Past

We study the definability of valuation rings in ordered fields (in the language of ordered rings). We show that any henselian valuation ring that is definable in the language of ordered rings is already definable in the language of rings. However, this does not hold when we drop the assumption of henselianity.

This is joint work with Philip Dittmann, Sebastian Krapp and Salma Kuhlmann.

Groupoid C*-algebras and twisted groupoid C*-algebras are introduced by Renault in the late ’70. Twisted groupoid C*-algebras have since proved extremely important in the study of structural properties for large classes of C*-algebras. On the other hand, Steinberg algebras are introduced independently by Steinberg and Clark, Farthing, Sims and Tomforde around 2010 which are a purely algebraic analogue of groupoid C*-algebras. Steinberg algebras provide useful insight into the analytic theory of groupoid C*-algebras and give rise to interesting examples of *-algebras. In this talk, I will first recall some relevant background on topological groupoids and twisted groupoid C*-algebras, then I will introduce twisted Steinberg algebras which generalise the Steinberg algebras and provide a purely algebraic analogue of twisted groupoid C*-algebras. If I have enough time, I will further introduce pair of algebras which consist of a Steinberg algebra and an algebra of locally constant functions on the unit space, it is an algebraic analogue of Cartan pairs

Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on the non-empty open subset of algebraically smooth points, through which every invariant morphism factors uniquely. Some applications to quotients by connected groups will be discussed.
 

The characteristic polynomial of a random Hermitian matrix induces naturally a field on the real line. In the case of the Gaussian Unitary ensemble (GUE), this fields is expected to have a very special correlation structure: the logarithm of this field is log-correlated and its maximum is at the heart of a conjecture from Fyodorov and Simm predicting its asymptotic behavior.   As a first step in this direction, we obtained in collaboration with R. Butez and O. Zeitouni, a central limit theorem for the logarithm of the characteristic polynomial of the Gaussian beta Ensembles and for a certain class of random Jacobi matrices. In this talk, I will explain how the tridiagonal representation of the GUE and orthogonal polynomials techniques allow us to analyse the fluctuations of the characteristic polynomial.

We investigate the maximum distance of a rank-two tensor to rank-one tensors. An equivalent problem is given by the minimal ratio of spectral and Frobenius norm of a tensor. For matrices the distance of a rank k matrix to a rank r matrices is determined by its singular values, but since there is a lack of a fitting analog of the singular value decomposition for tensors, this question is more difficult in the regime of tensors.
 

This talk is about the global structure of planar graphs and other more general graph classes. The starting point is the Lipton-Tarjan separator theorem, followed by Baker's decomposition of a planar graph into layers with bounded treewidth. I will then move onto layered treewidth, which is a more global version of Baker's decomposition. Layered treewidth is a precursor to the recent development of row treewidth, which has been the key to solving several open problems. Finally, I will describe generalisations for arbitrary minor-closed classes.

In this talk I will present network-theoretic tools [1-2] to filter information in large-scale datasets and I will show that these are powerful tools to study complex datasets. In particular I will introduce correlation-based information filtering networks and the planar filtered graphs (PMFG) and I will show that applications to financial data-sets can meaningfully identify industrial activities and structural market changes [3-4].

It has been shown that by making use of the 3-clique structure of the PMFG a clustering can be extracted allowing dimensionality reduction that keeps both local information and global hierarchy in a deterministic manner without the use of any prior information [5-6]. However, the algorithm so far proposed to construct the PMFG is numerically costly with O(N3) computational complexity and cannot be applied to large-scale data. There is therefore scope to search for novel algorithms that can provide, in a numerically efficient way, such a reduction to planar filtered graphs. I will introduce a new algorithm, the TMFG (Triangulated Maximally Filtered Graph), that efficiently extracts a planar subgraph which optimizes an objective function. The method is scalable to very large datasets and it can take advantage of parallel and GPUs computing [7]. Filtered graphs are valuable tools for risk management and portfolio optimization too [8-9] and they allow to construct probabilistic sparse modeling for financial systems that can be used for forecasting, stress testing and risk allocation [10].

