In silico study of macromolecular crowding effects on biochemical signaling
Abstract
***** PLEASE NOTE THAT THIS WILL TAKE PLACE ON TUESDAY 11TH JUNE ****
Signal transduction pathways are sophisticated information processing machinery in the cell that is arguably taking advantage of highly non-idealistic natures of intracellular environments for its optimum operations. In this study, we focused on effects of intracellular macromolecular crowding on signal transduction pathways using single-particle simulations. We have previously shown that rebinding of kinases to substrates can remarkably increase processivity of dual-phosphorylation reactions and change both steady-state and transient responses of the reaction network. We found that molecular crowding drastically enhances the rebinding effect, and it shows nonlinear time dependency although kinetics at the macroscopic level still follows the conventional model in dilute media. We applied the rate law revised on the basis of these calculations to MEK-ERK system and compared it with experimental measurements.
***** PLEASE NOTE THAT THIS WILL TAKE PLACE ON TUESDAY 11TH JUNE ****
On Sofic Groups
Abstract
The class of sofic groups was introduced by Gromov in 1999. It
includes all residually finite and all amenable groups. In fact, no group has been proved
not to be sofic, so it remains possible that all groups are sofic. Their
defining property is that, roughly speaking, for any finite subset F of
the group G, there is a map from G to a finite symmetric group, which is
approximates to an injective homomorphism on F. The widespread interest in
these group results partly from their connections with other branches of
mathematics, including dynamical systems. In the talk, we will concentrate
on their definition and algebraic properties.
Learning from the past, predicting the statistics for the future, learning an evolving system using Rough Paths Theory.
Abstract
In this talk, we consider the setting: a random realization of an evolving dynamical system, and explain how, using notions common in the theory of rough paths, such as the signature, and shuffle product, one can provide a new united approach to the fundamental problem of predicting the conditional distribution of the near future given the past. We will explain how the problem can be reduced to a linear regression and least squaresanalysis. The approach is clean and systematic and provides a clear gradation of finite dimensional approximations. The approach is also non-parametric and very general but still presents itself in computationally tractable and flexible restricted forms for concrete problems. Popular techniques in time series analysis such as GARCH can be seen to be restricted special cases of our approach but it is not clear they are always the best or most informative choices. Some numerical examples will be shown in order to compare our approach and standard time series models.
Simulation of BSDE’s and Wiener chaos expansions
Abstract
This talk is based on a joint work with Céline Labart. We are interested in this paper in the numerical simulation of solutions to Backward Stochastic Differential Equations. There are several existing methods to handle this problem and one of the main difficulty is always to compute conditional expectations.
Even though our approach can also be applied in the case of the dynamic programmation equation, our starting point is the use of Picard's iterations that we write in a forward way
In order to compute the conditional expectations, we use Wiener Chaos expansions of the underlying random variables. From a practical point of view, we keep only a finite number of terms in the expansions and we get explicit formulas.
We will present numerical experiments and results on the error analysis.
16:30
Langlands functoriality and non linear Poisson formulas
Abstract
"We introduce some type of generalized Poisson formula which is equivalent
to Langlands' automorphic transfer from an arbitrary reductive group over a
global field to a general linear group."
Martingale Optimal Transport and Robust Hedging
Abstract
The martingale optimal transportation problem is motivated by
model-independent bounds for the pricing and hedging exotic options in
financial mathematics.
In the simplest one-period model, the dual formulation of the robust
superhedging cost differs from the standard optimal transport problem by
the presence of a martingale constraint on the set of coupling measures.
The one-dimensional Brenier theorem has a natural extension. However, in
the present martingale version, the optimal coupling measure is
concentrated on a pair of graphs which can be obtained in explicit form.
These explicit extremal probability measures are also characterized as
the unique left and right monotone martingale transference plans, and
induce an optimal solution of the kantorovitch dual, which coincides
with our original robust hedging problem.
By iterating the above construction over n steps, we define a Markov
process whose distribution is optimal for the n-periods martingale
transport problem corresponding to a convenient class of cost functions.
Similarly, the optimal solution of the corresponding robust hedging
problem is deduced in explicit form. Finally, by sending the time step
to zero, this leads to a continuous-time version of the one-dimensional
Brenier theorem in the present martingale context, thus providing a new
remarkable example of Peacock, i.e. Processus Croissant pour l'Ordre
Convexe. Here again, the corresponding robust hedging strategy is
obtained in explicit form.
14:00
Recurrent Neural Networks in Modelling Biological Networks: Oscillatory p53 interaction dynamics
Macrophages drive tumour regrowth after chemotherapy: can we use zebrafish to model this and predict ways to block it?
Abstract
***** PLEASE NOTE THAT THIS WILL TAKE PLACE ON FRIDAY 7TH JUNE *****
Microelectromechanical Systems, Inverse Eigenvalue Analysis and Nonlinear Lattices
Abstract
Collective behaviours of coupled linear or nonlinear resonators have been of interest to engineers as well as mathematician for a long time. In this presentation, using the example of coupled resonant nano-sensors (which leads to a Linear pencil with a Jacobian matrix), I will show how previously feared and often avoided coupling between nano-devices along with their weak nonlinear behaviour can be used with inverse eigenvalue analysis to design multiple-input-single-output nano-sensors. We are using these matrices in designing micro/Nano electromechanical systems, particularly resonant sensors capable for measuring very small mass for use as environmental as well as biomedical monitors. With improvement in fabrication technology, we can design and build several such sensors on one substrate. However, this leads to challenges in interfacing them as well as introduces undesired parasitic coupling. More importantly, increased nonlinearity is being observed as these sensors reduce in size. However, this also presents an opportunity to experimentally study chains or matrices of coupled linear and/or nonlinear structures to develop new sensing modalities as well as to experimentally verify theoretically or numerically predicted results. The challenge for us is now to identify sensing modalities with chain of linear or nonlinear resonators coupled either linearly or nonlinearly. We are currently exploring chains of Duffing resonators, van der Pol oscillators as well as FPU type lattices.
