L2

We consider self-organizing systems, i.e. systems consisting of a large number of interacting entities which spontaneously coordinate and achieve a collective dynamics. Sush systems are ubiquitous in nature (flocks of birds, herds of sheep, crowds, ...). Their mathematical modeling poses a number of fascinating questions such as finding the conditions for the emergence of collective motion. In this talk, we will consider a simplified model first proposed by Vicsek and co-authors and consisting of self-propelled particles interacting through local alignment.
We will rigorously study the multiplicity and stability of its equilibria through kinetic theory methods. We will illustrate our findings by numerical simulations.

From a theoretical point of view, theta is a relatively simple quantity: the rate of change in value of a financial derivative with respect to time. In a Black-Scholes world, the theta of a delta hedged option can be viewed as `rent’ paid in exchange for gamma. This relationship is fundamental to the risk-management of a derivatives portfolio. However, in the real world, the situation becomes significantly more complicated. In practice the model is continually being recalibrated, and whereas in the Black-Scholes world volatility is not a risk factor, in the real world it is stochastic and carries an associated risk premium. With the heightened interest in automation and electronic trading, we increasingly need to attempt to capture trading, marking and risk management practice algorithmically, and this requires careful consideration of the relationship between the risk neutral and historical measures. In particular these effects need to be incorporated in order to make sense of theta and the time evolution of a derivatives portfolio in the historical measure. 

Cell migration is of vital importance in many biological studies, hence robust cell tracking algorithms are needed for inference of dynamic features from (static) in vivo and in vitro experimental imaging data of cells migrating. In recent years much attention has been focused on the modelling of cell motility from physical principles and the development of state-of-the art numerical methods for the simulation of the model equations. Despite this, the vast majority of cell tracking algorithms proposed to date focus solely on the imaging data itself and do not attempt to incorporate any physical knowledge on cell migration into the tracking procedure. In this study, we present a mathematical approach for cell tracking, in which we formulate the cell tracking problem as an inverse problem for fitting a mathematical model for cell motility to experimental imaging data. The novelty of this approach is that the physics underlying the model for cell migration is encoded in the tracking algorithm. To illustrate this we focus on an example of Zebrafish (Danio rerio's larvae} Neutrophil migration and contrast an ad-hoc approach to cell tracking based on interpolation with the model fitting approach we propose in this talk.

July 5

9:30-10:30

Robert Langlands (IAS, Princeton)

Problems in the theory of automorphic forms: 45 years later 

11:00-12:00

Christopher Deninger (Univ. Münster)

Zeta functions and foliations     

 13:30-14:30          

Christophe Soulé (IHES, Bures-sur-Yvette)

 A singular arithmetic Riemann-Roch theorem           

14:40-15:40

Minhyong Kim (Univ. Oxford)

 Non-abelian reciprocity laws and Diophantine geometry

 16:10-17:10  

Constantin Teleman (Berkeley/Oxford)           

Categorical representations and Langlands duality 

July 6

 9:30-10:30

Ted Chinburg (Univ. Pennsylvania, Philadelphia)

 Higher Chern classes in Iwasawa theory

11:00-12:00          

Yuri Tschinkel (Courant Institute, New York)

Introduction to almost abelian anabelian geometry

13:30-14:30

Ralf Meyer (Univ. Göttingen)

Groupoids and higher groupoids

14:40-15:40

Dennis Gaitsgory (Harvard Univ., Boston)

Picard-Lefschetz oscillators for Drinfeld-Lafforgue compactifications

16:10-17:10

François Loeser (Univ. Paris 6-7)

Motivic integration and representation theory

July 7

9:00-10:00

Matthew Morrow (Univ. Bonn)                                                  

On the deformation theory of algebraic cycles

10:30-11:30

Fedor Bogomolov (Courant Institute, New York/Univ. Nottingham)

On the section conjecture in anabelian geometry                 

13:15-14:15                                                                      

Kevin Buzzard (ICL, London)

p-adic Langlands correspondences

14:45-15:45                                                                      

Masatoshi Suzuki (Tokyo Institute of Technology)

