Title: A storm in a tea-cup(?)

Date:  2pm Tuesday 12th November 2024

Venue: LT907

Abstract: This talk has two parts. In the first part I will talk about the amazing results obtained recently by using techniques that originate with Nash and Gromov, and in the second part I will talk about implications of these results for admissibility criteria, which is ongoing work with H. Gimperlein, R. J. Knops and M. Slemrod.  Some audience participation will be required in the first part.

Title: The 1-11-representability of graphs 

Date: 2.00pm Tuesday 19th November 2024

Venue: LT907

Abstract: 

A graph is word-representable if it can be represented by rules concerning the alternation of letters in words. While many well-studied classes of graphs are word-representable, not all graphs can be represented in this way. By relaxing the conditions that define an edge, we can represent all graphs in a so-called 2-11 manner. In the new terminology, word-representable graphs are precisely 0-11-representable graphs. Whether all graphs can be 1-11-represented remains unknown, and the 1-11-representation of graphs has gained attention in the literature. In this talk, I will discuss methods and tools for proving the word-representability and 1-11-representability of graphs.

Title: Ribbon Tilings

Date: 2.00pm Tuesday 26th November 2024

Venue: LT907

Abstract: A ribbon tile consists of a finite edge-connected sequence of unit squares in the plane, each square either directly above or directly to the right of the previous square in the sequence. An n-ribbon tile is a ribbon tile made up of n unit squares. Which finite regions of the plane can be tiled with n-ribbon tiles? How many n-ribbon tilings can a region have? In this talk I will give an introduction to n-ribbon tilings, focussing on these questions. The new material in the talk is based on joint work with Yinsong Chen and Vladislav Kargin (arXiv:2408.09272).

Title: Topology and Transcription in 3D Chromatin Loop Networks

Date: 2.00pm Thursday 28th November 2024

Venue: LT908

Abstract: The 3D folding of a mammalian gene can be studied by polymer models, where the chromatin fiber is represented by a semiflexible polymer which interacts with multivalent proteins, representing complexes of DNA-binding transcription factors and RNA polymerases. This physical model leads to microphase separation and the natural emergence of clusters of proteins and binding sites, which can be understood via a mean field theory. The clustering is  accompanied by the folding of chromatin into a set of topologies, each associated with a different network of loops. By combining Brownian dynamic simulations, statistical mechanics and combinatorics, we first classify these networks, and then find their relative importance or statistical weight, when the properties of the underlying polymer are those relevant to chromatin. Unlike polymer networks previously studied, our chromatin networks have finite average distances between successive binding sites, and we show that this feature leads to giant differences between the weights of topologies with the same number of edges and nodes but different wiring. These weights strongly favor rosette-like structures with a local cloud of loops with respect to more complicated nonlocal topologies. Our results suggest that genes should overwhelmingly fold into a small fraction of all possible 3D topologies, which can be robustly characterized by the framework we propose here.

Title: Irrational Enumeration

Date: Tuesday 11th February 2025, 2.00pm

Venue: LT907

Abstract: It is well known that the asymptotic behaviour of the number of objects in a combinatorial class can be determined from the singularities on the circle of convergence of its generating function. In this talk we extend the scope of analytic combinatorics to classes whose objects have irrational sizes by establishing an analogous result when it is no longer required that the size of an object must be an integer.

The generating function for such a class is a power series that admits irrational exponents (which we call a Ribenboim series). A transformation then yields a generalised Dirichlet series from which the asymptotics of the coefficients can be extracted by singularity analysis using an appropriate Tauberian theorem. In practice, the asymptotics can often be determined directly from the original generating function. The technique will be illustrated with a variety of applications, including tilings with tiles of irrational area, lattice walks enumerated by Euclidean length, and trees with vertices of irrational size. We will also explore phase transitions in the asymptotics of families of irrational combinatorial classes.

Title: A Variational Perspective on Auxetic Metamaterials of Checkerboard Type

Date: Tuesday 18th March 2025, 2.00pm

Venue: LT907

Abstract:  Auxetic metamaterials have the counterintuitive property that they expand perpendicular to applied forces under stretching. In this talk, we discuss homogenization via Gamma-convergence for elastic materials with stiff checkerboard-type heterogeneities under the assumption of non-self-interpenetration. Our result rigorously confirms these structures as auxetic. The challenging part of the proof is determining the admissible macroscopic deformation behavior, or in other words, characterizing the weak Sobolev limits of deformation maps whose gradients are locally close to rotations on the stiff components. To this end, we establish an asymptotic rigidity result showing that, under suitable scaling assumptions, the attainable macroscopic deformations are affine conformal contractions. Our strategy is to tackle first an idealized model with full rigidity on the stiff tiles and then transfer the findings to a model with diverging elastic constants. The latter requires a new quantitative geometric rigidity estimate for non-connected touching squares and a tailored Poincaré-type inequality for checkerboard structures.

This is joint work with Wolf-Patrick Düll (University of Stuttgart) and Carolin Kreisbeck (KU Eichstätt-Ingolstadt).

Title: On the Connection of the Prandtl Equations and the Harmonic Oscillator

Date: Tuesday 25th March 2025, 2.00pm

Venue: LT907

Abstract:  A deep connection  is established between the Prandtl equations linearised around a quadratic shear flow, confluent hypergeometric functions of the first kind, and the Schrödinger operator. The first result concerns a spectral condition associated with unstable quasimodes of the Prandtl equations. We give all solutions of the governing ODE in terms of Kummer's functions. By classifying their asymptotic behaviour, we determine that the spectral condition has unique, determined eigenvalue and eigenfunction, the latter being expressible as a combination of elementary functions.

Moreover, we prove that any quasi-eigenmode solution of the linearised Prandtl equations around a quadratic shear flow can be explicitly determined from algebraic eigenfunctions of the Schrödinger operator with a quadratic potential. We show finally that the obtained analytical formulation of the velocity align with previous numerical simulations in the literature.

Title: Long time behavior for SIS model driven by pure-jump noise with Markov switching  

Wednesday 17th September 2025, 3.00-4.00pm

Venue: LT907

Abstract: In this talk, we focus on long time behavior for SIS epidemic model driven by an alpha-stable process with Markov switching. Necessary and sufficient conditions for recurrence or extinction have been established. To analyze the ergodic behavior, we establish the well-posedness of a nonlocal Dirichlet problem with regime switching and derive a strong maximum principle. Our results disclose heavy-tailed fluctuations and stationary distribution of Markov chain significantly affect the epidemic threshold and asymptotic dynamics.