results in this model for natural graph properties, such as connectivity, Hamiltonicity, and containment of fixed-size subgraphs. Specifically, I will describe a strategy to construct a Hamilton cycle at the hitting time for minimum degree 2 by retaining a linear number of edges. This extends the classical hitting time result for Hamiltonicity originally established by Ajtai–Komlós–Szemerédi and Bollobás.
The talk is based on joint work with Alan Frieze and Michael Krivelevich.