Past

Ax's theorem on the dimension of the intersection of an algebraic subvariety and a formal subgroup (Theorem 1F in "Some topics in differential algebraic geometry I...") implies Schanuel type transcendence results for a vast class of formal maps (including exp on a semi-abelian variety). Ax stated and proved this theorem in the characteristic 0 case, but the statement is meaningful for arbitrary characteristic and still implies positive characteristic transcendence results. I will discuss my work on positive characteristic version of Ax's theorem.

Motivated by the study of PDEs, we introduce the notion of a D-module on a variety X and give the basics of three perspectives on the theory: modules over the sheaf of differential operators on X; quasi-coherent modules with flat connection; and crystals on X. This talk will assume basic knowledge of algebraic geometry (such as rudimentary sheaf theory).

We discuss a simple variational principle for coherent material vortices

in two-dimensional turbulence. Vortex boundaries are sought as closed

stationary curves of the averaged Lagrangian strain. We find that

solutions to this problem are mathematically equivalent to photon spheres

around black holes in cosmology. The fluidic photon spheres satisfy

explicit differential equations whose outermost limit cycles are optimal

Lagrangian vortex boundaries. As an application, we uncover super-coherent

material eddies in the South Atlantic, which yield specific Lagrangian

transport estimates for Agulhas rings. We also describe briefly coherent

Lagrangian vortex detection to three-dimensional flows.

In this talk I will give a definition of cluster algebra and state some main results.

Moreover, I will explain how the combinatorics of certain cluster algebras can be modeled in geometric terms.

The accurate and stable numerical calculation of higher-order

derivatives of holomorphic functions (as required, e.g., in random matrix

theory to extract probabilities from a generating function) turns out to

be a surprisingly rich topic: there are connections to asymptotic analysis,

the theory of entire functions, and to algorithmic graph theory.

\textbf{James Newbury} \newline

Title: Heavy traffic diffusion approximation of the limit order book in a one-sided reduced-form model. \newline

Abstract: Motivated by a zero-intelligence approach, we try to capture the

dynamics of the best bid (or best ask) queue in a heavy traffic setting,

i.e when orders and cancellations are submitted at very high frequency.

We first prove the weak convergence of the discrete-space best bid/ask

queue to a jump-diffusion process. We then identify the limiting process

as a regenerative elastic Brownian motion with drift and random jumps to

the origin.

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\textbf{Zhaoxu Hou} \newline

Title: Robust Framework In Finance: Martingale Optimal Transport and

Robust Hedging For Multiple Marginals In Continuous Time

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Abstract: It is proved by Dolinsky and Soner that there is no duality

gap between the robust hedging of path-dependent European Options and a

martingale optimal problem for one marginal case. Motivated by their

work and Mykland's idea of adding a prediction set of paths (i.e.

super-replication of a contingent claim only required for paths falling

in the prediction set), we try to achieve the same type of duality

result in the setting of multiple marginals and a path constraint.

Working together with the Blue Brain Project at the EPFL, I'm trying to develop new topological methods for neural modelling. As a mathematician, however, I'm really motivated by how these questions in neuroscience can inspire new mathematics. I will introduce new work that I am doing, together with Kathryn Hess and Ran Levi, on brain plasticity and learning processes, and discuss some of the topological and geometric features that are appearing in our investigations.

I will look at the classical constructions that can be made using a straight edge and compass, I will then look at the limits of these constructions. I will then show how much further we can get with Origami, explaining how it is possible to trisect an angle or double a cube. Compasses not supplied.

Thoughts on the Burnside problem

Quasimaps provide compactifications, depending on a stability parameter epsilon, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter epsilon varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas in genus zero, and will explain why they can be viewed as generalizations (in several directions) of Givental's toric mirror theorems. I will also describe extensions of wall-crossing to higher genus, and (time permitting) to orbifold GIT targets as well.
The talk is based on joint works with Bumsig Kim, and partly also with Daewoong Cheong and with Davesh Maulik.

Hessians of functionals of PDE solutions have important applications in PDE-constrained optimisation (Newton methods) and uncertainty quantification (for accelerating high-dimensional Bayesian inference).  With current techniques, a typical cost for one Hessian-vector product is 4-11 times the cost of the forward PDE solve: such high costs generally make their use in large-scale computations infeasible, as a Hessian solve or eigendecomposition would have costs of hundreds of PDE solves.

