Past

How can we adapt the Topological Data Analysis (TDA) pipeline to use several filter functions at the same time? Two orthogonal approaches can be considered: (1) running the standard 1-parameter pipeline and doing statistics on the resulting barcodes; (2) running a multi-parameter version of the pipeline, still to be defined. In this talk I will present two recent contributions, one for each approach. The first contribution considers intrinsic compact metric spaces and shows that the so-called Persistent Homology Transform (PHT) is injective over a dense subset of those. When specialized to metric graphs, our analysis yields a stronger result, namely that the PHT is injective over a subset of full measurem which allows for sufficient statistics. The second contribution investigates the bi-parameter version of the TDA pipeline and shows a decomposition result "à la Crawley-Boevey" for a subcategory of the 2-parameter persistence modules called "exact modules". This result has an impact on the study of interlevel-sets persistence and on that of sheaves of vector spaces on the real line. 

This is joint work with Elchanan Solomon on the one hand, with Jérémy Cochoy on the other hand.

When dealing with geometric structures one natural question that arise is "when does a subset inherit the geometry of the ambient space"? In the case of hyperbolic space, the concept of quasi-convexity provides answer to this question. However, for a general metric space, being quasi-convex is not a quasi-isometric invariant. This motivates the notion of Morse subsets. In this talk we will motivate the definition and introduce some examples. Then we will introduce the class of hierarchically hyperbolic groups (HHG), and furnish a complete characterization of Morse subgroups of HHG. If time allows, we will discuss the relationship between Morse subgroups and hyperbolically-embedded subgroups. This is a joint work with Hung C. Tran and Jacob Russell.

Exponents of Diophantine approximation are defined to study specific sets of real numbers for which Dirichlet's pigeonhole principle can be improved. Khintchine stated a transference principle between the two exponents in the cases  of simultaneous approximation and approximation by linear forms. This shows that exponents of Diophantine approximation are related, and these relations can be studied via so called spectra. In this talk, we provide an optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation for both simultaneous approximation and approximation by linear forms. This is joint work with Nikolay Moshchevitin.

Abstract:
Highlights:

•We increasingly live in a digital world and commercial companies are not the only beneficiaries. The public sector can also use data to tackle pressing issues.
•Machine learning is starting to make an impact on the tools regulators use, for spotting the bad guys, for estimating demand, and for tackling many other problems.
•The speech uses an array of examples to argue that much regulation is ultimately about recognising patterns in data. Machine learning helps us find those patterns.
 
Just as moving from paper maps to smartphone apps can make us better navigators, Stefan’s speech explains how the move from using traditional analysis to using machine learning can make us better regulators.
 
Mini Biography:
 
Stefan Hunt is the founder and Head of the Behavioural Economics and Data Science Unit. He has led the FCA’s use of these two fields and designed several pioneering economic analyses. He is an Honorary Professor at the University of Nottingham and has a PhD in economics from Harvard University.
 

Superhydrophobic surfaces, formed by air entrapment within the cavities of a hydrophobic solid substrate, offer a promising potential for drag reduction in small-scale flows. It turns out that low-drag configurations are associated with singular limits, which to date have typically been addressed using numerical schemes. I will discuss the application of singular perturbations to several of the canonical problems in the field. 

The development of reduced order models for complex applications, offering the promise for rapid and accurate evaluation of the output of complex models under parameterized variation, remains a very active research area. Applications are found in problems which require many evaluations, sampled over a potentially large parameter space, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed.

However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large-scale applications, in particular for nonlinear and/or time-dependent problems.

After giving a brief general introduction to reduced order models, we discuss developments in two different directions. In the first part, we discuss recent developments of reduced methods that conserve chosen invariants for nonlinear time-dependent problems. We pay particular attention to the development of reduced models for Hamiltonian problems and propose a greedy approach to build the basis. As we shall demonstrate, attention to the construction of the basis must be paid not only to ensure accuracy but also to ensure stability of the reduced model. Time permitting, we shall also briefly discuss how to extend the approach to include more general dissipative problems through the notion of port-Hamiltonians, resulting in reduced models that remain stable even in the limit of vanishing viscosity and also touch on extensions to Euler and Navier-Stokes equations.

