Past

Market impact, leverage, systemic risk, and the perils of mark-to-market accounting

Market impact is the price change associated with new buy or sell orders entering the market. It provides a useful alternative to thinking in terms of supply and demand for several reasons, the most important being that there is theoretical and empirical evidence that it follows a universal law. Understanding market impact is essential for adjusting investment size, for optimizing execution tactics, and provides a useful tool for understanding market ecology and systemic risk. I will present a new method for impact-adjusted accounting, and show how it can avoid the serious problems of marking-to-market when leverage is used. Then I will discuss how market impact can be combined with network theory to understand the problem of overlapping portfolios and market crowding. Since I am a new faculty member, at the beginning of the talk I will say a bit about my interests and current projects.

Recent advances in engineered muscle tissue attached to a synthetic substrate motivates the development of appropriate constitutive and numerical models. Applications of active materials can be expanded by using robust, non-mammalian muscle cells, such as those of Manduca sexta. In this talk we present a   continuum model that accounts for the stimulation of muscle fibers by introducing multiple stress-free reference configurations and for the hysteretic response by specifying a pseudo-elastic energy function. A simple example representing uniaxial loading-unloading is used to validate and verify the characteristics of the model. Then, based on experimental data of muscular thin films, a more complex case shows the qualitative potential of Manduca muscle tissue in active biohybrid constructs.

 We present a new, more conceptual proof of our result that, when a finite group acts on a polynomial ring, the regularity of the ring of invariants is at most zero, and hence one can write down bounds on the degrees of the generators and relations. This new proof considers the action of the group on the Cech complex and looks at when it splits over the group algebra. It also applies to a more general class of rings than just polynomial ones.

How many nodal degree d plane curves are tangent to a given line? The celebrated Caporaso-Harris recursion formula gives a complete answer for any number of nodes, degrees, and all possible tangency conditions. In this talk, I will report my recent work on the generalization of the above problem to count singular curves with given tangency condition to a fixed smooth divisor on general surfaces. I will relate the enumeration to tautological integrals on Hilbert schemes of points and show the numbers of curves in question are given by universal polynomials. As a result, we can obtain infinitely many new formulas for nodal curves and understand the asymptotic behavior for all singular curves with any tangency conditions.

We will discuss the origin of the conjectured colour-kinematics

duality in perturbative gauge theory, according to which there is a

symmetry between the colour dependence and the kinematic dependence of the

S-matrix. Based on this duality, there is a prescription to obtain gravity

amplitudes as the "double copy" of gauge theory amplitudes. We will first

consider tree-level amplitudes, where a diffeomorphism algebra underlies

the structure of MHV amplitudes, mirroring the colour algebra. We will

then draw on the progress at tree-level to consider one-loop amplitudes.

Topological field theories give a connection between

topology and algebra. This connection can be exploited in both

directions: using algebra to construct topological invariants, or

using topology to prove algebraic theorems. In this talk, I will

explain an interesting example of the latter phenomena. Radford's

theorem, as generalized by Etingof-Nikshych-Ostrik, says that in a

finite tensor category the quadruple dual functor is easy to

understand. It's somewhat mysterious that the double dual is hard to

understand but the quadruple dual is easy. Using topological field

theory, we show that Radford's theorem is exactly the consequence of

the Dirac belt trick in topology. That is, the double dual

corresponds to the generator of $\pi_1(\mathrm{SO}(3))$ and so the

quadruple dual is trivial in an appropriate sense exactly because

$\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2$. This is part of a large

project, joint with Chris Douglas and Chris Schommer-Pries, to

understand local field theories with values in the 3-category of

tensor categories via the cobordism hypothesis.

Given a flat torus, we consider certain discrete graph approximations of

it and determine the asymptotics of the number of spanning trees

("complexity") of these graphs as the mesh gets finer. The constants in the

asymptotics involve various notions of determinants such as the

determinant of the Laplacian ("height") of the torus. The analogy between

the complexity of graphs and the height of manifolds was previously

commented on by Sarnak and Kenyon. In dimension two, similar asymptotics

were established earlier by Barber and Duplantier-David in the context of

statistical physics.

Our proofs rely on heat kernel analysis involving Bessel functions, which

in the torus case leads into modular forms and Epstein zeta functions. In

view of a folklore conjecture it also suggests that tori corresponding to

densest regular sphere packings should have approximating graphs with the

largest number of spanning trees, a desirable property in network theory.

Joint work with G. Chinta and J. Jorgenson.

We construct explicitly a bridge process whose distribution, in its own filtration, is the same as the difference of two independent Poisson processes with the same intensity and its time 1 value satisfies a specific constraint. This construction allows us to show the existence of Glosten-Milgrom equilibrium and its associated optimal trading strategy for the insider. In the equilibrium the insider employs a mixed strategy to randomly submit two types of orders: one type trades in the same direction as noise trades while the other cancels some of the noise trades by submitting opposite orders when noise trades arrive. The construction also allows us to prove that Glosten-Milgrom equilibria converge weakly to Kyle-Back equilibrium, without the additional assumptions imposed in \textit{K. Back and S. Baruch, Econometrica, 72 (2004), pp. 433-465}, when the common intensity of the Poisson processes tends to infinity. This is a joint work with Umut Cetin.

Ice-dammed lakes form next to, on the surface of, and beneath glaciers

and ice sheets. Some lakes are known to drain catastrophically,

creating hazards, wasting water resources and modulating the flow of

the adjacent ice. My work aims to increase our understanding of such

drainage. Here I will focus on lakes that form next to glaciers and

drain subglacially (between ice and bedrock) through a channel. I will

describe how such a system can be modelled and present results from

model simulations of a lake that fills due to an input of meltwater

and drains through a channel that receives a supply of meltwater along

its length. Simulations yield repeating cycles of lake filling and

drainage and reveal how increasing meltwater input to the system

affects these cycles: enlarging or attenuating them depending on how

the meltwater is apportioned between the lake and the channel. When

inputs are varied with time, simulating seasonal meteorological

cycles, the model simulates either regularly repeating cycles or

irregular cycles that never repeat. Irregular cycles demonstrate

sensitivity to initial conditions, a high density of periodic orbits

and topological mixing. I will discuss how these results enhance our

understanding of the mechanisms behind observed variability in these

systems.

This is the session for our industrial sponsors to propose project ideas. Academic staff are requested to attend to help shape the problem statements and to suggest suitable internal supervisors for the projects. 

Recent years have seen increasing attention to the subtle effects on

intracellular transport caused when multiple molecular motors bind to

a common cargo. We develop and examine a coarse-grained model which

resolves the spatial configuration as well as the thermal fluctuations

of the molecular motors and the cargo. This intermediate model can

accept as inputs either common experimental quantities or the

effective single-motor transport characterizations obtained through

systematic analysis of detailed molecular motor models. Through

stochastic asymptotic reductions, we derive the effective transport

properties of the multiple-motor-cargo complex, and provide analytical

explanations for why a cargo bound to two molecular motors moves more

slowly at low applied forces but more rapidly at high applied forces

than a cargo bound to a single molecular motor. We also discuss how

our theoretical framework can help connect in vitro data with in vivo

behavior.

The Discontinuous Galerkin (DG) method has been used to solve a wide range of partial differential equations. Especially for advection dominated problems it has proven very reliable and accurate. But even for elliptic problems it has advantages over continuous finite element methods, especially when parallelization and local adaptivity are considered.

In this talk we will first present a variation of the compact DG method for elliptic problems with varying coefficients. For this method we can prove stability on general grids providing a computable bound for all free parameters. We developed this method to solve the compressible Navier-Stokes equations and demonstrated its efficiency in the case of meteorological problems using our implementation within the DUNE software framework, comparing it to the operational code COSMO used by the German weather service.

After introducing the notation and analysis for DG methods in Euclidean spaces, we will present a-priori error estimates for the DG method on surfaces. The surface finite-element method with continuous ansatz functions was analysed a few years ago by Dzuik/Elliot; we extend their results to the interior penalty DG method where the non-smooth approximation of the surface introduces some additional challenges.

In 1977, Cline Parshall, Scott and van der Kallen wrote a seminal paper `Rational and generic cohomology' which exhibited a connection between the cohomology for algebraic groups and the cohomology for finite groups of Lie type, showing that in many cases one can conclude that there is an isomorphism of cohomology through restriction from the algebraic to the finite group.

One unfortunate problem with their result is that there remain infinitely many modules for which their theory---for good reason---tells us nothing. The main result of this talk (recent work with Parshall and Scott) is to show that almost all the time, one can manipulate the simple modules for finite groups of Lie type in such a way as to recover an isomorphism of its cohomology with that of the algebraic group.