In this talk, we demonstrate that it is possible to exploit the common structure of the adjoint, tangent linear and second-order adjoint equations to greatly accelerate the computation of Hessian-vector products, by trading a large amount of computation for a large amount of storage. In some cases of practical interest, the cost of a Hessian-
vector product is reduced to a small fraction of the forward solve, making it feasible to employ sophisticated algorithms which depend on them.

Perfect matchings are fundamental objects of study in graph theory. There is a substantial classical theory, which cannot be directly generalised to hypergraphs unless P=NP, as it is NP-complete to determine whether a hypergraph has a perfect matching. On the other hand, the generalisation to hypergraphs is well-motivated, as many important problems can be recast in this framework, such as Ryser's conjecture on transversals in latin squares and the Erdos-Hanani conjecture on the existence of designs. We will discuss a characterisation of the perfect matching problem for uniform hypergraphs that satisfy certain density conditions (joint work with Richard Mycroft), and a polynomial time algorithm for determining whether such hypergraphs have a perfect matching (joint work with Fiachra Knox and Richard Mycroft).

Finding a sparse signal solution of an underdetermined linear system of measurements is commonly solved in compressed sensing by convexly relaxing the sparsity requirement with the help of the l1 norm. Here, we tackle instead the original nonsmooth nonconvex l0-problem formulation using projected gradient methods. Our interest is motivated by a recent surprising numerical find that despite the perceived global optimization challenge of the l0-formulation, these simple local methods when applied to it can be as effective as first-order methods for the convex l1-problem in terms of the degree of sparsity they can recover from similar levels of undersampled measurements. We attempt here to give an analytical justification in the language of asymptotic phase transitions for this observed behaviour when Gaussian measurement matrices are employed. Our approach involves novel convergence techniques that analyse the fixed points of the algorithm and an asymptotic probabilistic analysis of the convergence conditions that derives asymptotic bounds on the extreme singular values of combinatorially many submatrices of the Gaussian measurement matrix under matrix-signal independence assumptions.

This work is joint with Andrew Thompson (Duke University, USA).

Quasimaps provide compactifications, depending on a stability parameter epsilon, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter epsilon varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas in genus zero, and will explain why they can be viewed as generalizations (in several directions) of Givental's toric mirror theorems. I will also describe extensions of wall-crossing to higher genus, and (time permitting) to orbifold GIT targets as well.
The talk is based on joint works with Bumsig Kim, and partly also with Daewoong Cheong and with Davesh Maulik.

Mixed motives turn up in number theory in various guises. Rather than discuss the rather deep foundational questions involved, this talk will aim

to give several illustrations of the ubiquity of mixed motives and their realizations. Along the way I hope to mention some of: the Mordell-Weil

theorem, the theory of height pairings, special values of L-functions, the Mahler measure of a polynomial, Galois deformations and the motivic

fundamental group.

We show that the positive mass theorem holds for

asymptotically flat, $n$-dimensional Riemannian manifolds with a metric

that is continuous, lies in the Sobolev space $W^{2, n/2}_{loc}$, and

has non-negative scalar curvature in the distributional sense. Our

approach requires an analysis of smooth approximations to the metric,

and a careful control of elliptic estimates for a related conformal

transformation problem. If the metric lies in $W^{2, p}_{loc}$ for

$p>n/2$, then we show that our metrics may be approximated locally

uniformly by smooth metrics with non-negative scalar curvature.

This talk is based on joint work with N. Tassotti and conversations with

J.J. Bevan.

With F. Johansson Viklund (Columbia) and A. Turner (Lancaster), we have studied a regularized version of the Hastings-Levitov model of random Laplacian growth. In addition to the usual feedback parameter $\alpha>0$, this regularized version of the growth process features a smoothing parameter $\sigma>0$.

We prove convergence of random clusters, in the limit as the size of the individual aggregating particles tends to zero, to deterministic limits, provided the smoothing parameter does not tend to zero too fast. We also study scalings limit of the harmonic measure flow on the boundary, and show that it can be described in terms of stopped Brownian webs on the circle. In contrast to the case $\alpha=0$, the flow does not always collapse into a single Brownian motion, which can be interpreted as a random number of infinite branches being present in the clusters.