Filtered graphs can be used not only to extract relevant and significant information but more importantly to extract knowledge from an overwhelming quantity of unstructured and structured data. I will provide a practitioner example by a successful Silicon Valley start-up, Yewno. The key idea underlying Yewno’s products is the concept of the Knowledge Graph, a framework based on filtered graph research, whose goal is to extract signals from evolving corpus of data. The common principle is that a methodology leveraging on developments in Computational linguistics and graph theory is used to build a graph representation of knowledge [11], which can be automatically analysed to discover hidden relations between components in many different complex systems. This Knowledge Graph based framework and inference engine has a wide range of applications, including finance, economics, biotech, law, education, marketing and general research.

[1] T. Aste, T. Di Matteo, S. T. Hyde, Physica A 346 (2005) 20.

[2] T. Aste, Ruggero Gramatica, T. Di Matteo, Physical Review E 86 (2012) 036109.

[3] M. Tumminello, T. Aste, T. Di Matteo, R. N. Mantegna, PNAS 102, n. 30 (2005) 10421.

[4] N. Musmeci, Tomaso Aste, T. Di Matteo, Journal of Network Theory in Finance 1(1) (2015) 1-22.

[5] W.-M. Song, T. Di Matteo, and T. Aste, PLoS ONE 7 (2012) e31929.

[6] N. Musmeci, T. Aste, T. Di Matteo, PLoS ONE 10(3): e0116201 (2015).

[7] Guido Previde Massara, T. Di Matteo, T. Aste, Journal of Complex networks 5 (2), 161 (2016).

[8] F. Pozzi, T. Di Matteo and T. Aste, Scientific Reports 3 (2013) 1665.

[9] N. Musmeci, T. Aste and T. Di Matteo, Scientific Reports 6 (2016) 36320.

[10] Wolfram Barfuss, Guido Previde Massara, T. Di Matteo, T. Aste, Phys.Rev. E 94 (2016) 062306.

[11] Ruggero Gramatica, T. Di Matteo, Stefano Giorgetti, Massimo Barbiani, Dorian Bevec and Tomaso Aste, PLoS One (2014) PLoS ONE 9(1): e84912.

iii) identify a special class of correlated compressors based on the idea of random permutations, for which we coin the term PermK. The use of this technique results in the strict improvement on the previous MARINA rate. In the low Hessian variance regime, the improvement can be as large as √n, when d > n, and 1 + √d/n, when n<=d, where n is the number of workers and d is the number of parameters describing the model we are learning.

Ardakov and Wadsley developed a theory of D-modules on rigid analytic spaces and established a Beilinson-Bernstein style localisation theorem for coadmissible modules over the locally analytic distribution algebra. Using this theory, they obtained admissible locally analytic representations of SL_2 by taking global sections of Drinfeld line bundles. In this talk, we will extend their techniques to SL_3 by studying the Drinfeld upper half-space \Omega^{(3)} of dimension 2.

In 2020, cirrhosis and other liver diseases were among the top five causes of death for
individuals aged 35-65 in Scotland, England and Wales. At present, the only curative
treatment for end-stage liver disease is through transplant which is unsustainable.
Stem cell therapies could provide an alternative. By encapsulating the stem cells we
can modulate the shear stress imposed on each cell to promote integrin expression
and improve the probability of engraftment. We model an individual, hydrogel-coated
stem cell moving along a fluid-filled channel due to a Stokes flow. The stem cell is
treated as a Newtonian fluid and the coating is treated as a poroelastic material with
finite thickness. In the limit of a stiff coating, a semi-analytical approach is developed
which exploits a decoupling of the fluids and solid problems. This enables the tractions
and pore pressures within the coating to be obtained, which then feed directly into a
purely solid mechanics problem for the coating deformation. We conduct a parametric
study to elucidate how the properties of the coating can be tuned to alter the stress
experienced by the cell.