The second part of the talk discusses the combination of reduced order modeling for nonlinear problems with the use of neural networks to overcome known problems of on-line efficiency for general nonlinear problems. We discuss the general idea in which training of the neural network becomes part of the offline part and demonstrate its potential through a number of examples, including for the incompressible Navier-Stokes equations with geometric variations.

This work has been done with in collaboration with B.F. Afkram (EPFL, CH), N. Ripamonti EPFL, CH) and S. Ubbiali (USI, CH).

Given $d \ge 1$, $T>0$ and a vector field $\mathbf b \colon [0,T] \times \mathbb R^d \to \mathbb R^d$, we study the problem of uniqueness of weak solutions to the associated transport equation $\partial_t u + \mathbf b \cdot \nabla u=0$ where $u \colon [0,T] \times \mathbb R^d \to \mathbb R$ is an unknown scalar function. In the classical setting, the method of characteristics is available and provides an explicit formula for the solution of the PDE, in terms of the flow of the vector field $\mathbf b$. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost.
In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian representation. This tool allows to represent a suitable class of vector fields as superposition of trajectories: we will then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a family of 1-dimensional equations. We will finally show that if $\mathbf b$ is locally of class $BV$ in the space variable, the decomposition satisfies this local structural assumption: this yields in particular the renormalization property for nearly incompressible $BV$ vector fields and thus gives a positive answer to the (weak) Bressan's Compactness Conjecture. This is a joint work with S. Bianchini.
 

Tight security is increasingly gaining importance in real-world
cryptography, as it allows to choose cryptographic parameters in a way
that is supported by a security proof, without the need to sacrifice
efficiency by compensating the security loss of a reduction with larger
parameters. However, for many important cryptographic primitives,
including digital signatures and authenticated key exchange (AKE), we
are still lacking constructions that are suitable for real-world deployment.

This talk will present the first first practical AKE protocol with tight
security. It allows the establishment of a key within 1 RTT in a
practical client-server setting, provides forward security, is simple
and easy to implement, and thus very suitable for practical deployment.
It is essentially the "signed Diffie-Hellman" protocol, but with an
additional message, which is crucial to achieve tight security. This
message is used to overcome a technical difficulty in constructing
tightly-secure AKE protocols.

The second important building block is a practical signature scheme with
tight security in a real-world multi-user setting with adaptive
corruptions. The scheme is based on a new way of applying the
Fiat-Shamir approach to construct tightly-secure signatures from certain
identification schemes.

For a theoretically-sound choice of parameters and a moderate number of
users and sessions, our protocol has comparable computational efficiency
to the simple signed Diffie-Hellman protocol with EC-DSA, while for
large-scale settings our protocol has even better computational per-
formance, at moderately increased communication complexity.

Given an endomorphism f:X --> X of a 'dualisable' object in a symmetric monoidal category, one can define its trace Tr(f). It turns out that the trace is 'universal' among the scalars we can produce from f. To prove this we will think of the 1d framed bordism category as the 'walking dualisable object' (using the cobordism hypothesis) and then apply the Yoneda lemma.
Employing similar techniques we can define 'hermitian' objects (generalising hermitian vector spaces) and prove that there is a 1-1 correspondence between Hermitian structures on a fixed object X and self-adjoint automorphisms of X. If time permits I will sketch how this relates to hermitian K-theory.

While all results of the talk hold for infinity-categories, they work equally well for ordinary categories. Therefore no knowledge of higher category theory is needed to follow the talk.

I will report on two recent papers with D. Ghioca and U. Zannier (joined by P. Corvaja and F. Hu, respectively) in which we consider variants of the Mordell-Lang conjecture.  In the first of these, we study the dynamical Mordell-Lang conjecture in positive characteristic, proving some instances, but also showing that in general the problem is at least as hard as a difficult diophantine problem over the integers.  In the second paper, we study the Mordell-Lang problem for extensions of abelian varieties by the additive group.  Here we have positive results in the function field case obtained by using the socle theorem in the form offered as an aside in Hrushovski's 1996 paper and in the number field case we relate this problem to the Bombieri-Lang conjecture.