Abstract: The boundary Harnack principle states that the ratio of any two functions, which are positive and harmonic on a domain, is bounded near some part of the boundary where both functions vanish. A given domain may or may not have this property, depending on the geometry of its boundary and the underlying metric measure space.

In this talk, we will consider a scale-invariant boundary Harnack principle on domains that are inner uniform. This has applications such as two-sided bounds on the Dirichlet heat kernel, or the identification of the Martin boundary and the topological boundary for bounded inner uniform domains.

The inner uniformity provides a large class of domains which may have very rough boundary as long as there are no cusps. Aikawa and Ancona proved the scale-invariant boundary Harnack principle on inner uniform domains in Euclidean space. Gyrya and Saloff-Coste gave a proof in the setting of non-fractal strictly local Dirichlet spaces that satisfy a parabolic Harnack inequality.

I will present a scale-invariant boundary Harnack principle for inner uniform domains in metric measure Dirichlet spaces that satisfy a parabolic Harnack inequality. This result applies to fractal spaces.

We analyse the impact of market makers' risk aversion on the equilibrium in a speculative market consisting of a risk neutral informed trader and noise traders. The unwillingness of market makers to bear risk causes the informed trader to absorb large shocks in their inventories. The informed trader's optimal strategy is to drive the market price to its fundamental value while disguising her trades as the ones of an uninformed strategic trader. This results in a mean reverting demand, price reversal, and systematic changes in the market depth. We also find that an increase in risk aversion leads to lower market depth, less efficient prices, stronger price reversal and slower convergence to fundamental value. The endogenous value of private information, however, is non-monotonic in risk aversion. We will mainly concentrate on the case when the private signal of the informed is static. If time permits, the implications of a dynamic signal will be discussed as well.

Based on a joint work with Albina Danilova.

Many years ago, Mark Kac was consulted by biologist colleague Lamont Cole regarding field-based observations of animal populations that suggested the existence of 3-4 year cycles in going from peak to peak. Kac provided an elegant argument for how purely random sequences of numbers could yield a mean value of 3 years, thereby establishing the notion that pattern can seemingly emerge in random processes. (This does not, however, mean that there could be a largely deterministic cause of such population cycles.)

By extending Kac's argument, we show how the distribution of cycle length can be analytically established using methods derived from random graph theory, etc. We will examine how such distributions emerge in other natural settings, including large earthquakes as well as colored Brownian noise and other random models and, for amusement, the Standard & Poor's 500 index for percent daily change from 1928 to the present.

We then show how this random model could be relevant to a variety of spatially-dependent problems and the emergence of clusters, as well as to memory and the aphorism "bad news comes in threes." The derivation here is remarkably similar to the former and yields some intriguing closed-form results. Importantly, the centroids or "centers of mass" of these clusters also yields clusters and a hierarchy then emerges. Certain "universal" scalings appear to emerge and scaling factors reminiscent of Feigenbaum numbers. Finally, as one moves from one dimension to 2, 3, and 4 dimensions, the scaling behaviors undergo modest change leaving this scaling phenomena qualitatively intact.

Finally, we will show how that an adaptation of the Langevin equation from statistical physics provides not simply a null-hypothesis for matching the observation of 3-4 year cycles, but a remarkably simple model description for the behavior of animal populations.

The transfer principle of Ax-Kochen-Ershov says that every first order sentence φ in the language of valued fields is, for p sufficiently big, true in ℚ_p iff it is true in \F_p((t)). Motivic integration allowed to generalize this to certain kinds of non-first order sentences speaking about functions from the valued field to ℂ. I will present some new transfer principles of this kind and explain how they are useful in representation theory. In particular, local integrability of Harish-Chandra characters, which previously was known only in ℚ_p, can be transferred to \F_p((t)) for p >> 1. (I will explain what this means.)

This is joint work with Raf Cluckers and Julia Gordon.