17:30
Strategy-Proof Auctions for Complex Procurement
Abstract
Some real resource allocation problems are so large and complex that optimization would computationally infeasible, even with complete information about all the relevant values. For example, the proposal in the US to use television broadcasters' bids to determine which stations go off air to make room for wireless broadband is characterized by hundreds of thousands of integer constraints. We use game theory and auction theory to characterize a class of simple, strategy-proof auctions for such problems and show their equivalence to a class of "clock auctions," which make the optimal bidding strategy obvious to all bidders. We adapt the results of optimal auction theory to reduce expected procurement costs and prove that the procurement cost of each clock auction is the same as that of the full information equilibrium of its related paid-as-bid (sealed-bid) auction.
Externally definable sets in real closed fields
Abstract
An externally definable set of a first order structure $M$ is a set of the form $X\cap M^n$ for a set $X$ that is parametrically definable in some elementary extension of $M$. By a theorem of Shelah, these sets form again a first order structure if $M$ is NIP. If $M$ is a real closed field, externally definable sets can be described as some sort of limit sets (to be explained in the talk), in the best case as Hausdorff limits of definable families. It is conjectured that the Shelah structure on a real closed field is generated by expanding the field with convex subsets of the line. This is known to be true in the archimedean case by van den Dries (generalised by Marker and Steinhorn). I will report on recent progress around this question, mainly its confirmation on real closed fields that are close to being maximally valued with archimedean residue field. The main tool is an algebraic characterisation of definable types in real closed valued fields. I also intend to give counterexamples to a localized version of the conjecture. This is joint work with Francoise Delon.
Paul Milgrom, Shirley and Leonard Ely Professor of Humanities and Sciences at Stanford University
An introduction to the invariant quaternion algebra associated to a hyperbolic 3-manifold.
Abstract
I will show how to associate a quaternion algebra to a hyperbolic 3-manifold. I will then go through some examples and applications of this theory
Discontinuous Galerkin Methods for Modeling the Coastal Ocean
Abstract
The coastal ocean contains a diversity of physical and biological
processes, often occurring at vastly different scales. In this talk,
we will outline some of these processes and their mathematical
description. We will then discuss how finite element methods are used
in coastal ocean modeling and recent research into
improvements to these algorithms. We will also highlight some of the
successes of these methods in simulating complex events, such as
hurricane storm surges. Finally, we will outline several interesting
challenges which are ripe for future research.
Hamiltonian propagation of monokinetic measures with rough momentum profiles (work in collaboration with Peter Markowich and Thierry Paul)
Abstract
Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure that is the image of the initial graph by the flow. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We study the conditions under which this projected measure can have atoms, and give an example in which the projected measure is singular with respect to the Lebesgue measure and diffuse. We discuss applications of our results to the classical limit of the Schrödinger equation. Finally we present various examples and counterexamples showing that our results are sharp.
Numerical approximations for a nonloncal model for sandpiles
Abstract
- In this talk we study numerical approximations of continuous solutions to a nonlocal $p$-Laplacian type diffusion equation,
\[
u_t (t, x) = \int_\Omega J(x − y)|u(t, y) − u(t, x)|^{p-2} (u(t, y) − u(t, x)) dy.
\]
-
First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one, as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties with the continuous problem: it preserves the total mass and the solution converges to the mean value of the initial condition, as $t$ goes to infinity.
-
Next, we discretize also the time variable and present a totally discrete method which also enjoys the above mentioned properties.
-
In addition, we investigate the limit as $p$ goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile.
- Finally, we present some numerical experiments that illustrate our results.
- This is a joint work with J. D. Rossi.
11:00
Positivity Problems for Linear Recurrence Sequences
Abstract
We consider two decision problems for linear recurrence sequences (LRS)
over the integers, namely the Positivity Problem (are all terms of a given
LRS positive?) and the Ultimate Positivity Problem (are all but finitely
many terms of a given LRS positive?). We show decidability of both
problems for LRS of order 5 or less, and for simple LRS (i.e. whose
characteristic polynomial has no repeated roots) of order 9 or less. Our
results rely on on tools from Diophantine approximation, including Baker's
Theorem on linear forms in logarithms of algebraic numbers. By way of
hardness, we show that extending the decidability of either problem to LRS
of order 6 would entail major breakthroughs on Diophantine approximation
of transcendental numbers.
This is joint with work with Joel Ouaknine and Matt Daws.
Decay for fields outside black holes
Abstract
The Einstein equation from general relativity is a
quasilinear hyperbolic, geometric PDE (when viewed in an appropriate
coordinate system) for a manifold. A particularly interesting set of
known, exact solutions describe black holes. The wave and Maxwell
equations on these manifolds are models for perturbations of the known
solutions and have attracted a significant amount of attention in the
last decade. Key estimates are conservation of energy and Morawetz (or
integrated local energy) estimates. These can be proved using both
Fourier analytic methods and more geometric methods. The main focus of
the talk will be on decay estimates for solutions of the Maxwell
equation outside a slowly rotating Kerr black hole.