Mon, 23 Oct 2023
15:30
L4

Khovanov homology and the Fukaya category of the three-punctured sphere

Claudius Zibrowius
(Durham University)
Abstract

About 20 years ago, Dror Bar-Natan described an elegant generalisation
of Khovanov homology to tangles with any number of endpoints, by
considering certain quotients of two-dimensional relative cobordism
categories.  I claim that these categories are in general not
well-understood (not by me in any case).  However, if we restrict to
tangles with four endpoints, things simplify and Bar-Natan's category
turns out to be closely related to the wrapped Fukaya category of the
four-punctured sphere.  This relationship gives rise to a symplectic
interpretation of Khovanov homology that is useful both for doing
calculations and for proving theorems.  I will discuss joint work in
progress with Artem Kotelskiy and Liam Watson where we investigate what
happens when we fill in one of the punctures.
 

Mon, 27 Feb 2023
15:30
L4

SL(2,C)-character varieties of knots and maps of degree 1

Raphael Zentner
(Durham University)
Abstract

We ask to what extend the SL(2,C)-character variety of the
fundamental group of the complement of a knot in S^3 determines the
knot. Our methods use results from group theory, classical 3-manifold
topology, but also geometric input in two ways: the geometrisation
theorem for 3-manifolds, and instanton gauge theory. In particular this
is connected to SU(2)-character varieties of two-component links, a
topic where much less is known than in the case of knots. This is joint
work with Michel Boileau, Teruaki Kitano, and Steven Sivek.

Tue, 22 Nov 2022

15:30 - 16:30
L6

Domino Shuffle and Matrix Refactorizations

Sunil Chhita
(Durham University)
Abstract

This talk is motivated by computing correlations for domino tilings of the Aztec diamond.  It is inspired by two of the three distinct methods that have recently been used in the simplest case of a doubly periodic weighting, that is the two-periodic Aztec diamond. This model is of particular probabilistic interest due to being one of the few models having a boundary between polynomially and exponentially decaying macroscopic regions in the limit. One of the methods to compute correlations, powered by the domino shuffle, involves inverting the Kasteleyn matrix giving correlations through the local statistics formula. Another of the methods, driven by a Wiener-Hopf factorization for two- by-two matrix valued functions, involves the Eynard-Mehta theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by an LU- and UL-decomposition, based on a matrix refactorization, for the product of the transition matrices. In this talk, we present results to say that the evolution of the face weights under the domino shuffle and the matrix refactorization is the same. This is based on joint work with Maurice Duits (Royal Institute of Technology KTH).  

 

Thu, 20 Oct 2022

12:00 - 13:00
L1

Revisiting Two Classic Surface Tension Problems: Rough Capillary Rise and Fluctuations of Cellular Droplets

Prof. Halim Kusumaatmaja
(Durham University)
Further Information

Prof Halim Kusumaatmaja is currently a Professor of Physics at Durham University and he also holds an EPSRC Fellowship in Engineering. Prof Kusumaatmaja graduated with a Master of Physics from the University of Leicester in 2004 and a PhD in Physics from the University of Oxford in 2008. He worked as a Postdoctoral Research Associate at the Max Planck Institute of Colloids and Interfaces (2008-2011) and at the University of Cambridge (2011-2013), before moving to Durham University and rising through the ranks from Assistant Professor (2013-2017) to Associate Professor (2017-2020) and Full Professor (2020-now). Prof Kusumaatmaja leads an interdisciplinary research group in the area of Soft Matter and Biophysics. Current research interests include wetting and interfacial phenomena, bio-inspired materials, liquid-liquid phase separation in biology, multistable elastic structures, colloidal and molecular self-assembly, and high performance computing.

Abstract

In this talk I will discuss our recent work on two problems. The first problem concerns with capillary rise between rough structures, a fundamental wetting phenomenon that is functionalised in biological organisms and prevalent in geological or man-made materials. Predicting the liquid rise height is more complex than currently considered in the literature because it is necessary to couple two wetting phenomena: capillary rise and hemiwicking. Experiments, simulations and analytic theory demonstrate how this coupling challenges our conventional understanding and intuitions of wetting and roughness. For example, the critical contact angle for hemiwicking becomes separation-dependent so that hemiwicking can vanish for even highly wetting liquids. The rise heights for perfectly wetting liquids can also be different in smooth and rough systems. The second problem concerns with droplets (or condensates) formed via a liquid-liquid phase separation process in biological cells. Despite the widespread importance of surface tension for the interactions between these droplets and other cellular components, there is currently no reliable technique for their measurement in live cells. To address this, we develop a high-throughput flicker spectroscopy technique. Applying it to a class of cellular droplets known as stress granules, we find their interface fluctuations cannot be described by surface tension alone. It is necessary to consider elastic bending deformation and a non-spherical base shape, suggesting that stress granules are viscoelastic droplets with a structured interface, rather than simple Newtonian liquids. Moreover, given the broad distributions of surface tension and bending rigidity observed, different types of stress granules can only be differentiated via large-scale surveys, which was not possible previously and our technique now enables.

 

Thu, 05 May 2022

16:00 - 17:00
L5

Gaussian distribution of squarefree and B-free numbers in short intervals

Alexander Mangerel
(Durham University)
Abstract
(Joint with O. Gorodetsky and B. Rodgers) It is a classical quest in analytic number theory to understand the fine-scale distribution of arithmetic sequences such as the primes. For a given length scale h, the number of elements of a "nice" sequence in a uniformly randomly selected interval $(x,x+h], 1 \leq x \leq X$, might be expected to follow the statistics of a normally distributed random variable (in suitable ranges of $1 \leq h \leq X$).  Following the work of Montgomery and Soundararajan, this is known to be true for the primes, but only if we assume several deep and long-standing conjectures among which the Riemann Hypothesis. In fact, previously such distributional results had not been proven for any (non-trivial) sequence of number-theoretic interest, unconditionally.