Consider a smooth, complex projective variety X inside P^n and an action of a reductive linear algebraic group G inside GL(n+1,C). On the one hand, we can view this as an algebra-geometric set-up and use geometric invariant theory (GIT) to construct a quotient variety X // G, which parameterises `most' of the closed orbits of X. On the other hand, X is naturally a symplectic manifold, and since G is reductive we can take a maximal real compact Lie subgroup K of G and consider the symplectic reduction of X by K with respect to an appropriate moment map. The Kempf-Ness theorem then says that the results of these two constructions are homeomorphic. In this talk I will define GIT and symplectic reduction and try to sketch the proof of the Kempf-Ness theorem.

We consider a random geometric graph model relevant to wireless mesh networks. Nodes are placed uniformly in a domain, and pairwise connections

are made independently with probability a specified function of the distance between the pair of nodes, and in a more general anisotropic model, their orientations. The probability that the network is (k-)connected is estimated as a function of density using a cluster expansion approach. This leads to an understanding of the crucial roles of

local boundary effects and of the tail of the pairwise connection function, in contrast to lower density percolation phenomena.

Convex regularization has become a popular approach to solve large scale inverse or data separation problems. A prominent example is the problem of identifying a sparse signal from linear samples my minimizing the l_1 norm under linear constraints. Recent empirical research indicates that many convex regularization problems on random data exhibit a phase transition phenomenon: the probability of successfully recovering a signal changes abruptly from zero to one as the number of constraints increases past a certain threshold. We present a rigorous analysis that explains why phase transitions are ubiquitous in convex optimization. It also describes tools for making reliable predictions about the quantitative aspects of the transition, including the location and the width of the transition region. These techniques apply to regularized linear inverse problems, to demixing problems, and to cone programs with random affine constraints. These applications depend on a new summary parameter, the statistical dimension of cones, that canonically extends the dimension of a linear subspace to the class of convex cones.

Joint work with Dennis Amelunxen, Mike McCoy and Joel Tropp.

Wei Wei

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Title: "Optimal Switching at Poisson Random Intervention Times"

(joint work with Dr Gechun Liang)

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Abstract: The paper introduces a new class of optimal switching problems, where

the player is allowed to switch at a sequence of exogenous Poisson

arrival times, and the underlying switching system is governed by an

infinite horizon backward stochastic differential equation system. The

value function and the optimal switching strategy are characterized by

the solution of the underlying switching system. In a Markovian setting,

the paper gives a complete description of the structure of switching

regions by means of the comparison principle.

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Julen Rotaetxe

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Title: Applicability of interpolation based finite difference method to problems in finance

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Abstract:

I will present the joint work with Christoph Reisinger on

the applicability of a numerical scheme relying on finite differences

and monotone interpolation to discretize linear and non-linear diffusion

equations. We propose suitable transformations to the process modeling

the underlying variable in order to overcome issues stemming from the

width of the stencil near the boundaries of the discrete spatial domain.

Numerical results would be given for typical diffusion models used in

finance in both the linear and non-linear setting.

We study the nonlinear wave equations on a class of asymptotically flat Lorentzian manifolds $(\mathbb{R}^{3+1}, g)$ with time dependent inhomogeneous metric g. Based on a new approach for proving the decay of solutions of linear wave equations, we give several applications to nonlinear problems. In particular, we show the small data global existence result for quasilinear wave equations satisfying the null condition on a class of time dependent inhomogeneous backgrounds which do not settle to any particular stationary metric.

Let $G$ be an addable minor-closed class of graphs. We prove that a zero-one law holds in monadic second-order logic (MSO) for connected graphs in G, and a convergence law in MSO for all graphs in $G$. For each surface $S$, we prove the existence of a zero-one law in first order logic (FO) for connected graphs embeddable in $S$, and a convergence law in FO for all graphs embeddable in $S$. Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface $S$. If $G$ is an addable minor-closed class, we prove that the closure of the set of limiting probabilities is a finite union of intervals, and it is the same for FO and MSO. For the class of planar graphs it consists of exactly 108 intervals. We give examples of non-addable classes where the results are quite different: for instance, the zero-one law does not hold for caterpillars, even in FO. This is joint work with Peter Heinig, Tobias Müller and Anusch Taraz.