Holography in asymptotically flat spaces is one of the most coveted goals of modern mathematical physics. In this talk, I will motivate a novel holographic description of self-dual SO(8) Yang-Mills + self-dual conformal gravity on a Euclidean signature, asymptotically flat background called Burns space. The holographic dual lives on a stack of D1-branes wrapping a CP^1 cycle in the twistor space of R^4 and is given by a gauged beta-gamma system with SO(8) flavor and a pair of defects at the north and south poles. It provides the first example of a stringy realization of (asymptotically) flat holography and is a Euclidean signature variant of celestial holography. This is based on ongoing work with Kevin Costello and Natalie Paquette.

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.

In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.

The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

Many classical arithmetic problems ranging from the elementary ones such as the density of square-free numbers to more difficult such as the density of primes, can be extended to integer matrices. Arithmetic problems over higher dimensions are typically much more difficult. Indeed, the Bateman-Horn conjecture predicting the density of numbers giving prime values of multivariate polynomials is very much open. In this talk I give an overview of the unfortunately brief history of integer random matrices.

Let X be a closed Riemannian manifold, and represent the algebra C(X) of continuous functions on X on the Hilbert space L^2(X) by multiplication.  Inspired by the heat kernel proof of the Atiyah-Singer index theorem, I'll explain how to describe K-homology (i.e. the dual theory to Atiyah-Hirzebruch K-theory) in terms of parametrized families of operators on L^2(X) that get more and more 'local' in X as time tends to infinity.

I'll then switch perspectives from C(X) -- the prototypical example of a commutative C*-algebra -- to noncommutative C*-algebras built from discrete groups, and explain how the underlying large-scale geometry of the groups can give rise to approximate 'decompositions' of the C*-algebras.  I'll then explain how to use these decompositions and localization in the sense above to compute K-homology, and the connection to some conjectures in topology, geometry, and C*-algebra theory.

Let $N$ be a compact Riemannian manifold of dimension 3 or higher, and $g$ a Lipschitz non-negative (or non-positive) function on $N$. In joint works with Neshan Wickramasekera we prove that there exists a closed hypersurface $M$ whose mean curvature attains the values prescribed by $g$. Except possibly for a small singular set (of codimension 7 or higher), the hypersurface $M$ is $C^2$ immersed and two-sided (it admits a global unit normal); the scalar mean curvature at $x$ is $g(x)$ with respect to a global choice of unit normal. More precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections: around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when $g$ is a constant, in which the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature.

Moving from persistent homology in one parameter to multiparameter persistence comes at a significant increase in complexity. In particular, the notion of a barcode does not generalize straightforwardly. However, in this talk, I will show how it is possible to assign a unique barcode to a multiparameter persistence module if one is willing to take Z-linear combinations of intervals. The theoretical discussion will be complemented by numerical experiments. This is joint work with Steffen Oppermann and Steve Oudot.

In embryonic development, formation of the retinal vasculature is  critically dependent on prior establishment of a mesh of astrocytes.  
Astrocytes emerge from the optic nerve head and then migrate over the retinal surface in a radially symmetric manner and mature through 
differentiation.  We develop a PDE model describing the migration and  differentiation of astrocytes, and numerical simulations are compared to 
experimental data to assist in elucidating the mechanisms responsible for the distribution of astrocytes via parameter analysis. This is joint 
work with Timothy Secomb.

In his 1976 proof of the converse of Herbrand’s theorem, Ribet used Eisenstein-cuspidal congruences to produce unramified degree-p extensions of the p-th cyclotomic field when p is an odd prime. After reviewing Ribet’s strategy, we will discuss recent work with Preston Wake in which we apply similar techniques to produce unramified degree-p extensions of Q(N^{1/p}) when N is a prime that is congruent to -1 mod p. This answers a question posed on Frank Calegari’s blog.

Tensor structured linear operators play an important role in matrix equations and low-rank modelling. Motivated by this we consider the problem of approximating a matrix by a sum of Kronecker products. It is known that an optimal approximation in Frobenius norm can be obtained from the singular value decomposition of a rearranged matrix, but when the goal is to approximate the matrix as a linear map, an operator norm would be a more appropriate error measure. We present an alternating optimization approach for the corresponding approximation problem in spectral norm that is based on semidefinite programming, and report on its practical performance for small examples.
This is joint work with Venkat Chandrasekaran and Mareike Dressler.