As a model for the primes, in this talk I will address such statistical questions for the sequence of squarefree numbers, i.e., numbers not divisible by the square of any prime, among other related ``sifted'' sequences called B-free numbers. I hope to further motivate and explain our main result that shows, unconditionally, that short interval counts of squarefree numbers do satisfy Gaussian statistics, answering several questions of R.R. Hall.

Fri, 04 Mar 2022

15:00 - 16:00
L6

Open questions on protein topology in its natural environment.

Christopher Prior
(Durham University)
Abstract

Small angle x-ray scattering is one of the most flexible and readily available experimental methods for obtaining information on the structure of proteins in solution. In the advent of powerful predictive methods such as the alphaFold and rossettaFold algorithms, this information has become increasingly in demand, owing to the need to characterise the more flexible and varying components of proteins which resist characterisation by these and more standard experimental techniques. To deal with structures about little of which is known a parsimonious method of representing the tertiary fold of a protein backbone as a discrete curve has been developed. It represents the fundamental local Ramachandran constraints through a pair of parameters and is able to generate millions of potentially realistic protein geometries in a short space of time. The data obtained from these methods provides a treasure trove of information on the potential range of topological structures available to proteins, which is much more constrained that that available to self-avoiding walks, but still far more complex than currently understood from existing data. I will introduce this method and its considerations then attempt to pose some questions I think topological data analysis might help answer. Along the way I will ask why roadies might also help give us some insight….

Mon, 10 Jun 2019
15:45
L6

Unitary group integrals, surfaces, and mapping class groups

Michael Magee
(Durham University)
Abstract


For any word w in a free group of rank r>0, and any compact group G, w induces a `word map' from G^r to G by substitutions of elements of G for the letters of w. We may also choose the r elements of G independently with respect to Haar measure on G, and then apply the word map. This gives a random element of G whose distribution depends on w. An interesting observation is that this distribution doesn't change if we change w by an automorphism of the free group. It is a wide open question whether the measures induced by w on compact groups determine w up to automorphisms.
My talk will be mostly about the case G = U(n), the n by n complex unitary matrices. The technical tool we use is a precise formula for the moments of the distribution induced by w on U(n). In the formula, there is a surprising appearance of concepts from infinite group theory, more specifically, Euler characteristics of mapping class groups of surfaces. I'll explain how our formula allows us to make progress on the question described above.
This is joint work with Doron Puder (Tel Aviv).
 

Mon, 25 Feb 2019
15:45
L6

Twisted Blanchfield pairings and Casson-Gordon invariants

Anthony Conway
(Durham University)
Abstract

 In the late seventies, Casson and Gordon developed several knot invariants that obstruct a knot from being slice, i.e. from bounding a disc in the 4-ball. In this talk, we use twisted Blanchfield pairings to define twisted generalisations of the Levine-Tristram signature function, and describe their relation to the Casson-Gordon invariants. If time permits, we will present some obstructions to algebraic knots being slice. This is joint work with Maciej Borodzik and Wojciech Politarczyk.

Thu, 07 Mar 2019

14:00 - 15:00
L4

Flexible computational abstractions for complex preconditioners

Dr Lawrence Mitchell
(Durham University)
Abstract

Small block overlapping, and non-overlapping, Schwarz methods are theoretically highly attractive as multilevel smoothers for a wide variety of problems that are not amenable to point relaxation methods.  Examples include monolithic Vanka smoothers for Stokes, overlapping vertex-patch decompositions for $H(\text{div})$ and  $H(\text{curl})$ problems, along with nearly incompressible elasticity, and augmented Lagrangian schemes.

 While it is possible to manually program these different schemes,  their use in general purpose libraries has been held back by a lack   of generic, composable interfaces. We present a new approach to the   specification and development such additive Schwarz methods in PETSc  that cleanly separates the topological space decomposition from the  discretisation and assembly of the equations. Our preconditioner is  flexible enough to support overlapping and non-overlapping additive  Schwarz methods, and can be used to formulate line, and plane smoothers, Vanka iterations, amongst others. I will illustrate these new features with some examples utilising the Firedrake finite element library, in particular how the design of an approriate computational interface enables these schemes to be used as building blocks inside block preconditioners.

This is joint work with Patrick Farrell and Florian Wechsung (Oxford), and Matt Knepley (Buffalo).

Thu, 07 Jun 2018
12:00
L5

On singular limits for the Vlasov-Poisson system

Mikaela Iacobelli
(Durham University)
Abstract

The Vlasov-Poisson system is a kinetic equation that models collisionless plasma. A plasma has a characteristic scale called the Debye length, which is typically much shorter than the scale of observation. In this case the plasma is called ‘quasineutral’. This motivates studying the limit in which the ratio between the Debye length and the observation scale tends to zero. Under this scaling, the formal limit of the Vlasov-Poisson system is the Kinetic Isothermal Euler system. The Vlasov-Poisson system itself can formally be derived as the limit of a system of ODEs describing the dynamics of a system of N interacting particles, as the number of particles approaches infinity. The rigorous justification of this mean field limit remains a fundamental open problem. In this talk we present the rigorous justification of the quasineutral limit for very small but rough perturbations of analytic initial data for the Vlasov-Poisson equation in dimensions 1, 2, and 3. Also, we discuss a recent result in which we derive the Kinetic Isothermal Euler system from a regularised particle model. Our approach uses a combined mean field and quasineutral limit.

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