The last decade or so has seen substantial progress in the theory of (outer) automorphism groups of right-angled Artin groups (RAAGs), spearheaded by work of Charney and Vogtmann. Many of the techniques used for RAAGs also apply to a wider class of groups, graph products of finitely generated abelian groups, which includes right-angled Coxeter groups (RACGs). In this talk, I will give an introduction to automorphism groups of such graph products, and describe recent developments surrounding the outer automorphism groups of RACGs, explaining the links to what we know in the RAAG case.

# Past Forthcoming Seminars

Let $X$ be a K3 surface and let $Z_X(q)$ be the generating series of the topological Euler characteristics of the Hilbert scheme of points on $X$. It is known that $q/Z_X(q)$ equals the discriminant form $\Delta(\tau)$ after the change of variables $q=e^{2 \pi i \tau}$. In this talk we consider the equivariant generalization of this result, when a finite group $G$ acts on $X$ symplectically. Mukai and Xiao has shown that there are exactly 81 possibilities for such an action in terms of types of the fixed points. The analogue of $q/Z_X(q)$ in each of the 81 cases turns out to be a cusp form (after the same change of variables). Knowledge of modular forms is not assumed in the talk; I will introduce all necessary concepts. Joint work with Jim Bryan.

Neural Network algorithms have achieved unprecedented performance in image recognition over the past decade. However, their application in real world use-cases, such as self driving cars, raises the question of whether it is safe to rely on them.

We generally associate the robustness of these algorithms with how easy it is to generate an adversarial example: a tiny perturbation of an image which leads it to be misclassified by the Neural Net (which classifies the original image correctly). Neural Nets are strongly susceptible to such adversarial examples, but when the architecture of the target neural net is unknown to the attacker it becomes more difficult to generate these examples efficiently.

In this Black-Box setting, we frame the generation of an adversarial example as an optimisation problem solvable via derivative free optimisation methods. Thus, we introduce an algorithm based on the BOBYQA model-based method and compare this to the current state of the art algorithm.

## Further Information:

The Hales–Jewett Theorem states that any r–colouring of [m]^n contains a monochromatic combinatorial line if n is large enough. Shelah’s proof of the theorem implies that for m = 3 there always exists a monochromatic combinatorial line whose set of active coordinates is the union of at most r intervals. I will present some recent findings relating to this observation. This is joint work with Nina Kamcev.

Unitary highest weight modules were classified in the 1980s by Enright-Howe-Wallach and independently by Jakobsen. The classification is based on a version of the Dirac inequality, but the proofs also require a number of other techniques and are quite involved. We present a much simpler proof based on a different version of the Dirac inequality. This is joint work with Vladimir Soucek and Vit Tucek.

## Further Information:

Complexity Oxford Imperial College, COXIC, is a series of workshops aiming at bringing together researchers in Oxford and Imperial College interested in complex systems. The events take place twice a year, alternatively in Oxford and in London, and give the possibility to PhD students and young postdocs to present their research.

Schedule:

2:00: Welcome

2:15: Maria del Rio Chanona (OX), On the structure and dynamics of the job market

2:35: Max Falkenberg McGillivray (IC), Modelling the broken heart

2:55: Fernando Rosas (OX), Quantifying high-order interdependencies

3:15 - 4:00: Coffee break

4:00: Rishi Nalin Kumar (IC), Building scalable agent based models using open source technologies

4:20: Rodrigo Leal Cervantes (OX) Greed Optimisation of Modularity with a Self-Adaptive Resolution Parameter

4:40: TBC

5:00: Social event at the Lamb & Flag

We adapt convergent look-ahead and backward finite difference formulas to compute future eigenvectors and eigenvalues of piecewise smooth time-varying matrix flows $A(t)$. This is based on the Zhang Neural Network model for time-varying problems and uses the associated error function

$E(t) =A(t)V(t)−V(t)D(t)$

with the Zhang design stipulation

$\dot{E}(t) =−\eta E(t)$.

Here $E(t)$ decreased exponentially over time for $\eta >0$. It leads to a discrete-time differential equation of the form $P(t_k)\dot{z}(t_k) = q(t_k)$ for the eigendata vector $z(t_k)$ of $A(t_k)$. Convergent high order look-ahead difference formulas then allow us to express $z(t_k+1)$ in terms of earlier discrete $A$ and $z$ data. Numerical tests, comparisons and open questions follow.

The time bottleneck in the manufacturing process of Besi (company involved in ESGI 149 Innsbruck) is the extraction of undamaged dies from a component wafer. The easiest way for them to speed up this process is to reduce the number of 'selections' made by the robotic arm. Each 'selection' made by this robotic arm can be thought of as choosing a 2x2 submatix of a large binary matrix, and editing the 1's in this submatrix to be 0's. The quesiton is: what is the fewest number of 2x2 submatrices required to cover the full matrix, and how can we find this number. This problem can be solved exactly using integer programming methods, although this approach proves to be prohibitively expensive for realistic sizes. In this talk I will describe the approach taken by my team at EGSI 149, as well as directions for further improvement.

Social surveys are widely used in today's society as a method for obtaining opinions and other information from large groups of people. The questions in social surveys are usually presented in either multiple-choice or free-response formats. Despite their advantages, free-response questions are employed less commonly in large-scale surveys, because in such situations, considerable effort is needed to categorise and summarise the resulting large dataset. This is the so-called coding problem. Here we propose a survey framework in which, respondents not only write down their own opinions, but also input information characterising the similarity between their individual responses and those of other respondents. This is done in much the same way as ``likes" are input in social network services. The information input in this simple procedure constitutes relational data among opinions, which we call the opinion graph. The diversity of typical opinions can be identified as a modular structure of such a graph, and the coding problem is solved through graph clustering in a statistically principled manner. We demonstrate our approach using a poll on the 2016 US presidential election and a survey given to graduates of a particular university.

Our understanding of physical phenomena is intimately linked to the way we understand the relevant observables describing them. While a big deal of progress has been made for processes occurring in flat space-time, much less is known in cosmological settings. In this context, we have processes which happened in the past and which we can detect the remnants of at present time. Thus, the relevant observable is the late-time wavefunction of the universe. Questions such as "what properties they ought to satisfy in order to come from a consistent time evolution in cosmological space-times?", are still unanswered, and are compelling given that in these quantities time is effectively integrated out. In this talk I will report on some recent progress in this direction, aiming towards the idea of a formulation of cosmology "without time". Amazingly enough, a new mathematical structure, we called "cosmological polytope", which has its own first principle definition, encodes the singularity structure we ascribe to the perturbative wavefunction of the universe, and makes explicit its (surprising) relation to the flat-space S-matrix. I will stress how the cosmological polytopes allow us to: compute the wavefunction of the universe at arbitrary points and arbitrary loops (with novel representations for it); interpret the residues of its poles in terms of flat-space processes; provide a general geometrical proof for the flat-space cutting rules; reconstruct the perturbative wavefunction from the knowledge of the flat-space S-matrix and a subset of symmetries enjoyed by the wavefunction.