Past

The holographic entanglement entropy formula identifies the generalized entropy of the bulk AdS spacetime with the entanglement entropy of the boundary CFT. However the bulk microstate interpretation of the generalized entropy remains poorly understood. Progress along this direction requires understanding how to define Hilbert space factorization and entanglement entropy in the bulk closed string theory.   As a toy model for AdS/CFT, we study the entanglement entropy of closed strings in the topological A model, which enjoys a gauge-string duality.   We define a notion of generalized entropy for closed strings on the resolved conifold using the replica trick.   As in AdS/CFT, we find this is dual to (defect) entanglement entropy in the dual Chern Simons gauge theory.   Our main result is a bulk microstate interpretation of generalized entropy in terms of open strings and their edge modes, which we identify as entanglement branes.   

More precisely, we give a self consistent factorization of the closed string Hilbert space which introduces open string edge modes transforming under a q-deformed surface symmetry group. Compatibility with this symmetry requires a q-deformed definition of entanglement entropy. Using the topological vertex formalism, we define the Hartle Hawking state for the resolved conifold and compute its q-deformed entropy directly from the reduced density matrix.   We show that this is the same as the generalized entropy.   Finally, we relate non local aspects of our factorization map to analogous phenomenon recently found in JT gravity.

Abstract: In this work, we analyze a class of stochastic inverse optimal control problems with entropy regularization. We first characterize the set of solutions for the inverse control problem. This solution set exemplifies the issue of degeneracy in generic inverse control problems that there exist multiple reward or cost functions that can explain the displayed optimal behavior. Then we establish one resolution for the degeneracy issue by providing one additional optimal policy under a different discount factor. This resolution does not depend on any prior knowledge of the solution set. Through a simple numerical experiment with deterministic transition kernel, we demonstrate the ability of accurately extracting the cost function through our proposed resolution.

Joint work with Sam Cohen (Oxford) and Lukasz Szpruch (Edinburgh).

I will discuss modified Newton methods for solving nonlinear systems of PDEs. These methods introduce additional variables before deriving the Newton linearization. These variables can then often be eliminated analytically before solving the Newton system, such that existing solvers can be adapted easily and the computational cost does not increase compared to a standard Newton method. The resulting algorithms yield favorable convergence properties. After illustrating the ideas on a simple example, I will show its application for the solution of incompressible Stokes flow problems with viscoplastic constitutive relation, where the additionally introduced variable is the stress tensor. These models are commonly used in earth science models. This is joint work with Johann Rudi (Argonne) and Melody Shih (NYU).

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Synchronized activity of neurons is important for many aspects of brain function. Synchronization is affected by both network-level parameters, such as connectivity between neurons, and neuron-level parameters, such as firing rate. Many of these parameters are not static but may vary slowly in time. In this talk we focus on neuron-level parameters. Our work centres on the neurotransmitter acetylcholine, which has been shown to modulate the firing properties of several types of neurons through its affect on potassium currents such as the muscarine-sensitive M-current.  In the brain, levels of acetylcholine change with activity.  For example, acetylcholine is higher during waking and REM sleep and lower during slow wave sleep. We will show how the M-current affects the bifurcation structure of a generic conductance-based neural model and how this determines synchronization properties of the model.  We then use phase-model analysis to study the effect of a slowly varying M-current on synchronization.  This is joint work with Victoria Booth, Xueying Wang and Isam Al-Darbasah

The integral of mean curvature squared is a conformal invariant that measures the distance from a given immersion to the standard embedding of a round sphere. Following work of Robert Bryant who showed that all Willmore spheres in the 3-sphere are conformally minimal, Robert Kusner proposed in the early 1980s to use the Willmore energy to obtain an “optimal” sphere eversion, called the min-max sphere eversion.

We will present a method due to Tristan Rivière that permits to tackle a wide variety of min-max problems, including ones about the Willmore energy. An important step to solve Kusner’s conjecture is to determine the Morse index of branched Willmore spheres, and we show that the Morse index of conformally minimal branched Willmore spheres is equal to the index of a canonically associated matrix whose dimension is equal to the number of ends of the dual minimal surface.

