Past

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.

-

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.

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.

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.

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.

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.

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.

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.

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

--

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.