Past

In this talk we will introduce Leary and Minasyan's CAT(0) but not biautomatic groups as lattices in a product of a Euclidean space and a tree.  We will then investigate properties of general lattices in that product space.  We will also consider a construction of lattices in a Salvetti complex for a right-angled Artin group and a Euclidean space.  Finally, if time permits we will also discuss a "hyperbolic Leary–Minasyan group" and some work in progress with Motiejus Valiunas towards an application.

Free boundary minimal surfaces are a notoriously elusive object in geometric analysis. From 2011, Fraser and Schoen's research program found a relationship between free boundary minimal surfaces in unit balls and metrics which maximise the first nontrivial Steklov eigenvalue. In this talk, I will explain how we can adapt homogenisation theory, a branch of applied mathematics, to a geometric setting in order to obtain surfaces with first Steklov eigenvalue as large as possible, and how it leads to the existence of free boundary minimal surfaces which were previously thought not to exist.

Scattering amplitudes in N=4 super-Yang-Mills theory are known to be functions of cluster variables of Gr(4,n) and certain algebraic functions of cluster variables. In this talk we give an overview of the known cluster algebraic structure of both tree amplitudes and the symbol of loop amplitudes. We suggest an algorithm for computing symbol alphabets by solving matrix equations of the form C.Z = 0 associated with plabic graphs. These matrix equations associate functions on Gr(m,n) to parameterizations of certain cells of Gr_+ (k,n) indexed by plabic graphs. We are able to reproduce all known algebraic functions of cluster variables appearing in known symbol alphabets. We further show that it is possible to obtain all rational symbol letters (in fact all cluster variables) by solving C.Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of Gr_+ (n-4,n). Finally we discuss a property of the symbol called cluster adjacency.

The SI model is the most basic of all compartmental models used to describe the spreading of information through a population. In this talk we will present a mathematical formalism to solve the SI model on generic networks. Our methods rely on a tensor product formulation of the dynamical spreading process, inspired by many-body quantum systems. Here we will focus on time-dependent expectation values for the state of individual nodes, which can be obtained from contributions of subgraphs of the network. We show how to compute these contributions systematically and derive a set of symmetry relations among subgraphs of differing topologies. We conclude by comparing our results for small sample networks to Monte-Carlo simulations and mean-field approximations.

arXiv link: https://arxiv.org/abs/2109.03530

Stochastic differential equations (SDEs) on compact foliated spaces were introduced a few years ago. As a corollary, a leafwise Brownian motion on a compact foliated space was obtained as a solution to an SDE. In this work we construct stochastic flows associated with the SDEs by using rough path theory, which is something like a 'deterministic version' of Ito's SDE theory.

This is joint work with Kiyotaka Suzaki.

Although a multidimensional data array can be very large, it may contain coherence patterns much smaller in size. For example, we may need to detect a subset of genes that co-express under a subset of conditions. In this presentation, we discuss our recently developed co-clustering algorithms for the extraction and analysis of coherent patterns in big datasets. In our method, a co-cluster, corresponding to a coherent pattern, is represented as a low-rank tensor and it can be detected from the intersection of hyperplanes in a high dimensional data space. Our method has been used successfully for DNA and protein data analysis, disease diagnosis, drug therapeutic effect assessment, and feature selection in human facial expression classification. Our method can also be useful for many other real-world data mining, image processing and pattern recognition applications.

We will present the precise late-time asymptotics for scalar fields on both extremal and sub-extremal black holes including the full Reissner-Nordstrom family and the subextremal Kerr family. Asymptotics for higher angular modes will be presented for all cases. Applications in observational signatures will also be discussed. This work is joint with Y. Angelopoulos (Caltech) and D. Gajic (Cambridge)

I will present work in progress with Erik Panzer, Matteo Parisi and Ömer Gürdoğan on the analytic structure of Feynman(esque) integrals: We consider integrals of meromorphic differential forms over relative cycles in a compact complex manifold, the underlying geometry encoded in a certain (parameter dependant) subspace arrangement (e.g. Feynman integrals in their parametric representation). I will explain how the analytic struture of such integrals can be studied via methods from differential topology; this is the seminal work by Pham et al (using tools and methods developed by Leray, Thom, Picard-Lefschetz etc.). Although their work covers a very general setup, the case we need for Feynman integrals has never been worked out in full detail. I will comment on the gaps that have to be filled to make the theory work, then discuss how much information about the analytic structure of integrals can be derived from a careful study of the corresponding subspace arrangement.