I will discuss compressible types and relate them to uniform definability of types over finite sets (UDTFS), to uniformity of honest definitions and to the construction of compressible models in the context of (local) NIP. All notions will be defined during the talk.
Joint with Martin Bays and Pierre Simon.

We introduce the notion of systolic complexes and give conditions on presentations to construct such complexes using Cayley graphs.

We consider Garside groups to find examples of groups admitting such a presentation.
 

The general Structural Reflection (SR) principle asserts that for every definable, in the first-order language of set theory, possibly with parameters, class $\mathcal{C}$ of relational structures of the same type there exists an ordinal $\alpha$ that reflects $\mathcal{C}$, i.e.,  for every $A$ in $\mathcal{C}$ there exists $B$ in $\mathcal{C}\cap V_\alpha$ and an elementary embedding from $B$ into $A$. In this form, SR is equivalent to Vopenka’s Principle (VP). In my talk I will present some different natural variants of SR which are equivalent to the existence some well-known large cardinals weaker than VP. I will also consider some forms of SR, reminiscent of Chang’s Conjecture, which imply the existence of large cardinal principles stronger than VP, at the level of rank-into-rank embeddings and beyond. The latter is a joint work with Philipp Lücke.

For five decades, mathematicians have exploited the beauties of random matrix theory (RMT) in the hope of discovering principles which govern complex ecosystems. While RMT lies at the heart of the ideas, their translation toward biological reality requires some heavy lifting: dynamical systems theory, statistics, and large-scale computations are involved, and any predictions should be challenged with empirical data. This can become very awkward.

In addition to a morose journey through some of my personal failures to make RMT meet reality, I will try to sketch out some more constructive future perspectives. In particular, new methods for microbial community composition, dynamics and evolution might allow us to apply RMT ideas to the treatment of cystic fibrosis. In addition, in fisheries I will argue that sometimes the very absence of an empirical dataset can add to the practical value of models as tools to influence policy.

Numerical relativity allows us to simulate the behaviour of regions of space and time where gravity is strong and dynamical. For example, it allows us to calculate precisely the gravitational waveform that should be generated by the merger of two inspiralling black holes. Since the first detection of gravitational waves from such an event in 2015, banks of numerical relativity “templates” have been used to extract further information from noisy data streams. In this talk I will give an overview of the field - what are we simulating, why, and what are the main challenges, past and future.

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The Harish-Chandra Lefschetz principle asserts representation theory for real groups, p-adic groups and automorphic forms should be placed on an equal footing. A particular example in this aspect is that Ciubotaru and Trapa constructed Arakawa-Suzuki type functors between category of Harish-Chandra modules and category of graded Hecke algebra modules, giving an explicit connection on the representation categories between p-adic and real sides. 

This talk plans to begin with comparing the representation theory between real and p-adic general linear groups, such as unitary and unipotent representations. Then I shall explain results in more details on the p-adic branching law from GL(n+1) to GL(n), including branching laws for Arthur type representations (one of the non-tempered Gan-Gross-Prasad conjectures). The analogous results and predictions on the real group side will also be discussed. Time permitting, I will explain a notion of left-right Bernstein-Zelevinsky derivatives and its applications on branching laws.
 

We present a probabilistic generative model and efficient algorithm to model reciprocity in directed networks. Unlike other methods that address this problem such as exponential random graphs, it assigns latent variables as community memberships to nodes and a reciprocity parameter to the whole network rather than fitting order statistics. It formalizes the assumption that a directed interaction is more likely to occur if an individual has already observed an interaction towards her. It provides a natural framework for relaxing the common assumption in network generative models of conditional independence between edges, and it can be used to perform inference tasks such as predicting the existence of an edge given the observation of an edge in the reverse direction. Inference is performed using an efficient expectation-maximization algorithm that exploits the sparsity of the network, leading to an efficient and scalable implementation. We illustrate these findings by analyzing synthetic and real data, including social networks, academic citations and the Erasmus student exchange program. Our method outperforms others in both predicting edges and generating networks that reflect the reciprocity values observed in real data, while at the same time inferring an underlying community structure. We provide an open-source implementation of the code online.

arXiv link: https://arxiv.org/abs/2012.08215

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Event shape observables describe how energy is distributed in the final state in scattering processes. Recent years have seen increasing interest from different physics areas in event shapes, in particular the energy correlators. They define a class of observable quantities which admit a simple and unified formulation in quantum  field theory.

Three-point energy correlators (EEEC) measure the energy flow through three detectors as a function of the three angles between them. We analytically compute the one-loop EEEC in maximally supersymmetric Yang-Mills theory. The result is a linear combination of logarithms and dilogarithms, decomposed onto a basis of single-valued transcendental functions. Its symbol contains 16 alphabet letters, revealing a dihedral symmetry of the three-point event shape.  Our results represent the first perturbative computation of a three-parameter event-shape observable, providing information on the function space at higher-loop order, and valuable input to the study of conformal light-ray OPE.

The Alfven waves are fundamental wave phenomena in magnetized plasmas and the dynamics of Alfven waves are governed by the MHD system. In the talk,  we construct and study the long time behavior of (viscous and non-viscous) Alfven waves.

As applications, (1) We provide a rigorous justification for the following dynamical phenomenon observed in many contexts: the solution at the beginning behave like non-dispersive waves and the shape of the solution persists for a very long time (proportional to the Reynolds number); thereafter, the solution will be damped due to the long-time accumulation of the diffusive effects;

(2) We prove the rigidity aspects of the scattering problem for the MHD equations: We prove that the Alfven waves must vanish if their scattering fields vanish at infinities.

We present linear-quadratic stochastic differential games on directed chains inspired by the directed chain stochastic differential equations introduced by Detering, Fouque, and Ichiba in a previous work. We solve explicitly for Nash equilibria with a finite number of players and we study more general finite-player games with a mixture of both directed chain interaction and mean field interaction. We investigate and compare the corresponding games in the limit when the number of players tends to infinity. 

The limit is characterized by Catalan functions and the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain, with or without the presence of mean field interaction.

Joint work with Yichen Feng and Tomoyuki Ichiba.

Together with Alex Degtyarev and Vincent Florence we introduced a new link invariant, called slope, of a colored link in an integral homology sphere. In this talk I will define the invariant, highlight some of its most interesting properties as well as its relationship to Conway polynomials and to the  Kojima–Yamasaki eta-function. The stress in this talk will be on our latest computational progress: a formula to calculate the slope from a C-complex.

Jaclyn Lang
Explicit Class Field Theory
Class field theory was a major achievement in number theory
about a century ago that presaged many deep connections in mathematics
that today are known as the Langlands Program.  Class field theory
associates to each number field an special extension field, called the
Hilbert class field, whose ring of integers satisfies unique
factorization, mimicking the arithmetic in the usual integers.  While
the existence of this field is always guaranteed, it is a difficult
problem to find explicit generators for the Hilbert class field in
general.  The theory of complex multiplication of elliptic curves is
essentially the only setting where there is an explicit version of class
field theory.  We will briefly introduce class field theory, highlight
what is known in the theory of complex multiplication, and end with an
example for the field given by a fifth root of 19.  There will be many
examples!

Jan Sbierski
The strength of singularities in general relativity
One of the many curious features of Einstein’s theory of general relativity is that the theory predicts its own breakdown at so-called gravitational singularities. The gravitational field in general relativity is modelled by a Lorentzian manifold — and thus a gravitational singularity is signalled by the geometry of the Lorentzian manifold becoming singular. In this talk I will first review the classical definition of a gravitational singularity along with a classification of their strengths. I will conclude with outlining newly developed techniques which capture the singularity at the level of the connection of Lorentzian manifolds.

Many machine learning applications involve non-Euclidean data, such as graphs, strings or matrices. In such cases, exploiting Riemannian geometry can deliver algorithms that are computationally superior to standard(Euclidean) nonlinear programming approaches. This observation has resulted in an increasing interest in Riemannian methods in the optimization and machine learning community.

In the first part of the talk, we consider the task of learning a robust classifier in hyperbolic space. Such spaces have received a surge of interest for representing large-scale, hierarchical data, due to the fact that theyachieve better representation accuracy with fewer dimensions. We present the first theoretical guarantees for the (robust) large margin learning problem in hyperbolic space and discuss conditions under which hyperbolic methods are guaranteed to surpass the performance of their Euclidean counterparts. In the second part, we introduce Riemannian Frank-Wolfe (RFW) methods for constrained optimization on manifolds. Here, we discuss matrix-valued tasks for which such Riemannian methods are more efficient than classical Euclidean approaches. In particular, we consider applications of RFW to the computation of Riemannian centroids and Wasserstein barycenters, both of which are crucial subroutines in many machine learning methods.

The classification of finite simple groups shows that many (those of Lie type) are obtained as (projectivisations of) subgroups of some $GL_{n}(\mathbb{F}_{q})$.

Green first determined the character table of any $GL_{n}(\mathbb{F}_{q})$ in his influential 1955 paper, while others have since given more explicit constructions of certain `cuspidal' representations.

In this talk, I will introduce parabolic induction as a means of obtaining representations of $GL_{n}(\mathbb{F}_{q})$ from those of $GL_{m}(\mathbb{F}_{q})$ where $m<n$.

Finding the irreducible representations of any $GL_{n}(\mathbb{F}_{q})$ then becomes inductive on $n$ for fixed $q$, with the cuspidal representations serving as atoms for this process.

Harish-Chandra's philosophy of cusp forms reduces the problem to the following two steps:

•  Find the cuspidal representations of any $GL_{n}(\mathbb{F}_{q})$
•  Determine the irreducible components of any representation $\sigma_{1}\circ\dots\circ\sigma_{k}$ parabolically induced from cuspidals $\sigma_{i}$

The majority of my talk will then aim to address each of these points.

Natural killer (NK) cells are part of the innate immune system and are capable of killing diseased cells. As a result, NK cells are being used for adoptive cell therapies for cancer patients. The activation of NK cell stimulatory receptors leads to a cascade of intracellular phosphorylation reactions, which activates key signaling species that facilitate the secretion of cytolytic molecules required for cell killing. Strategies that maximize the activation of such intracellular species can increase the likelihood of NK cell killing upon contact with a cancer cell and thereby improve efficacy of NK cell-based therapies. However, NK cell exhaustion, a phenotype characterized by reduced effector functionality, can limit the NK cell’s capacity for cell lysis. Due to the complexity of intracellular signaling, it is difficult to deduce a priori which strategies can enhance species activation.  

To aid in the development of strategies to enhance NK cell activation and limit the NK cell exhaustion, we constructed a mechanistic model of the signaling pathways activated by stimulatory receptors in NK cells. We then extended the model to describe the dynamics of the cytolytic molecules granzyme B (GZMB) and perforin-1 (PRF1). We implemented an information-theoretic approach to perform a global sensitivity analysis and optimal control theory to investigate strategies to enhance intracellular signaling and maximize GZMB and PRF1 secretion. We recently expanded the modeling to investigate the role of NK cell heterogeneity on tumor cell killing. In total, we developed a theoretical framework that provides actionable insight into engineering robust NK cells for clinical applications.

I will discuss a string theory perspective on the Bethe/Gauge correspondence for the XXX superspin chain. I explain how to realize 4d Chern-Simons theory with gauge supergroup using branes, and how the brane configurations for the superspin chain get mapped to 2d N = (2,2) quiver gauge theories proposed by Nekrasov. This is based on my ongoing work with Nafiz Ishtiaque, Faroogh Moosavian and Surya Raghavendran.

Elastic strings are among the simplest one-dimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (tensile force produces elongation) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible), and after a certain amount of tensile force is applied the stretch of the string is maximized (becomes inextensible). In this talk, we discuss a model for these stretch-limited elastic strings, in what way they model elastic behavior, the well-posedness and asymptotic stability of certain simple motions, and (many) open questions.

