L4

I will speak about recent work with Michael Wemyss (arXiv:1309.0698), applying noncommutative deformation theory to study the birational geometry of 3-folds. In particular, I will explain how every flippable or floppable rational curve in a 3-fold has a naturally associated algebra of noncommutative deformations, even in the singular setting. We investigate the properties of this algebra, and indicate how to calculate it in examples using quiver techniques. This gives new information about the (commutative) geometry of 3-folds, and in particular provides a new tool to differentiate between flops.

As a further application, we show how the noncommutative deformation algebra controls the homological properties of a floppable curve. In this setting, work of Bridgeland and Chen yields a Fourier-Mukai flop-flop functor which acts on the derived category of the 3-fold (assuming any singularities are at worst Gorenstein terminal). We show that this functor can be described as a spherical twist about the universal family over the noncommutative deformation algebra.

In the second part, I will talk about further work in progress, and explain some more technical details, such as the use of noncommutative deformation functors, and the categorical mutations of Iyama and Wemyss. If there is time, I will also give some higher-dimensional examples, and discuss situations involving chains of intersecting floppable curves. In this latter case, deformations, intersections and homological properties are encoded by the path algebra of a quiver, generalizing the algebra of noncommutative deformations.

I will speak about recent work with Michael Wemyss (arXiv:1309.0698), applying noncommutative deformation theory to study the birational geometry of 3-folds. In particular, I will explain how every flippable or floppable rational curve in a 3-fold has a naturally associated algebra of noncommutative deformations, even in the singular setting. We investigate the properties of this algebra, and indicate how to calculate it in examples using quiver techniques. This gives new information about the (commutative) geometry of 3-folds, and in particular provides a new tool to differentiate between flops.

As a further application, we show how the noncommutative deformation algebra controls the homological properties of a floppable curve. In this setting, work of Bridgeland and Chen yields a Fourier-Mukai flop-flop functor which acts on the derived category of the 3-fold (assuming any singularities are at worst Gorenstein terminal). We show that this functor can be described as a spherical twist about the universal family over the noncommutative deformation algebra.

In the second part, I will talk about further work in progress, and explain some more technical details, such as the use of noncommutative deformation functors, and the categorical mutations of Iyama and Wemyss. If there is time, I will also give some higher-dimensional examples, and discuss situations involving chains of intersecting floppable curves. In this latter case, deformations, intersections and homological properties are encoded by the path algebra of a quiver, generalizing the algebra of noncommutative deformations.

Recent years have seen rapid expansion of the capabilities 
to recreate in vivo conditions using in vitro microfluidic assays.  
A wide range of single cell and cell population behaviors can now 
be replicated, controlled and imaged for detailed studies to gain 
new insights.  Such experiments also provide useful fodder for 
computational models, both in terms of estimating model parameters 
and for testing model-generated hypotheses.  Our experiments have 
focused in several different areas.  
1) Single cell migration experiments in 3D collagen gels have 
revealed that interstitial flow can lead to biased cell migration 
in the upstream direction, with important implications to cancer 
invasion.  We show this phenomenon to be a consequence of 
integrin-mediated mechanotransduction.  
2) Endothelial cells seeded in fibrin gels form perfusable 
vascular networks within 2-3 days through a process termed 
“vasculogenesis”.  The process by which cells sense their 
neighbours, extend projections and form anastomoses, and 
generate interconnected lumens can be observed through time-lapse 
microscopy.  
3) These vascular networks, once formed, can be perfused with 
medium containing cancer cells that become lodged in the 
smaller vessels and proceed to transmigrate across the endothelial 
barrier and invade into the surrounding matrix.  High resolution 
imaging of this process reveals a fascinating sequence of events 
involving interactions between a tumour cell, endothelial cells, 
and underlying matrix.  These three examples will be presented 
with a view toward gaining new insights through computational 
modelling of the associated phenomena.

Worst-case portfolio optimization has been introduced in Korn and Wilmott

(2002) and is based on distinguishing between random stock price

fluctuations and market crashes which are subject to Knightian

uncertainty. Due to the absence of full probabilistic information, a

worst-case portfolio problem is considered that will be solved completely.

The corresponding optimal strategy is of a multi-part type and makes an

investor indifferent between the occurrence of the worst possible crash

and no crash at all.

We will consider various generalizations of this setting and - as a very

recent result - will in particular answer the question "Is it good to save

for bad times or should one consume more as long as one is still rich?"

We construct a model for asset price in a limit order book, which captures on one hand main stylized facts of microstructure effects, and on the other hand is tractable for dealing with optimal high frequency trading by stochastic control methods. For this purpose, we introduce a model for describing the fluctuations of a tick-by-tick single asset price, based on Markov renewal process.

We consider a point process associated to the timestamps of the price jumps, and marks associated to price increments. By modeling the marks with a suitable Markov chain, we can reproduce the strong mean-reversion of price returns known as microstructure noise. Moreover, by using Markov renewal process, we can model the presence of spikes in intensity of market activity, i.e. the volatility clustering. We also provide simple parametric and nonparametric statistical procedures for the estimation of our model. We obtain closed-form formulae for the mean signature plot, and show the diffusive behavior of our model at large scale limit. We illustrate our results by numerical simulations, and find that our model is consistent with empirical data on futures Euribor and Eurostoxx. In a second part, we use a dynamic programming approach to our semi Markov model applied to the problem of optimal high frequency trading with a suitable modeling of market order flow correlated with the stock price, and taking into account in particular the adverse selection risk. We show a reduced-form for the value function of the associated control problem, and provide a convergent and computational scheme for solving the problem. Numerical tests display the shape of optimal policies for the market making problem.

This talk is based on joint works with Pietro Fodra.