The second series of our short films, Me and My Maths, is now up and running on our social media channels and you can watch a compilation of the first four films via the video below. Me and My Maths: short films about people who also do maths.
Starring in order of appearance: Kylie and Chloe, Andrea, Doyne, and Kate Wenqi.
Seminars in the building:
Wednesday 15th February (5th week), 17:00, L4
Rogério Monteiro de Siqueira (Universidade de São Paolo) - ‘On Green’s theorem: a visual history through textbooks and other printed matter’
Wednesday 22nd February (6th week), 17:00, L4
Nicolas Michel (Bergische Universität Wuppertal) - ‘On the uses and abuses of the history of mathematics’
Correlated stochastic block models: graph matching and community recovery
Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.
Abstract
I will discuss statistical inference problems on edge-correlated stochastic block models. We determine the information-theoretic threshold for exact recovery of the latent vertex correspondence between two correlated block models, a task known as graph matching. As an application, we show how one can exactly recover the latent communities using multiple correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph. Furthermore, we obtain the precise threshold for exact community recovery using multiple correlated graphs, which captures the interplay between the community recovery and graph matching tasks. This is based on joint work with Julia Gaudio and Anirudh Sridhar.
Some combinatorial applications of guided random processes
Abstract
Random greedy algorithms became ubiquitous in Combinatorics after Rödl's nibble (semi-random method), which was repeatedly refined for various applications, such as iterative graph colouring algorithms (Molloy-Reed) and lower bounds for the Ramsey number $R(3,t)$ via the triangle-free process (Bohman-Keevash / Fiz Pontiveros-Griffiths-Morris). More recently, when combined with absorption, they have played a key role in many existence and approximate counting results for combinatorial structures, following a paradigm established by my proofs of the Existence of Designs and Wilson's Conjecture on the number of Steiner Triple Systems. Here absorption (converting approximate solutions to exact solutions) is generally the most challenging task, which has spurred the development of many new ideas, including my Randomised Algebraic Construction method, the Kühn-Osthus Iterative Absorption method and Montgomery's Addition Structures (for attacking the Ryser-Brualdi-Stein Conjecture). The design and analysis of a suitable guiding mechanism for the random process can also come with major challenges, such as in the recent proof of Erdős' Conjecture on Steiner Triple Systems of high girth (Kwan-Sah-Sawhney-Simkin). This talk will survey some of this background and also mention some recent results on the Queens Problem (Bowtell-Keevash / Luria-Simkin / Simkin) and the Existence of Subspace Designs (Keevash-Sah-Sawhney). I may also mention recent solutions of the Talagrand / Kahn-Kalai Threshold Conjectures (Frankston-Kahn-Narayanan-Park / Park-Pham) and thresholds for Steiner Triple Systems / Latin Squares (Keevash / Jain-Pham), where the key to my proof is constructing a suitable spread measure via a guided random process.
Mixed Hodge modules and real groups
Abstract
I will explain an ongoing program, joint with Kari Vilonen, that aims to study unitary representations of real reductive Lie groups using mixed Hodge modules on flag varieties. The program revolves around a conjecture of Schmid and Vilonen that natural filtrations coming from the geometry of flag varieties control the signatures of Hermitian forms on real group representations. This conjecture is expected to facilitate new progress on the decades-old problem of determining the set of unitary irreducible representations by placing it in a more conceptual context. Our results to date centre around the interaction of Hodge theory with the unitarity algorithm of Adams, van Leeuwen, Trapa, and Vogan, which calculates the signature of a canonical Hermitian form on an arbitrary representation by reducing to the case of tempered representations using deformations and wall crossing. Our results include a Hodge-theoretic proof of the ALTV wall crossing formula as a consequence of a more refined result and a verification of the Schmid-Vilonen conjecture for tempered representations.
15:30
Multivalued Dir-Minimizing Functions
The course will serve as an introduction to the theory of multivalued Dir-minimizing functions, which can be viewed as harmonic functions which attain multiple values at each point.
Aimed at Postgraduate students interested in geometric measure theory and its link with elliptic PDEs, a solid knowledge of functional analysis and Sobolev spaces, acquaintance with variational
methods in PDEs and some basic geometric measure theory are recommended.
Sessions led by Dr Immanuel Ben Porat will take place on
09 May 2023 15:30 - 17:30 C4
16 May 2023 15:30 - 17:30 C4
23 May 2023 15:30 - 17:30 C4
30 May 2023 15:30 - 17:30 C4
Should you be interested in taking part in the course, please send an email to @email.
Abstract
The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers.
15:30
Multivalued Dir-Minimizing Functions
The course will serve as an introduction to the theory of multivalued Dir-minimizing functions, which can be viewed as harmonic functions which attain multiple values at each point.
Aimed at Postgraduate students interested in geometric measure theory and its link with elliptic PDEs, a solid knowledge of functional analysis and Sobolev spaces, acquaintance with variational
methods in PDEs and some basic geometric measure theory are recommended.
Sessions led by Dr Immanuel Ben Porat will take place on
09 May 2023 15:30 - 17:30 C4
16 May 2023 15:30 - 17:30 C4
23 May 2023 15:30 - 17:30 C4
30 May 2023 15:30 - 17:30 C4
Should you be interested in taking part in the course, please send an email to @email.
Abstract
The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers.