Past

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.