In this talk I will discuss a number of topics at the interface between C* algebras and Geometric Group Theory, with an emphasis on Kazhdan projections, various versions of amenability and their connection to the geometry of groups. This is based on joint work with P. Nowak and J. Mackay.

We showcase some newly emerging connections between the theory of random walks and operator algebras, obtained by associating concrete algebras of operators to random walks. The C*-algebras we obtain give rise to new and interesting notions of ratio limit space and boundary, which are computed by appealing to various works on the asymptotic behavior of transition probabilities for random walks. Our results are leveraged to shed light on a question of Viselter on symmetry-unique quotients of Toeplitz C*-algebras of subproduct systems arising from random walks.

Techniques that address sequential data have been a central theme in machine learning research in the past years. More recently, such considerations have entered the field of finance-related ML applications in several areas where we face inherently path dependent problems: from (deep) pricing and hedging (of path-dependent options) to generative modelling of synthetic market data, which we refer to as market generation.

We revisit Deep Hedging from the perspective of the role of the data streams used for training and highlight how this perspective motivates the use of highly-accurate generative models for synthetic data generation. From this, we draw conclusions regarding the implications for risk management and model governance of these applications, in contrast to risk management in classical quantitative finance approaches.

Indeed, financial ML applications and their risk management heavily rely on a solid means of measuring and efficiently computing (similarity-)metrics between datasets consisting of sample paths of stochastic processes. Stochastic processes are at their core random variables with values on path space. However, while the distance between two (finite dimensional) distributions was historically well understood, the extension of this notion to the level of stochastic processes remained a challenge until recently. We discuss the effect of different choices of such metrics while revisiting some topics that are central to ML-augmented quantitative finance applications (such as the synthetic generation and the evaluation of similarity of data streams) from a regulatory (and model governance) perspective. Finally, we discuss the effect of considering refined metrics which respect and preserve the information structure (the filtration) of the market and the implications and relevance of such metrics on financial results.

Orders in major electronic stock markets are executed through centralised limit order books (LOBs). Large amounts of historical data have led to extensive research modeling LOBs, for the purpose of better understanding their dynamics and building simulators as a framework for controlled experiments, when testing trading algorithms or execution strategies.Most work in the literature models the aggregate view of the limit order book, which focuses on the volume of orders at a given price level, using a point process. In addition to this information, brokers and exchanges also have information on the identity of the agents submitting the order. This leads to a more granular view of limit order book dynamics, which we attempt to model using a heterogeneous model of order flow.

We present a granular representation of the limit order book that allows to account for the origins of different orders. Using client order flow from a major broker, we analyze the properties of variables in this representation. The heterogeneity of order flow is modeled by segmenting clients into different clusters, for which we identify representative prototypes. This segmentation appears to be stable both over time as well as over different stocks. Our findings can be leveraged to build more realistic order flow models that account for the diversity of the market participants.

An insightful exercise might be to ask what is the most important idea in linear algebra. Our first answer would not be eigenvalues or linearity, it would be “matrix factorizations.”  We will discuss a blueprint to generate  53 inter-related matrix factorizations (times 2) most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927), Harish-Chandra (1956), and Flensted-Jensen (1978) . We will discuss the interesting history. One anecdote is that Eugene Wigner (1968) discovered factorizations such as the svd in passing in a way that was buried and only eight authors have referenced that work. Ironically Wigner referenced Sigurður Helgason (1962) but Wigner did not recognize his results in Helgason's book. This work also extends upon and completes open problems posed by Mackey²&Tisseur (2003/2005).

Classical results of Random Matrix Theory concern exact formulas from the Hermite, Laguerre, Jacobi, and Circular distributions. Following an insight from Freeman Dyson (1970), Zirnbauer (1996) and Duenez (2004/5) linked some of these classical ensembles to Cartan's theory of Symmetric Spaces. One troubling fact is that symmetric spaces alone do not cover all of the Jacobi ensembles. We present a completed theory based on the generalized Cartan distribution. Furthermore, we show how the matrix factorization obtained by the generalized Cartan decomposition G=K₁AK₂ plays a crucial role in sampling algorithms and the derivation of the joint probability density of A.