Statics and dynamics of droplets on lubricated surfaces

Zhaohe Dai

The abstract is "Slippery liquid infused porous surfaces are formed by coating surface with a thin layer of oil lubricant. This thin layer prevents other droplets from reaching the solid surface and allows such deposited droplets to move with ultra-low friction, leading to a range of applications. In this talk, we will discuss the static and dynamic behaviour of droplets placed on lubricated surfaces. We will show that the layer thickness and the size of the substrate are key parameters in determining the final equilibrium. However, the evolution towards the equilibrium is extremely slow (on the order of days for typical experimental parameter values). As a result, we suggest that most previous experiments with oil films lubricating smooth substrates are likely to have been in an evolving, albeit slowly evolving, transient state.

Modeling and Design Optimization for Pleated Membrane Filters

Yixuan Sun

Membrane filtration is widely used in many applications, ranging from industrial processes to everyday living activities. With growing interest from both industrial and academic sectors in understanding the various types of filtration processes in use, and in improving filter performance, the past few decades have seen significant research activity in this area. Experimental studies can be very valuable, but are expensive and time-consuming, therefore theoretical studies offer potential as a cost-effective and predictive way to improve on current filter designs. In this work, mathematical models, derived from first principles and simplified using asymptotic analysis, are proposed for pleated membrane filters, where the macroscale flow problem of Darcy flow through a pleated porous medium is coupled to the microscale fouling problem of particle transport and deposition within individual pores of the membrane. Asymptotically-simplified models are used to describe and evaluate the membrane performance numerically and filter design optimization problems are formulated and solved for industrially-relevant scenarios. This study demonstrates the potential of such modeling to guide industrial membrane filter design for a range of applications involving purification and separation.

The question of when you can quasiisometrically embed a solvable Baumslag-Solitar group into another turns out to be equivalent to the question of when you can (1,A)-quasiisometrically embed a rooted tree into another rooted tree. We will briefly describe the geometry of the solvable Baumslag-Solitar groups before attacking the problem of embedding trees. We will find that the existence of (1,A)-quasiisometric embeddings between trees is intimately related to the boundedness of a family of integer sequences. 

The dynamics of quantum many-body systems is intricately related to random matrix theory (RMT), to such a degree that quantum chaos is even defined through random matrix level statistics. However, exact results on this connection are typically precluded by the exponentially large Hilbert space. After a short introduction to the role of RMT in many-body dynamics, I will show how dual-unitary circuits present a minimal model of quantum chaos where this connection can be made rigorous. This will be illustrated using a new kind of emergent random matrix behaviour following a quantum quench: starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design, which can be shown to hold exactly in dual-unitary circuit dynamics.

Pair combination repurposing of drugs is a common challenge faced by researchers in the pharmaceutical industry. Network biology and molecular machine learning based drug pair scoring techniques offer computation tools to predict the interaction, polypharmacy side effects and synergy of drugs. In this talk we overview of three things: (a) the theory and unified model of drug pair scoring (b) a relational machine learning model that can solve the pair scoring task (c) the design of large-scale machine learning systems needed to tackle the pair scoring task.

ArXiv links: https://arxiv.org/abs/2111.02916, https://arxiv.org/abs/2110.15087, https://arxiv.org/abs/2202.05240.

ML library: https://github.com/AstraZeneca/chemicalx

Can the S-matrix be complexified in a way consistent with causality? Since the 1960's, the affirmative answer to this question has been well-understood for 2→2 scattering of the lightest particle in theories with a mass gap at low momentum transfer, where the S-matrix is analytic everywhere except at normal-threshold branch cuts. We ask whether an analogous picture extends to realistic theories, such as the Standard Model, that include massless fields, UV/IR divergences, and unstable particles. Especially in the presence of light states running in the loops, the traditional iε prescription for approaching physical regions might break down, because causality requirements for the individual Feynman diagrams can be mutually incompatible. We demonstrate that such analyticity problems are not in contradiction with unitarity. Instead, they should be thought of as finite-width effects that disappear in the idealized 2→2 scattering amplitudes with no unstable particles, but might persist at higher multiplicity. To fix these issues, we propose an iε-like prescription for deforming branch cuts in the space of Mandelstam invariants without modifying the analytic properties. This procedure results in a complex strip around the real part of the kinematic space, where the S-matrix remains causal. To help with the investigation of related questions, we introduce holomorphic cutting rules, new approaches to dispersion relations, as well as formulae for local behavior of Feynman integrals near branch points, all of which are illustrated on explicit examples.

