Past

The torus T of projective space also acts on the Hilbert
scheme of subschemes of projective space, and the T-graph of the
Hilbert scheme has vertices the fixed points of this action, and edges
the closures of one-dimensional orbits. In general this graph depends
on the underlying field. I will discuss joint work with Rob
Silversmith, in which we construct of a subgraph, which we call the
spine, of the T-graph of Hilb^N(A^2) that is independent of the choice
of field. The key technique is an understanding of the tropical ideal,
in the sense of tropical scheme theory, of the ideal of the universal
family of an edge in the spine.

Recent developments on the black hole information paradox have shown that Euclidean wormholes — so called “replica wormholes’’  — can dominate the von Neumann entropy as computed by a gravitational path integral, and that inclusion of these wormholes results in a unitary Page curve. This development raises some puzzles from the perspective of factorization, and has raised questions regarding what the gravitational path integral is computing. In this talk, I will focus on understanding the relationship between the gravitational path integral and the partition function via the gravitational free energy (more generally the generating functional). A proper computation of the free energy requires a replica trick distinct from the usual one used to compute the entropy. I will show that in JT gravity there is a regime where the free energy computed without replica wormholes is pathological. Interestingly, the inclusion of replica wormholes is not quite sufficient to resolve the pathology: an alternative analytic continuation is required. I will discuss the implications of this for various interpretations of the gravitational path integral (e.g. as computing an ensemble average) and also mention some parallels with spin glasses. 

In this session, William Durham from the Bank of England will give a presentation about working as a mathematician in the BoE, and will give advice on interviewing for non-academic jobs. He has previously provided mock interviews in our department for jobs aimed at mathematicians with PhDs, and is happy to conduct some mock interviews (remotely, of course) for individuals as well.

Please email Helen McGregor (@email) by Monday 22 February if you might be interested in having a mock interview with William Durham on 5 March.
 

 I will be reviewing our recent article arXiv:2101.02218 where we propose a simple method for detecting global (a.k.a. non-perturbative) anomalies for generic quantum field theories. The basic idea is to study how the symmetries are realized on the Hilbert space of the theory. I will present several elementary examples where everything can be solved explicitly. After that, we will use these results to make non-trivial predictions about strongly interacting theories.

We propose a mathematical model that unifies the psychiatric concepts of drug-induced incentive salience (IST), reward prediction error

(RPE) and opponent process theory (OPT) to describe the emergence of addiction within substance abuse. The biphasic reward response (initially

positive, then negative) of the OPT is activated by a drug-induced dopamine release, and evolves according to neuro-adaptative brain

processes.  Successive drug intakes enhance the negative component of the reward response, which the user compensates for by increasing the

drug dose.  Further neuroadaptive processes ensue, creating a positive feedback between physiological changes and user-controlled drug

intake. Our drug response model can give rise to qualitatively different pathways for an initially naive user to become fully addicted.  The

path to addiction is represented by trajectories in parameter space that depend on the RPE, drug intake, and neuroadaptive changes.

We will discuss how our model can be used to guide detoxification protocols using auxiliary substances such as methadone, to mitigate withdrawal symptoms.

If this is useful here are my co-authors:
Davide Maestrini, Tom Chou, Maria R. D'Orsogna

For $G$ a finite group, $p$ a prime and $(K, \mathcal{O}_K, k)$ a $p$-modular system the group ring $\mathcal{O}_K G$ is an $\mathcal{O}_k$-order in the $K$-algebra $KG.$ Graduated $\mathcal{O}_K$-orders are a particularly nice class of $\mathcal{O}_K$-orders first introduced by Zassenhaus. In this talk will see that an $\mathcal{O}_K$-order $\Lambda$ in a split $K$-algebra $A$ is graduated if the decomposition numbers for the regular $A$-module are no greater than $1$. Furthermore will see that graduated orders can be described (not uniquely) by a tuple $n$ and a matrix $M$ called the exponant matrix. Finding a suitable $n$ and $M$ for a graduated order $\Lambda$ in the $K$-algebra $A$ provides a parameterisation of the $\Lambda$-lattices inside the regular $A$-module. Understanding the $\mathcal{O}_K G$-lattices inside representations of certain groups $G$ is of interest to those involved in the Langlands programme as well as of independent interest to algebraists.