A group is said to be quasirandom if all its unitary representations have “large” dimension. After introducing quasirandom groups and their basic properties, I shall turn to recent applications in two directions: constructions of expanders and non-existence of large product-free sets.

Let F be a field. A symplectic alternating algebra over F

consists of a symplectic vector space V over F with a non-degenerate

alternating form that is also equipped with a binary alternating

product · such that the law (u·v, w)=(v·w, u) holds. These algebraic

structures have arisen from the study of 2-Engel groups but seem also

to be of interest in their own right with many beautiful properties.

We will give an overview with a focus on some recent work on the

structure of nilpotent symplectic alternating algebras.

The concept of a matrix factorization was originally introduced by Eisenbud to study syzygies over local rings of singular hypersurfaces. More recently, interactions with mathematical physics, where matrix factorizations appear in quantum field theory, have provided various new insights. I will explain how matrix factorizations can be studied in the context of noncommutative algebraic geometry based on differential graded categories. We will see the relevance of the noncommutative analogue of de Rham cohomology in terms of classical singularity theory. Finally, I will outline how the Kapustin-Li formula for the noncommutative Serre duality pairing (originally computed via path integral methods) can be mathematically explained using a combination of homological perturbation theory and local duality.
Partly based on joint work with Daniel Murfet.

We propose a new algorithm for the approximate solution of large-scale high-dimensional tensor-structured linear systems. It can be applied to high-dimensional differential equations, which allow a low-parametric approximation of the multilevel matrix, right-hand side and solution in a tensor product format. We apply standard one-site tensor optimisation algorithm (ALS), but expand the tensor manifolds using the classical iterative schemes (e.g. steepest descent).  We obtain the rank--adaptive algorithm with the theoretical convergence estimate not worse than the one of the steepest descent, and fast practical convergence, comparable or even better than the convergence of more expensive two-site optimisation algorithm (DMRG).
The method is successfully applied for a high--dimensional problem of quantum chemistry, namely the NMR simulation of a large peptide.

This is a joint work with S.Dolgov (Max-Planck Institute, Leipzig, Germany), supported by RFBR and EPSRC grants.

Keywords: high--dimensional problems, tensor train format, ALS, DMRG, steepest descent, convergence rate, superfast algorithms, NMR.

The concept of a matrix factorization was originally introduced by Eisenbud to study syzygies over local rings of singular hypersurfaces. More recently, interactions with mathematical physics, where matrix factorizations appear in quantum field theory, have provided various new insights. I will explain how matrix factorizations can be studied in the context of noncommutative algebraic geometry based on differential graded categories. We will see the relevance of the noncommutative analogue of de Rham cohomology in terms of classical singularity theory. Finally, I will outline how the Kapustin-Li formula for the noncommutative Serre duality pairing (originally computed via path integral methods) can be mathematically explained using a combination of homological perturbation theory and local duality.
Partly based on joint work with Daniel Murfet.

We study a system of partial differential equations describing a steady flow of an incompressible generalized Newtonian fluid, wherein the Cauchy stress depends on concentration. Namely, we consider a coupled system of the generalized Navier-Stokes equations (viscosity of power-law type with concentration dependent power index) and convection-diffusion equation with non-linear diffusivity. We focus on the existence analysis of a weak solution for certain class of models by using a generalization of the monotone operator theory which fits into the framework of generalized Sobolev spaces with variable exponent (class of Sobolev-Orlicz spaces). Such results is then adapted for a suitable FEM approximation, for which the main tool of proof is a generalization of the Lipschitz approximation method.

In the first JAM seminar of 2013/2014, I will discuss the topic of singular perturbed hyperbolic systems of PDE arising in physical phenomena, particularly the St Venant equations of shallow water theory. Using a mixture of analytical and numerical techniques, I will demonstrate the dangers of approximating the dynamics of a system by the equations obtained upon taking a singular limit $\epsilon\rightarrow 0$ and furthermore how the dynamics of the system change when the parameter $\epsilon$ is taken to be small but finite. Problems of this type are ubiquitous in the physical sciences, and I intend to motivate another example arising in elastoplasticity, the subject of my DPhil study.

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Note: This seminar is not intended for faculty members, and is available only to current undergraduate and graduate students.