Joint work with Sungwoo Jeong.

An insightful exercise might be to ask what is the most important idea in linear algebra. Our first answer would not be eigenvalues or linearity, it would be “matrix factorizations.” We will discuss a blueprint to generate 53 inter-related matrix factorizations (times 2) most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927), Harish-Chandra (1956), and Flensted-Jensen (1978) . We will discuss the interesting history. One anecdote is that Eugene Wigner (1968) discovered factorizations such as the SVD in passing in a way that was buried and only eight authors have referenced that work. Ironically Wigner referenced Sigurður Helgason (1962) but Wigner did not recognize his results in Helgason's book. This work also extends upon and completes open problems posed by Mackey² & Tisseur (2003/2005).

Classical results of Random Matrix Theory concern exact formulas from the Hermite, Laguerre, Jacobi, and Circular distributions. Following an insight from Freeman Dyson (1970), Zirnbauer (1996) and Duenez (2004/5) linked some of these classical ensembles to Cartan's theory of Symmetric Spaces. One troubling fact is that symmetric spaces alone do not cover all of the Jacobi ensembles. We present a completed theory based on the generalized Cartan distribution. Furthermore, we show how the matrix factorization obtained by the generalized Cartan decomposition G=K₁AK₂ plays a crucial role in sampling algorithms and the derivation of the joint probability density of A.

Joint work with Sungwoo Jeong

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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Understanding the motion of small bodies at a fluid interface has relevance to a range of natural systems and technological applications. In this talk, we discuss two systems where capillarity and fluid inertia govern the dynamics of millimetric particles at a fluid interface.

In the first part, we present a study of superhydrophobic spheres impacting a quiescent water bath.  Under certain conditions particles may rebound completely from the interface - an outcome we characterize in detail through a synthesis of experiments, modeling, and direct numerical simulation.  In the second half, we introduce a system wherein millimetric disks trapped at a fluid interface are vertically oscillated and spontaneously self-propel.  Such "capillary surfers" interact with each other via their collective wavefield and self-assemble into a myriad of cooperative dynamic states.  Our experimental observations are well captured by a first theoretical model for their dynamics, laying the foundation for future investigations of this highly tunable active system.

The goal of this talk is to present some elementary examples of fusion 2-categories whilst doing as little higher category theory as possible. More precisely, it turns out that up to a canonical completion operation, certain higher fusion categories are entirely described by their maximal subspaces. I will briefly motivate this completion operation in the 1-categorical case, and go on to explain why working with spaces is good enough in this particular case. Then, we will review some fact about $E_n$-algebras, and why they come into the picture. Finally, we will have a look at some small examples arising from finite groups.

Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if $K$ is a field and $G$ is a torsion-free group, then the group ring $K[G]$ has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and present my recent counterexample to the unit conjecture.

We consider the problem of clustering in two important families of networks: signed and directed, both relatively less well explored compared to their unsigned and undirected counterparts. Both problems share an important common feature: they can be solved by exploiting the spectrum of certain graph Laplacian matrices or derivations thereof. In signed networks, the edge weights between the nodes may take either positive or negative values, encoding a measure of similarity or dissimilarity. We consider a generalized eigenvalue problem involving graph Laplacians, with performance guarantees under the setting of a signed stochastic block model. The second problem concerns directed graphs. Imagine a (social) network in which you spot two subsets of accounts, X and Y, for which the overwhelming majority of messages (or friend requests, endorsements, etc) flow from X to Y, and very few flow from Y to X; would you get suspicious? To this end, we also discuss a spectral clustering algorithm for directed graphs based on a complex-valued representation of the adjacency matrix, which is able to capture the underlying cluster structures, for which the information encoded in the direction of the edges is crucial. We evaluate the proposed algorithm in terms of a cut flow imbalance-based objective function, which, for a pair of given clusters, it captures the propensity of the edges to flow in a given direction. Experiments on a directed stochastic block model and real-world networks showcase the robustness and accuracy of the method, when compared to other state-of-the-art methods. Time permitting, we briefly discuss potential extensions to the sparse setting and regularization, applications to lead-lag detection in time series and ranking from pairwise comparisons.