Low rank orthogonal tensor approximation (LROTA) is an important problem in tensor computations and their applications. A classical and widely used algorithm is the alternating polar decomposition method (APD). In this talk, I will first give very a brief introduction to tensors and their decompositions. After that, an improved version named iAPD of the classical APD will be proposed and all the following four fundamental properties of iAPD will be discussed : (i) the algorithm converges globally and the whole sequence converges to a KKT point without any assumption; (ii) it exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual O(1/k) for first order methods in optimization; (iii) more importantly, it converges R-linearly for a generic tensor without any assumption; (iv) for almost all LROTA problems, iAPD reduces to APD after finitely many iterations if it converges to a local minimizer. If time permits, I will also present some numerical experiments.

Our understanding and ability to compute the solutions to nonlinear partial differential equations has been strongly curtailed by our inability to effectively parameterize the inertial manifold of their solutions.  I will discuss our ongoing efforts for using machine learning to advance the state of the art, both for developing a qualitative understanding of "turbulent" solutions and for efficient computational approaches.  We aim to learn parameterizations of the solutions that give more insight into the dynamics and/or increase computational efficiency. I will discuss our recent work using machine learning to develop models of the small scale behavior of spatio-temporal complex solutions, with the goal of maintaining accuracy albeit at a highly reduced computational cost relative to a full simulation.  References: https://www.pnas.org/content/116/31/15344 and https://arxiv.org/pdf/2102.01010.pdf 

We explore reinforcement learning methods for finding the optimal policy in the linear quadratic regulator (LQR) problem. In particular, we consider the convergence of policy gradient methods in the setting of known and unknown parameters. We are able to produce a global linear convergence guarantee for this approach in the setting of finite time horizon and stochastic state dynamics under weak assumptions. The convergence of a projected policy gradient method is also established in order to handle problems with constraints. We illustrate the performance of the algorithm with two examples. The first example is the optimal liquidation of a holding in an asset. We show results for the case where we assume a model for the underlying dynamics and where we apply the method to the data directly. The empirical evidence suggests that the policy gradient method can learn the global optimal solution for a larger class of stochastic systems containing the LQR framework and that it is more robust with respect to model mis-specification when compared to a model-based approach. The second example is an LQR system in a higher-dimensional setting with synthetic data.

This talk concerns applications of differential geometry in numerical optimization. They arise when the optimization problem can be formulated as finding an optimum of a real-valued cost function defined on a smooth nonlinear search space. Oftentimes, the search space is a "matrix manifold", in the sense that its points admit natural representations in the form of matrices. In most cases, the matrix manifold structure is due either to the presence of certain nonlinear constraints (such as orthogonality or rank constraints), or to invariance properties in the cost function that need to be factored out in order to obtain a nondegenerate optimization problem. Manifolds that come up in practical applications include the rotation group SO(3) (generation of rigid body motions from sample points), the set of fixed-rank matrices (low-rank models, e.g., in collaborative filtering), the set of 3x3 symmetric positive-definite matrices (interpolation of diffusion tensors), and the shape manifold (morphing).

In the recent years, the practical importance of optimization problems on manifolds has stimulated the development of geometric optimization algorithms that exploit the differential structure of the manifold search space. In this talk, we give an overview of geometric optimization algorithms and their applications, with an emphasis on the underlying geometric concepts and on the numerical efficiency of the algorithm implementations.

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 pandemic has had a deleterious influence on the Hollywood film
industry.  Fortunately,  however, the thin film industry continues to
flourish.  A host of effects are responsible for confined liquids
exhibiting properties that differ from their bulk counterparts. For
example, the dominant polarization and surface forces across a layered
system can control the material behavior on length scales vastly larger
than the film thickness.  This basic class of phenomena, wherein
volume-volume interactions create large pressures, are at play in,
amongst many other settings, wetting, biomaterials, ceramics, colloids,
and tribology.  When the films so created involve phase change and are
present in disequilibrium, the forces can be so large that they destroy
the setting that allowed them to form in the first place. I will
describe the connection between such films in a semi-traditional wetting
dynamics geometry and active brownian dynamics.  I then explore their
power to explain a wide range of processes from materials- to astro- to
geo-science.

