Tue, 19 Apr 2022

14:00 - 15:00
C6

Epidemics on networks: From complicated structures to simple dynamics

Bastian Prasse
(European Centre for Disease Prevention and Control)
Abstract

The spread of an infectious disease crucially depends on the contact patterns of individuals, which range from superspreaders and clustered communities to isolated individuals with only a few regular contacts. The contact network specifies all contacts either between individuals in a population or, on a coarser scale, the contacts between groups of individuals, such as households, age groups or geographical regions. The structure of the contact network has a decisive impact on the viral dynamics. However, in most scenarios, the precise network structure is unknown, which constitutes a tremendous obstacle to understanding and predicting epidemic outbreaks.

This talk focusses on a stark contrast: network structures are complicated, but viral dynamics on networks are simple. Specifically, denote the N x 1 viral state vector by I(t) = (I_1(t), ..., I_N(t)), where N is the network size and I_i(t) is the infection probability of individual i at time t. The dynamics are “simple” in the way that the state I(t) evolves in a subspace X of R^N of astonishingly low dimension dim(X) << N. The low dimensionality of the viral dynamics has far-reaching consequences. First, it is possible to predict an epidemic outbreak, even without knowing the network structure. Second, provided that the basic reproduction number R_0 is close to one, the Susceptible-Infectious-Susceptible (SIS) epidemic model has a closed-form solution for arbitrarily large and heterogeneous contact networks.

Tue, 15 Mar 2022
14:00
C6

Colouring locally sparse graphs with the first moment method

Eoin Hurley
(Heidelberg University)
Abstract

A classical theorem of Molloy and Johansson states that if a graph is triangle free and has maximum degree at most $\Delta$, then it has chromatic number at most $\frac{\Delta}{\log \Delta}$. This was extended to graphs with few edges in their neighbourhoods by Alon-Krivelevich and Sudakov, and to list chromatic number by Vu. I will give a full and self-contained proof of these results that relies only on induction and the first moment method.

Thu, 03 Mar 2022
11:30
C6

Monadic Second Order interpretations

Mikołaj Bojańczyk
(University of Warsaw/University of Oxford)
Abstract

MSO can be used not only to accept/reject words, but also to transform words into other words, e.g. the doubling function w $\mapsto$ ww. The traditional model for this is called MSO transductions; the idea is that each position of the output word is interpreted in some position of the input word, and MSO is used to define the order on output positions and their labels. I will explain that an extension, where output positions are interpreted using $k$-tuples of input positions, is (a) is also well behaved; and (b) this is surprising.

Fri, 11 Feb 2022
16:00
C6

Renormalization Group Flows on Line Defects

Avia Raviv-Moshe
(Simons Center Stony Brook)
Further Information

It is also possible to join virtually via zoom.

Abstract

We will consider line defects in d-dimensional CFTs. The ambient CFT places nontrivial constraints on renormalization group flows on such line defects. We will see that the flow on line defects is consequently irreversible and furthermore a canonical decreasing entropy function exists. This construction generalizes the g theorem to line defects in arbitrary dimensions. We will demonstrate this generalization in some concrete examples, including a flow between Wilson loops in 4 dimensions, and an O(3) bosonic theory coupled to an impurity in the large spin representation of the bulk global symmetry.

Fri, 05 Nov 2021

15:30 - 16:30
C6

Short talks from Algebra PhDs

Algebra DPhil students
(University of Oxford)
Further Information

A collection of bite-size 10-15 minute talks from current DPhil students in the Algebra group. The talks will be accessible to masters students and above.

With plenty of opportunity to chat to current students about what doing a PhD in algebra and representation theory is like!

Mon, 01 Jul 2019

15:00 - 16:00
C6

The role of polyconvexity in dynamical problems of thermomechanics

Athanasios Tzavaras
(KAUST)
Abstract

The stabilization of thermo-mechanical systems is a classical problem in thermodynamics and well

understood in a context of gases. The objective of this talk is to indicate the role of null-Lagrangians and

certain transport/stretching identities in stabilizing thermomechanical systems associated with general

thermoelastic free energies. This allows to prove various convergence results among thermomechan-

ical theories, and suggests a variational scheme for the approximation of the equations of adiabatic

thermoelasticity.

Mon, 01 Jul 2019

16:00 - 17:00
C6

Uniqueness of regular shock reflection

Wei Xiang
(City University of Hong Kong)
Abstract

We will talk about our recent results on the uniqueness of regular reflection solutions for the potential flow equation in a natural class of self-similar solutions. The approach is based on a nonlinear version of method of continuity. An important property of solutions for the proof of uniqueness is the convexity of the free boundary.

Tue, 22 Jan 2019

14:30 - 15:30
C6

Testing for an odd hole

Paul Seymour
Abstract

There was major progress on perfect graphs in the early 2000's: Chudnovsky, Robertson, Thomas and I proved the ``strong perfect graph theorem'' that a graph is perfect if and only if it has no odd hole or odd antihole; and Chudnovsky, Cornuejols, Liu, Vuscovic and I found a polynomial-time algorithm to test whether a graph has an odd hole or odd antihole, and thereby test if it is perfect. (A ``hole'' is an induced cycle of length at least four, and an ``antihole'' is a hole in the complement graph.)

What we couldn't do then was test whether a graph has an odd hole, and this has stayed open for the last fifteen years, despite some intensive effort. I am happy to report that in fact it can be done in poly-time (in time O(|G|^{12}) at the last count), and in this talk we explain how.

Joint work with Maria Chudnovsky, Alex Scott, and Sophie Spirkl.

Tue, 15 Jan 2019

14:30 - 15:30
C6

Two Erdos-Hajnal-type theorems in hypergraphs

Mykhaylo Tyomkyn
Abstract

The Erdos-Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs.

1. A theorem of R\"odl asserts that if an n-vertex graph is non-universal then it contains an almost homogeneous set (i.e one with edge density either very close to 0 or 1) of size \Omega(n). We prove that if a 3-uniform hypergraph is non-universal then it contains an almost homogeneous set of size \Omega(log n). An example of R\"odl from 1986 shows that this bound is tight.

2. Let R_r(t) denote the size of the largest non-universal r-graph G so that neither G nor its complement contain a complete r-partite subgraph with parts of size t. We prove an Erd\H{o}s--Hajnal-type stepping-up lemma, showing how to transform a lower bound for R_r(t) into a lower bound for R_{r+1}(t). As an application of this lemma, we improve a bound of Conlon-Fox-Sudakov by showing that R_3(t) \geq t^{ct).

Joint work with M. Amir and A. Shapira

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