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.

In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.

The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

The distribution of primes in arithmetic progressions (AP) s a central question of analytic number theory. It is closely connected to the additive behaviour of primes (for example in the Goldbach problem) and application of sieves (for example in the Twin Prime problem). In this talk I will outline the basic results without going into technical details. The central questions I will consider are: What are the different tools used to study primes in AP? In what ranges of moduli are they useful? What error terms can be achieved? How do recent developments fit into the bigger picture?

Knot theory is divided into several subfields. One of these is hyperbolic knot theory, which is focused on the hyperbolic structure that exists on many knot complements. Another branch of knot theory is concerned with invariants that have connections to 4-manifolds, for example the knot signature and Heegaard Floer homology. In my talk, I will describe a new relationship between these two fields that was discovered with the aid of machine learning. Specifically, we show that the knot signature can be estimated surprisingly accurately in terms of hyperbolic invariants. We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. Our main result is that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This theorem has applications to Dehn surgery and to 4-ball genus. We will also present a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that have odd linking number with the knot. My talk will outline the proofs of these results, as well as describing the role that machine learning played in their discovery.

This is joint work with Alex Davies, Andras Juhasz, and Nenad Tomasev

We consider, in a framework of iterated random conformal maps, a two-parameteraggregation model of Hastings-Levitov type, in which the size and intensity of new particles are each chosen to vary as a power of the density of harmonic measure. Then we consider a limit
in which the overall intensity of particles become large, while the particles themselves become
small. For a certain `sub-critical' range of parameter values, we can show a law of large numbers and fluctuation central limit theorem. The admissible range of parameters includes an off-lattice version of the Eden model, for which we can show that disk-shaped clusters are stable.
Many open problem remain, not least because the limit PDE does not yet have a satisfactory mathematical theory.

This is joint work with Vittoria Silvestri and Amanda Turner.

Most Ricci flow theory takes the short-time existence of solutions as a starting point and ends up concerned with understanding the long-time limiting behaviour and the structure of any finite-time singularities that may develop along the way. In this talk I will look at what you can think of as singularities at time zero. I will describe some of the situations in which one would like to start a  Ricci flow with a space that is rougher than a smooth bounded curvature Riemannian manifold, and some of the situations in which one considers smooth initial data that is only achieved in a non-smooth way. A particularly interesting and useful case is the problem of starting a Ricci flow on a Riemann surface equipped with a measure. I will not be assuming expertise in Ricci flow theory. Parts of the talk are joint with either Hao Yin (USTC) or ManChun Lee (CUHK).

We consider the four-point correlator of the stress-energy tensor in N=4 SYM, to leading order in inverse powers of the central charge, but including all order corrections in 1/lambda. This corresponds to the AdS version of the Virasoro-Shapiro amplitude to all orders in the small alpha'/low energy expansion. Using dispersion relations in Mellin space, we derive an infinite set of sum rules. These sum rules strongly constrain the form of the amplitude, and determine all coefficients in the low energy expansion in terms of the CFT data for heavy string operators, in principle available from integrability. For the first set of corrections to the flat space amplitude we find a unique solution consistent with the results from integrability and localisation.

A few years back, Smale spaces were shown to exhibit noncommutative Poincaré duality (with Jerry Kaminker and Ian Putnam). The fundamental class was represented as an extension by the compacts. In current work we describe a Fredholm module representation of the fundamental class. The proof uses delicate approximations of the Smale space arising from a refining sequence of (open) Markov partition covers. I hope to explain all these notions in an elementary manner. This is joint work with Dimitris Gerontogiannis and Joachim Zacharias.