Point counting on definable sets in non-archimedean settings has many faces. For sets living in Q_p^n, one can count actual rational points of bounded height, but for sets in C((t))^n, one rather "counts" the polynomials in t of bounded degree. What if the latter is of infinite cardinality? We treat three settings, each with completely different behaviour for point counting : 1) the setting of subanalytic sets, where we show finiteness of point counting but growth can be aribitrarily fast with the degree in t ; 2) the setting of Pfaffian sets, which is new in the non-archimedean world and for which we show an analogue of Wilkie's conjecture in all dimensions; 3) the Hensel minimal setting, which is most general and where finiteness starts to fail, even for definable transcendental curves! In this infinite case, one bounds the dimension rather than the (infinite) cardinality. This represents joint work with Binyamini, Novikov, with Halupczok, Rideau, Vermeulen, and separate work by Cantoral-Farfan, Nguyen, Vermeulen.

Since its introduction in 1978 the curve complex has become one of the most important objects to study surfaces and their homeomorphisms. The curve complex is defined only using data about curves and their disjointness: a stunning feature of it is the fact that this information is enough to give it a rigid structure, that is every symplicial automorphism is induced topologically. Ivanov conjectured that this rigidity is a feature of most objects naturally associated to surfaces, if their structure is rich enough.

During the talk we will introduce the curve complex, then we will focus on its rigidity, giving a sketch of the topological constructions behind the proof. At last we will talk about generalisations of the curve complex, and highlight some rigidity results which are clues that Ivanov's Metaconjecture, even if it is more of a philosophical statement than a mathematical one, could be "true".

A uniform spanning tree of $\mathbb{Z}^4$ can be thought of as the "uniform measure" on trees of $\mathbb{Z}^4$. The past of 0 in the uniform spanning tree is the finite component that is disconnected from infinity when 0 is deleted from the tree. We establish the logarithmic corrections to the probabilities that the past contains a path of length $n$, that it has volume at least $n$ and that it reaches the boundary of the box of side length $n$ around 0. Dimension 4 is the upper critical dimension for this model in the sense that in higher dimensions it exhibits "mean-field" critical behaviour. An important part of our proof is the study of the Newtonian capacity of a loop erased random walk in 4 dimensions. This is joint work with Tom Hutchcroft.

We combine the notions of graded Clifford algebras and Drinfeld algebras. This gives us a framework to study algebras with a PBW property and underlying vector space $\mathbb{C}[G] \# Cl(V) \otimes S(U) $ for $G$-modules $U$ and $V$. The class of graded Clifford-Drinfeld algebras contains the Hecke-Clifford algebras defined by Nazarov, Khongsap-Wang. We give a new example of a GCD algebra which plays a role in an Arakawa-Suzuki duality involving the Clifford algebra.

We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To establish this, we work directly at the level of Benjamini-Schramm limits. More precisely, we exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible 'at infinity'. We then transfer this result to finite graphs via local weak convergence and a relative compactness argument. We believe that this 'local weak limit' approach to mixing properties of Markov chains will have many other applications.

How can a random walker on a network be helpful for patients suffering from amyotrophic lateral sclerosis (ALS)? Clinical trials in ALS continue to rely on survival or clinical functional scales as endpoints, since anatomical patterns of disease spread in ALS are poorly characterized in vivo. In this study, we generated individual brain networks of patients and controls based on cerebral magnetic resonance imaging (MRI) data. Then, we applied a computational model with a random walker to the brain MRI scan of patients to simulate this progressive network degeneration. We observe that computer‐simulated aggregation levels of the random walker mimic true disease patterns in ALS patients. Our results demonstrate the utility of computational network models in ALS to predict disease progression and underscore their potential as a prognostic biomarker.