Dark matter is known to exist, but no-one knows what it is or where it came
from.  We describe a new production mechanism of particle dark matter, which
hinges on a first-order cosmological phase transition.  We then show that
this mechanism can be slightly modified to produce primordial black holes.

While solar mass and supermassive black holes are now known to exist,
primordial black holes have not yet been seen but could solve a number of
problems in cosmology.  Finally, we demonstrate that if an evaporating
primordial black hole is observed, it will provide a unique window onto
Beyond the Standard Model physics.

There is a striking relationship between Willmore surfaces of revolution and elastic curves in hyperbolic half-space. Here the term elastic curve refer to a critical point of the energy given by the integral of the curvature squared. In the talk we will discuss this relationship and use it to study long-time existence and asymptotic behavior for the L2-gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. As in the case of Willmore flow of spheres, we show that if an initial datum has Willmore energy below 8 \pi then the solution of the Willmore flow converges to the Clifford Torus, possibly rescaled and translated. The energy threshold of 8 \pi turns out to be optimal for such a convergence result. 

The lecture is based on joint work with M. Müller (Univ. Freiburg), R. Schätzle (Univ. Tübingen) and A. Spener (Univ. Ulm).

The action of the automorphisms of a formal group on its deformation space is crucial to understanding periodic families in the homotopy groups of spheres and the unsolved Hecke orbit conjecture for unitary Shimura varieties. We can explicitly pin down this squirming action by geometrically modelling it as coming from an action on a moduli space, which we construct using inverse Galois theory and some representation theory (a Hurwitz space). I will show you pretty pictures.

Following the risk-taking model of Seel and Strack, n players decide when to stop privately observed Brownian motions with drift and absorption at zero. They are then ranked according to their level of stopping and paid a rank-dependent reward. We study the optimal reward design where a principal is interested in the average performance and the performance at a given rank. While the former can be related to reward inequality in the Lorenz sense, the latter can have a surprising shape. Next, I will present the mean-field version of this problem. A particular feature of this game is to be tractable without necessarily being smooth, which turns out to offer a cautionary tale. We show that the mean field equilibrium induces n-player ε-Nash equilibria for any continuous reward function— but not for discontinuous ones. We also analyze the quality of the mean field design (for maximizing the median performance) when used as a proxy for the optimizer in the n-player game. Surprisingly, the quality deteriorates dramatically as n grows. We explain this with an asymptotic singularity in the induced n-player equilibrium distributions. (Joint work with M. Nutz)

Most of the talk will be about the Farrell-Jones conjecture from the point of view of an outsider. I'll try to explain what the conjecture is about, why one wants to know it, and how to prove it in some cases. The motivation for the talk is my recent work with Fujiwara and Wigglesworth where we prove this conjecture for (virtually torsion-free hyperbolic)-by-cyclic groups. If there is time I will outline the proof of this result.

A celebrated result in geometry is the Kobayashi-Hitchin correspondence, which states that a holomorphic vector bundle on a compact Kähler manifold admits a Hermite-Einstein metric if and only if the bundle is slope polystable. Recently, Dervan and Sektnan have conjectured an analogue of this correspondence for fibrations whose fibres are compact Kähler manifolds admitting Kähler metrics of constant scalar curvature. Their conjecture is that such a fibration is polystable in a suitable sense, if and only if it admits an optimal symplectic connection. In this talk, I will provide an introduction to this theory, and describe my recent work on the conjecture. Namely, I show that existence of an optimal symplectic connection implies polystability with respect to a large class of fibration degenerations. The techniques used involve analysing geodesics in the space of relatively Kähler metrics of fibrewise constant scalar curvature, and convexity of the log-norm functional in this setting. This is work for my PhD thesis, supervised by Frances Kirwan and Ruadhaí Dervan.

A proposal for the worldsheet string theory that is dual to free N=4 SYM in 4d will be explained. It is described by a free field sigma model on the twistor space of AdS5 x S5, and is a direct generalisation of the corresponding model for tensionless string theory on AdS3 x S3. I will explain how our proposal fits into the general framework of AdS/CFT, and review the various checks that have been performed.
 