Translation invariant subspaces and GRH for zeta functions

16:00-17:00

Edward Frenkel (Univ. California Berkeley)

"Love and Math", the Langlands programme - Public presentation

July 8

9:15-10:15

Mikhail Kapranov (Kavli IMPU, Tokyo)

Lie algebras and E_n-algebras associated to secondary polytopes                                 

10:45-11:45          

Sergey Oblezin (Univ. Nottingham)

Whittaker functions, mirror symmetry and the Langlands correspondence

13:30-14:30                                                                      

Edward Frenkel (Univ. California Berkeley)

The Langlands programme and quantum dualities

14:40-15:40                                                                                              

Dominic Joyce (Univ. Oxford)

Derived symplectic geometry and categorification

16:10-17:10  

Urs Schreiber (Univ. Nijmegen, The Netherlands)

Correspondences of cohesive linear homotopy types and quantization

The main result is to give a separable, Cech-complete, 0-dimensional Moore space that is not Scott-domain representable. This result answered questions in the literature; it is known that each complete mertrisable space is Scott-domain representable. The talk will give a history of the techniques involved.

The most valuable asset that people in a sovereign state can have is good, sustainable governance. Setting up a system of good, sustainable governance is not easy. The big and well-known problem is time inconsistency of optimal policies. A mechanism that has proven valuable in mitigating the time inconsistency problem is rule by law. The too-big-to-fail problem in banking is the result of the time inconsistency problem. In this lecture I will argue there is an alternative financial system that is not subject to the too-big-to-fail problem. The alternative arrangement I propose is a pure transaction banking system. Transaction banks are required to hold 100$\%$ interest bearing reserves and can pay tax-free interest on demand deposits. With this system, there cannot be a bank run as there is no place to run to. Mutual arrangements would finance all business investment, which is not currently the case.

We study lower- and dual upper-bounds for Bermudan options in a MonteCarlo/MC setting and provide four contributions. 1) We introduce a local least-squares MC method, based on maximizing the Bermudan price and which provides a lower-bound, which "also" minimizes (not the dual upper-bound itself, but) the gap between these two bounds; where both bounds are specified recursively. 2) We confirm that this method is near optimal, for both lower- and upper-bounds, by pricing Bermudan max-call options subject to an up-and-out barrier; state-of-the-art methods including Longstaff-Schwartz produce a large gap of 100--200 basis points/bps (Desai et al. (2012)), which we reduce to just 5--15 bps (using the same linear basis of functions). 3) For dual upper-bounds based on continuation values (more biased but less time intensive), it works best to reestimate the continuation value in the continuation region only. And 4) the difference between the Bermudan option Delta and the intrinsic value slope at the exercise boundary gives the sensitivity to suboptimal exercise (up to a 2nd-order Taylor approximation). The up-and-out feature flattens the Bermudan price, lowering the Bermudan Delta well below one when the call-payoff slope is equal to one, which implies that optimal exercise "really" matters.

In this work, we want to construct the solution $(Y,Z,K)$ to the following BSDE

$$\begin{array}{l}

Y_t=\xi+\int_t^Tf(s,Y_s,Z_s)ds-\int_t^TZ_sdB_s+K_T-K_t, \quad 0\le t\le T, \\

{\mathbf E}[l(t, Y_t)]\ge 0, \quad 0\le t\le T,\\

\int_0^T{\mathbf E}[l(t, Y_t)]dK_t=0, \\

\end{array}

$$

where $x\mapsto l(t, x)$ is non-decreasing and the terminal condition $\xi$

is such that ${\mathbf E}[l(T,\xi)]\ge 0$.

This equation is different from the (classical) reflected BSDE. In particular, for a solution $(Y,Z,K)$,

we require that $K$ is deterministic. We will first study the case when $l$ is linear, and then general cases.

We also give some application to mathematical finance. This is a joint work with Philippe Briand and Romuald Elie.