Part of the series 'What do historians of mathematics do?'

In 1873 the Personal Recollections from Early Life to Old Age of Mary Somerville were published, containing detailed descriptions of her life as a 19th century philosopher, mathematician and advocate of women's rights. In an early draft of this work, Somerville reiterated the widely held view that a fundamental difference between men and women was the latter's lack of originality, or 'genius'.

In my talk I will examine how Somerville's view was influenced by the historic treatment of women, both within scientific research, scientific institutions and wider society. By building on my doctoral research I will also suggest an alternative viewpoint in which her work in the differential calculus can be seen as original, with a focus on her 1834 treatise On the Theory of Differences.

We consider numerical methods for solving  time dependent partial differential equations with convection-diffusion terms and anti-diffusive fractional operator of order $\alpha \in (1,2)$. These equations are motivated by two distinct applications: a dune morphodynamics model and a signal filtering method. 
We propose numerical schemes based on local discontinuous Galerkin methods to approximate the solutions of these equations. Numerical stability and convergence of these schemes are investigated. 
Finally numerical experiments are given to illustrate qualitative behaviors of solutions for both applications and to confirme the convergence results. 

Numerous problems in extremal hypergraph theory ask to determine the maximal size of a k-uniform hypergraph on n vertices that does not contain an 'enlarged' copy H^+ of a fixed hypergraph H. These include well-known  problems such as the Erdős-Sós 'forbidding one intersection' problem and the Frankl-Füredi 'special simplex' problem.

In this talk we present a general approach to such problems, using a 'junta approximation method' that originates from analysis of Boolean functions. We prove that any (H^+)-free hypergraph is essentially contained in a 'junta' -- a hypergraph determined by a small number of vertices -- that is also (H^+)-free, which effectively reduces the extremal problem to an easier problem on juntas. Using this approach, we obtain, for all C<k<n/C, a complete solution of the extremal problem for a large class of H's, which includes  the aforementioned problems, and solves them for a large new set of parameters.

Based on joint works with David Ellis and Nathan Keller
 

We introduce and analyze a discontinuous Galerkin method for the Oseen equations in two dimension spaces. The boundary conditions are mixed and they are assumed to be of three different types:
the vorticity  and the normal component of the velocity are given on a first part of the boundary, the pressure and the tangential component of the velocity are given on a second part of the boundary and the Dirichlet condition is given on the remainder part . We establish a priori error estimates in the energy norm for the velocity and in the L2 norm for the pressure. An a posteriori error estimate is also carried out yielding optimal convergence rate. The analysis is based on rewriting the method in a non-consistent manner using lifting operators in the spirit of Arnold, Brezzi, Cockburn and Marini.

We discuss the recent achievements in the attractors theory for damped wave equations in bounded domains which are related with Strichartz type estimates. In particular, we present the results related with the well-posedness and asymptotic smoothness of the solution semigroup in the case of critical quintic nonlinearity. The non-autonomous case will be also considered.
 

When studying a group, it is natural and often useful to try to cut it up 
onto simpler pieces. Sometimes this can be done in an entirely canonical 
way analogous to the JSJ decomposition of a 3-manifold, in which the 
collection of tori along which the manifold is cut is unique up to isotopy. 
It is a theorem of Brian Bowditch that if the group acts nicely on a metric 
space with a negative curvature property then a canonical decomposition can 
be read directly from the large-scale geometry of that space. In this talk 
we shall explore an algorithmic consequence of this relationship between 
the large-scale geometry of the group and is algebraic decomposition.

I present recently developed iterated residue formulas for tautological integrals over Hilbert schemes of points on  smooth  manifolds. Applications include curve and hypersurface counting formulas. Joint work with Andras Szenes.

 Suppose A is a nice abelian category (such as coherent sheaves coh(X) on a smooth complex projective variety X, or representations mod-CQ of a quiver Q) or T is a nice triangulated category (such as D^bcoh(X) or D^bmod-CQ) over C. Let M be the moduli stack of objects in A or T. Consider the homology H_*(M) over some ring R.
  Given a little extra data on M, for which there are natural choices in our examples, I will explain how to define the structure of a graded vertex algebra on H_*(M). By a standard construction, one can then define a graded Lie algebra from the vertex algebra; roughly speaking, this is a Lie algebra structure on the homology H_*(M^{pl}) of a "projective linear” version M^{pl} of the moduli stack M.
  For example, if we take T = D^bmod-CQ, the vertex algebra H_*(M) is the lattice vertex algebra attached to the dimension vector lattice Z^{Q_0} of Q with the symmetrized intersection form. The degree zero part of the graded Lie algebra contains the associated Kac-Moody algebra.
  The construction appears to be new, but is connected with a lot of work in Geometric Representation Theory, to do with Ringel-Hall-type algebras and their representations, such as the results of Grojnowski-Nakajima on Hilbert schemes. The vertex algebra construction is enormously general, and applies in huge classes of examples. There is a differential-geometric version too.
  The question I am hoping someone in the audience will answer is this: what is the physical interpretation of these vertex algebras?
  It is in some sense an "even Calabi-Yau” construction: when applied to coh(X) or D^bcoh(X), it is most natural for X a Calabi-Yau 2-fold or Calabi-Yau 4-fold, and is essentially trivial for X a Calabi-Yau 3-fold. I discovered it when I was investigating wall-crossing for Donaldson-Thomas type invariants for Calabi-Yau 4-folds. So perhaps one should look for an explanation in the physics of Calabi-Yau 2-folds or 4-folds, with M the moduli space of boundary conditions for the associated SCFT.

DNA computing is emerging as a versatile technology that promises a vast range of applications, including biosensing, drug delivery and synthetic biology. DNA logic circuits can be achieved in solution using strand displacement reactions, or by decision-making molecular robots-so called 'walkers'-that traverse tracks placed on DNA 'origami' tiles.

 Similarly to conventional silicon technologies, ensuring fault-free DNA circuit designs is challenging, with the difficulty compounded by the inherent unreliability of the DNA technology and lack of scientific understanding. This lecture will give an overview of computational models that capture DNA walker computation and demonstrate the role of quantitative verification and synthesis in ensuring the reliability of such systems. Future research challenges will also be discussed.

Leandro Sánchez Betancourt
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The Cost of Latency: Improving Fill Ratios in Foreign Exchange Markets

Latency is the time delay between an exchange streaming market data to a trader, the trader processing information and deciding to trade, and the exchange receiving the order from the trader.  Liquidity takers  face  a  moving target problem as a consequence of their latency in the marketplace -- they send marketable orders that aim at a price and quantity they observed in the LOB, but by the time their order was processed by the Exchange, prices (and/or quantities) may have worsened, so the  order  cannot  be  filled. If liquidity taking orders can walk the limit order book (LOB), then orders that arrive late may still be filled at worse prices. In this paper we show how to optimally choose the discretion of liquidity taking orders to walk the LOB. The optimal strategy balances the tradeoff between the costs of walking the LOB and targeting  a desired percentage of filled orders over a period of time.  We employ a proprietary data set of foreign exchange trades to analyze the performance of the strategy. Finally, we show the relationship between latency and the percentage of filled orders, and showcase the optimal strategy as an alternative investment to reduce latency.

Jasdeep Kalsi
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An SPDE model for the Limit Order Book

I will introduce a microscopic model for the Limit Order Book in a static setting i.e. in between price movements. Here, order flow at different price levels is given by Poisson processes which depend on the relative price and the depth of the book. I will discuss how reflected SPDEs can be obtained as scaling limits of such models. This motivates an SPDE with reflection and a moving boundary as a model for the dynamic Order Book. An outline for how to prove existence and uniqueness for the equation will be presented, as well as some simple simulations of the model.

Steady state chemical reaction models can be thought of as algebraic varieties whose properties are determined by the network structure. In experimental set-ups we often encounter the problem of noisy data points for which we want to find the corresponding steady state predicted by the model. Depending on the network there may be many such points and the number of which is given by the euclidean distance degree (ED degree). In this talk I show how certain properties of networks relate to the ED degree and how the runtime of numerical algebraic geometry computations scales with the ED degree.