In this talk I will present a gravitational interpretation for the superconformal index of N = 4 SYM theory in the large N limit. I will start by reviewing the so-called Bethe Ansatz formulation of the field theory index and its large N expansion (which includes both perturbative and non-perturbative corrections in 1/N). In the gravity side, according the rules of AdS/CFT correspondence, the index—interpreted as the supersymmetric partition function of N = 4 SYM—should be equivalent to the gravitational partition function on AdS_5 x S^5. The latter is classically evaluated as a sum over Euclidean gravity solutions with appropriate boundary conditions. In this context, I will show that (in the case of equal angular momenta) the contribution to the index of each Bethe Ansatz solution that admits a proper large N limit is captured by a complex black hole solution in the gravity side, which reproduces both its perturbative and non-perturbative behavior. More specifically, the number of putative black hole solutions turns out to be much larger than the number of Bethe Ansatz solutions. A resolution of this issue is found by requiring the gravity solutions to be “stable” under the non-perturbative corrections. This ensures that all the extra gravity solutions are ruled out and leads to a precise match between field theory and gravity.

In recent years, topological and geometric data analysis (TGDA) has emerged as a new and promising field for processing, analyzing and understanding complex data. Indeed, geometry and topology form natural platforms for data analysis, with geometry describing the ''shape'' behind data; and topology characterizing / summarizing both the domain where data are sampled from, as well as functions and maps associated with them. In this talk, I will show how topological (and geometric ideas) can be used to analyze graph data, which occurs ubiquitously across science and engineering. Graphs could be geometric in nature, such as road networks in GIS, or relational and abstract. I will particularly focus on the reconstruction of hidden geometric graphs from noisy data, as well as graph matching and classification. I will discuss the motivating applications, algorithm development, and theoretical guarantees for these methods. Through these topics, I aim to illustrate the important role that topological and geometric ideas can play in data analysis.

Let $G$ be a group acting transitively on a finite set $\Omega$. Then $G$ acts on $\Omega \times \Omega$ componentwise. Define the orbitals to be the orbits of $G$ on $\Omega \times \Omega$. The diagonal orbital is the orbital of the form $\Delta = \{(\alpha, \alpha) \mid \alpha \in \Omega \}$. The others are called non-diagonal orbitals. Let $\Gamma$ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set $\Omega$ and edge set $(\alpha,\beta) \in \Gamma$ with $\alpha, \beta \in \Omega$. If the action of $G$ on $\Omega$ is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs.

There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding specific bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups. 

The amazing results of DeepMind's AlphaFold2 in the last CASP experiment  caused a huge stir in both the AI and biology fields, and this was of 
course widely reported in the general media. The claim is that the  protein folding problem has finally been solved, but has it really? Not 
to spoil the ending, but of course not. In this talk I will not be  talking (much) about AlphaFold2 itself, but instead what inspiration we 
can take from it about future directions we might want to take in protein structure bioinformatics research using modern AI techniques. 
Along the way, I'll illustrate my thoughts with some recent and current  machine-learning-based projects from my own lab in the area of protein 
structure and folding.
 

From latent variable models in machine learning to inverse problems in computational imaging, tensors pervade the data sciences.  Often, the goal is to decompose a tensor into a particular low-rank representation, thereby recovering quantities of interest about the application at hand.  In this talk, I will present a recent method for low-rank CP symmetric tensor decomposition.  The key ingredients are Sylvester’s catalecticant method from classical algebraic geometry and the power method from numerical multilinear algebra.  In simulations, the method is roughly one order of magnitude faster than existing CP decomposition algorithms, with similar accuracy.  I will state guarantees for the relevant non-convex optimization problem, and robustness results when the tensor is only approximately low-rank (assuming an appropriate random model).  Finally, if the tensor being decomposed is a higher-order moment of data points (as in multivariate statistics), our method may be performed without explicitly forming the moment tensor, opening the door to high-dimensional decompositions.  This talk is based on joint works with João Pereira, Timo Klock and Tammy Kolda.