After presenting this study on characterizing the structural changes in neurodegenerative diseases with network science, I will give an outlook on my new work on characterizing the dynamic changes in brain networks for Parkinson’s disease and counteracting these with (simulated) deep brain stimulation using the neuroinformatics platform The Virtual Brain (www.thevirtualbrain.org) .

Article link: https://onlinelibrary.wiley.com/doi/full/10.1002/ana.25706

The conformal bootstrap program for CFTs in d>2 dimensions is
based on well-defined rules and in principle it could be easily included
into rigorous mathematical physics. I will explain some interesting
conjectures which emerged from the program, but which so far lack rigorous
proof. No prior knowledge of CFTs or conformal bootstrap will be assumed.

Prof. Franco Rampazzo ‘Mathematical Control Theory’ (Department of Mathematics of the University of Padova, as part of Oxford Padova connection) TT 2021
Aimed at: Any DPhil students with interest in learning about Mathematical Control Theory
Course Length:     24 hours total (to be in English) 
Dates and Times:  starts 2 March 2021 

We will discuss the problem of deciding (algorithmically) whether a variety over a local field K has a K-rational point, surveying some known results. I will then allow K to be an infinite extension (of some arithmetic interest) of a local field and present some recent work.
 

Nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations This talk is about joint work with Viorel Barbu. We consider a class of nonlinear Fokker-Planck (- Kolmogorov) equations of type \begin{equation*} \frac{\partial}{\partial t} u(t,x) - \Delta_x\beta(u(t,x)) + \mathrm{div} \big(D(x)b(u(t,x))u(t,x)\big) = 0,\quad u(0,\cdot)=\mu, \end{equation*} where $(t,x) \in [0,\infty) \times \mathbb{R}^d$, $d \geq 3$ and $\mu$ is a signed Borel measure on $\mathbb{R}^d$ of bounded variation. In the first part of the talk we shall explain how to construct a solution to the above PDE based on classical nonlinear operator semigroup theory on $L^1(\mathbb{R}^d)$ and new results on $L^1- L^\infty$ regularization of the solution semigroups in our case. In the second part of the talk we shall present a general result about the correspondence of nonlinear Fokker-Planck equations (FPEs) and McKean-Vlasov type SDEs. In particular, it is shown that if one can solve the nonlinear FPE, then one can always construct a weak solution to the corresponding McKean-Vlasov SDE. We would like to emphasize that this, in particular, applies to the singular case, where the coefficients depend "Nemytski-type" on the time-marginal law of the solution process, hence the coefficients are not continuous in the measure-variable with respect to the weak topology on probability measures. This is in contrast to the literature in which the latter is standardly assumed. Hence we can cover nonlinear FPEs as the ones above, which are PDEs for the marginal law densities, realizing an old vision of McKean.

References V. Barbu, M. Röckner: From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE, Ann. Prob. 48 (2020), no. 4, 1902-1920. V. Barbu, M. Röckner: Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations, J. Funct. Anal. 280 (2021), no. 7, 108926.

One of the main questions in the theory of the linear transport equation is whether uniqueness of solutions to the Cauchy problem holds in the case the given vector field is not smooth. We will show that even for incompressible, Sobolev (thus quite “well-behaved”) vector fields, uniqueness of solutions can drastically fail. This result can be seen as a counterpart to DiPerna and Lions’ well-posedness theorem, and, more generally, it can be interpreted as an instance of the “flexibility vs. rigidity” duality, which is a common feature of PDEs appearing in completely different fields, such as differential geometry and fluid dynamics (joint with G. Sattig and L. Székelyhidi). 

Studying quasi-isometries between groups is a major theme in geometric group theory. Of particular interest are the situations where the existence of a quasi-isometry between two groups implies that the groups are equivalent in a stronger algebraic sense, such as being commensurable. I will survey some results of this type, and then talk about recent work with Daniel Woodhouse where we prove quasi-isometric rigidity for certain graphs of virtually free groups, which include "generic" cyclic HNN extensions of free groups.