In this talk, I will introduce our investigations on how to make deep learning easier to optimize, faster to train, and more robust to out-of-distribution prediction. To be specific, we design a group-invariant optimization framework for ReLU neural networks; we compensate the gradient delay in asynchronized distributed training; and we improve the out-of-distribution prediction by incorporating “causal” invariance.

Dr Oliver Sheridan-Methven from Citadel Securities, (an InFoMM and MScMCF alumni), will be talking about his experiences from studying at the Mathematical Institute, interviewing for jobs, to working in finance. Now in Zurich, Oliver is a quantitative developer in the advanced scientific computing team at Citadel Securities, a world leading market maker. Citadel Securities specialises in ultra high frequency trading, low latency execution, and their researchers tackle cutting edge machine learning and data science problems on colossal data sets with humongous computational resources. Oliver will be talking about his own experiences, and also how mathematicians are naturally great fits for a huge number of roles at Citadel Securities.

We discuss linear convolutional neural networks (LCNs) and their critical points. We observe that the function space (that is, the set of functions represented by LCNs) can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the network's architecture on the geometry of the function space.

For instance, for LCNs with one-dimensional convolutions having stride one and arbitrary filter sizes, we provide a full description of the boundary of the function space. We further study the optimization of an objective function over such LCNs: We characterize the relations between critical points in function space and in parameter space and show that there do exist spurious critical points. We compute an upper bound on the number of critical points in function space using Euclidean distance degrees and describe dynamical invariants for gradient descent.

This talk is based on joint work with Thomas Merkh, Guido Montúfar, and Matthew Trager.

The blow-up B of a scheme X in a closed subscheme Z enjoys the universal property that for any scheme X' over X such that the pullback of Z to X' is an effective Cartier divisor, there is a unique morphism of X' into B over X. It is well-known that the blow-up commutes along flat base change.

In this talk, I will discuss a derived enhancement B' of B, namely the derived blow-up, which enjoys a universal property against all schemes over X, satisfies arbitrary (derived) base-change, and contains B as a closed subscheme. To this end, we will need some elements from derived algebraic geometry, which I will review along the way. This will allow us to construct the derived blow-up as the projective spectrum of the derived Rees algebra, and state its functor of points in terms of virtual Cartier divisors, using Weil restrictions.

This is based on ongoing joint work with Adeel Khan and David Rydh.

We consider close-packed tiling models of geometric objects -- a mixture of hardcore dimers and plaquettes -- as a generalisation of the familiar dimer models. Specifically, on an anisotropic cubic lattice, we demand that each site be covered by either a dimer on a z-link or a plaquettein the x-y plane. The space of such fully packed tilings has an extensive degeneracy. This maps onto a fracton-type `higher-rank electrostatics', which can exhibit a plaquette-dimer liquid and an ordered phase. We analyse this theory in detail, using height representations and T-duality to demonstrate that the concomitant phase transition occurs due to the proliferation of dipoles formed by defect pairs. The resultant critical theory can be considered as a fracton version of the Kosterlitz-Thouless transition. A significant new element is its UV-IR mixing, where the low energy behavior of the liquid phase and the transition out of it is dominated by local (short-wavelength) fluctuations, rendering the critical phenomenon beyond the renormalization group paradigm.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

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.

Random walks are a common model for the exploration and discovery of complex networks. While numerous algorithms have been proposed to map out an unknown network, a complementary question arises: in a known network, which nodes and edges are most likely to be discovered by a random walker in finite time? In this talk we introduce exposure theory, a statistical mechanics framework that predicts the learning of nodes and edges across several types of networks, including weighted and temporal, and show that edge learning follows a universal trajectory. While the learning of individual nodes and edges is noisy, exposure theory produces a highly accurate prediction of aggregate exploration statistics. As a specific application, we extend exposure theory to better understand human learning with its typical mental errors, and thus account for distortions of learned networks.

This talk is based on https://arxiv.org/abs/2202.11262