Ewch i’r prif gynnwys

Intradisciplinary lecture series

Mae'r cynnwys hwn ar gael yn Saesneg yn unig.

Audience sitting in lecture theatre

A series of talks from our staff, highlighting their current research within a wide mathematical context.

These talks are accessible to final year undergraduate students, postgraduate students, and all staff in the School.

The talks take place between 15:10-16:10 in lecture room M/0.40 on the ground floor of the School. All are welcome to attend.

Upcoming lectures

The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. Thus, Poonen's Bertini theorem over finite fields has a motivic analog due to Vakil and Wood, which expresses the motivic density of smooth hypersurface sections as the degree goes to infinity in terms of a special value of Kapranov's zeta function.

I will report on joint work with Sean Howe, providing a broad generalization of Vakil and Wood's result, which implies in particular a motivic analog of Poonen's Bertini theorem with Taylor conditions, as well as motivic analogs of many generalizations and variants of Poonen's theorem. A key ingredient for this is a notion of motivic Euler product which allows us to write down candidate motivic probabilities.

Past lectures

In this general audience talk, we will combine tools from analysis (and in particular spectral analysis) and probability theory to investigate time decay properties of various physical systems (such as the spreading of heat inside some volume). It is well-known that a spectral gap is equivalent to a so called Poincaré inequality and to an exponential rate of decay. After explaining these notions, we will see how some of these results can be extended to cases where there is no spectral gap. This is joint work with Amit Einav (Graz).

For natural and industrial elastic materials, uncertainties in the mechanical responses arise from the inherent micro-structural inhomogeneity, sample-to-sample intrinsic variability, and observational data, which are sparse, inferred from indirect measurements, and polluted by noise. For these materials, deterministic approaches, which are based on average data values, can greatly underestimate or overestimate their properties, and stochastic representations accounting also for data dispersion are needed.

In this talk, I will present an explicit strategy for constructing stochastic hyperelastic models described by a strain-energy function where the parameters are random variables characterised by probability density functions. These models are able to propagate uncertainties from input data to output quantities of interest. In particular, for a stochastic hyperelastic body with a simple geometry, I will show analytically that, by contrast to the deterministic elastic problem where a single critical value strictly separates the cases where an instability can or cannot occur, for the stochastic problem, there is a probabilistic interval where the stable and unstable states always compete in the sense that both have a quantifiable chance to be found. More complex but still tractable problems can be treated in a similar manner.

This talk will be in two parts. In the first part I will give a brief overview of some of my research into how graph theoretical models can be used to solve (or at least approximately solve) some real-world operational research (OR) problems.

The second half will be a more detailed case study on the maximum happy vertices problem, stemming from some recent work carried out with colleagues in Australia [1]. This is a new type of problem that involves determining a vertex colouring of a graph such that the number of vertices assigned to the same colour as all of their neighbours is maximised. This problem is trivial if no vertices are precoloured, though in general it is NP-hard.

[1] Lewis, R., D. Thiruvady and K. Morgan (2019) 'Finding Happiness: An Analysis of the Maximum Happy Vertices Problem'. Computers and Operations Research, vol. 103, pp. 265-276.

An alternating sign matrix (ASM) is a square matrix in which each entry is -1, 0 or 1, and along each row and column the nonzero entries alternate in sign, starting and ending with a 1. This talk will provide an elementary introduction to the combinatorics of ASMs, and will include an outline of a recent proof of a conjecture for the number of odd-order diagonally and antidiagonally symmetric ASMs.

I will give an overview and introduction to conformal symmetry. Structures exhibiting conformal symmetry are invariant with respect to changes of coordinates which preserve angles but not necessarily lengths. In two dimensions the algebra of local conformal transformations is infinite-dimensional and thus leads to rich mathematical structures with fascinating implications for physics, such as, exactly solvable quantum field theories.

The single-arm trial is a very simple design: everyone gets a treatment (mostly within oncology) and wait to see if it works, in terms of a binary response. It assumes that those not treated will not get better or have a small chance (p0) of getting better with the hope that the treatment improves response to the chance p1. One improvement to this design is an adaptive version where an interim analysis is carried out after observing the outcomes of the first, n1 participants. The intention of the interim analysis is to stop the trial early as the treatment is not working.

This adaptive version of the trial design is often labelled Simon’s two-stage design and it is perhaps the most used adaptive design in practice. However, from results of a review it shows how a design can be abused with poor reporting and adherence to a design.  I will describe different ways of improving this design such as doing many interims, being Bayesian, adding in biomarkers and comparing this approach to randomised designs. Perhaps in the end I might grow to like this design.

The talk touches upon several topics from different subjects. First, I will talk about uniformly distributed sequences and different characteristics of uniformity. Second, I will argue that, contrarily to a wide-spread belief, the sequences with good space-filling properties do not resemble uniformly distributed sequences, especially if the dimension of the space is large. Most of the time I will spend on the problem of covering of a high-dimensional cube by n balls and will demonstrate several unexpected phenomena of good covering schemes. I will also mention the problem of quantisation also known as the problem of optimal facility location.

Contact us

Please contact Dr Jonathan Ben-Artzi with any questions about this event series.

Dr Jonathan Ben-Artzi

Dr Jonathan Ben-Artzi

Senior Lecturer

Email
ben-artzij@caerdydd.ac.uk
Telephone
02920 875624