This is a joint GR-QFT seminar, to celebrate in advance the 90th birthday of Roger Penrose later in the summer, comprising 9 talks on conformal cyclic cosmology.  The provisional schedule is as follows:

11:00 Roger Penrose (Oxford, UK) : The Initial Driving Forces Behind CCC

11:10 Paul Tod (Oxford, UK) : Questions for CCC

11:20 Vahe Gurzadyan (Yerevan, Armenia): CCC predictions and CMB

11:30 Krzysztof Meissner (Warsaw, Poland): Perfect fluids in CCC

11:40 Daniel An (SUNY, USA) : Finding information in the Cosmic Microwave Background data

11:50 Jörg Frauendiener (Otago, New Zealand) : Impulsive waves in de Sitter space and their impact on the present aeon

12:00 Pawel Nurowski (Warsaw, Poland and Guangdong Technion, China): Poincare-Einstein expansion and CCC

12:10 Luis Campusano (FCFM, Chile) : (Very) Large Quasar Groups

12:20 Roger Penrose (Oxford, UK) : What has CCC achieved; where can it go from here?

The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

Abstract: In this talk we try to establish an analytic framework for studying Set-Valued Backward Stochastic Differential Equations (SVBSDE for short), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will be based on the notion of Hukuhara difference between sets, in order to compensate the lack of “inverse” operation of the traditional Minkowski addition, whence the vector space structure, in traditional set-valued analysis. We shall examine and establish a useful foundation of set-valued stochastic analysis under this algebraic framework, including some fundamental issues regarding Aumann-Itˆo integrals, especially when it is connected to the martingale representation theorem. We shall identify some fundamental challenges and propose some extensions of the existing theory that are necessary to study the SVBSDEs. This talk is based on the joint works with C¸ a˘gın Ararat and Wenqian Wu.

I will describe joint work with Abouzaid which constructs a stable homotopy theory refinement of Floer homology that has coefficients in the Morava K-theory spectra. The classifying spaces of finite groups satisfy Poincare duality for the Morava K-theories, which allows us to use this version of Floer homology to produce virtual fundamental chains for moduli spaces of Floer trajectories. As an application, we prove the Arnold conjecture for ordinary cohomology with coefficients in finite fields.

How aware should we be of letting AI make decisions on prison sentences? Or what is our responsibility in ensuring that mathematics does not predict another global stock crash?

In this talk, Helena will outline how we can view ethics and responsibility as central to processes of innovation and describe her experiences applying this perspective to teaching in the Department of Computer Science. There will be a chance to open up discussion about how this same approach can be applied in other Departments here in Oxford.

Helena is an interdisciplinary researcher working in the Department of Computer Science. She works on projects that involve examining the social impacts of computer-based innovations and identifying the ways in which these innovations can better meet societal needs and empower users. Helena is very passionate about the need to embed ethics and responsibility into processes of learning and research in order to foster technologies for the social good.

It is a well-known fact that Boolean rings, those rings in which $x^2 = x$ for all $x$, are necessarily commutative. There is a short and completely elementary proof of this. One may wonder what the situation is for rings in which $x^n = x$ for all $x$, where $n > 2$ is some positive integer. Jacobson and Herstein proved a very general theorem regarding these rings, and the proof follows a widely applicable strategy that can often be used to reduce questions about general rings to more manageable ones. We discuss this strategy, but will also focus on a different approach: can we also find ''elementary'' proofs of some special cases of the theorem? We treat a number of these explicit computations, among which a few new results.

Our society is witnessing an exponential growth of data being generated. Among the various data types being routinely collected, event logs are available in a wide variety of domains. Despite historical and structural digitalisation challenges, healthcare is an example where the analysis of event logs might bring a new revolution.

In this talk, I will present our recent efforts in analysing and exploring temporal event data sequences extracted from event logs. Our visual analytics approach is able to summarise and seamlessly explore large volumes of complex event data sequences. We are able to easily derive observations and findings that otherwise would have required significant investment of time and effort.  To facilitate the identification of findings, we use a hierarchical clustering approach to cluster sequences according to time and a novel visualisation environment.  To control the level of detail presented to the analyst, we use a hierarchical aggregation tree and an Align-Score-Simplify strategy based on an information score.   To show the benefits of this approach, I will present our results in three real world case studies: CUREd, Outpatient clinics and MIMIC-III. These will respectively cover the analysis of calls and responses of emergency services, the efficiency of operation of two outpatient clinics, and the evolution of patients with atrial fibrillation hospitalised in an acute and critical care unit. To finalise the talk, I will share our most recent work in the analysis of clinical events extracted from Electronic Health Records for the study of multimorbidity.

Tendon tissue engineering aims to grow functional tissue in the lab. Tissue is grown inside a bioreactor which controls both the mechanical and biochemical environment. As tendon cells alter their behaviour in response to mechanical stresses, designing suitable bioreactor loading regimes forms a key component in ensuring healthy tissue growth.  

Linking the forces imposed by the bioreactor to the shear stress experienced by individual cell is achieved by homogenisation using multiscale asymptotics. We will present a continuum model capturing fluid-structure interaction between the nutrient media and the fibrous scaffold where cells grow. Solutions reflecting different experimental conditions will be discussed in view of the implications for shear stress distribution experienced by cells across the bioreactor.  

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.