L1

We review the theory of $E_n$-algebras (roughly, algebras with $n$ compatible multiplications) and discuss $E_n$-deformation theory in the sense of Lurie. We then describe, to the best of our ability, the use of $E_n$-deformation theory in the on-going work of Calaque, Pantev, Toen, Vezzosi, and Vaquie about deformation quantization of derived stacks with shifted Poisson structure.

Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE.

A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials.

The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.

The original transport problem is to optimally move a pile of soil to an excavation.

Mathematically, given two measures of equal mass, we look for an optimal bijection that takes

one measure to the other one and also minimizes a given cost functional. Kantorovich relaxed

this problem by considering a measure whose marginals agree with given two measures instead of

a bijection. This generalization linearizes the problem. Hence, allows for an easy existence

result and enables one to identify its convex dual.

In robust hedging problems, we are also given two measures. Namely, the initial and the final

distributions of a stock process. We then construct an optimal connection. In general, however,

the cost functional depends on the whole path of this connection and not simply on the final value.

Hence, one needs to consider processes instead of simply the maps S. The probability distribution

of this process has prescribed marginals at final and initial times. Thus, it is in direct analogy

with the Kantorovich measure. But, financial considerations restrict the process to be a martingale

Interestingly, the dual also has a financial interpretation as a robust hedging (super-replication)

problem.

In this talk, we prove an analogue of Kantorovich duality: the minimal super-replication cost in

the robust setting is given as the supremum of the expectations of the contingent claim over all

martingale measures with a given marginal at the maturity.

This is joint work with Yan Dolinsky of Hebrew University.

Please see http://www.cs.ox.ac.uk/seminars/CompBioPublicSeminars/

Please see http://www.cs.ox.ac.uk/seminars/CompBioPublicSeminars/

Mechanics plays an important role during several biological phenomena such as morphogenesis,

wound healing, bone remodeling and tumorogenesis. Each one of these events is triggered by specific

elementary cell deformations or movements that may involve single cells or populations of cells. In

order to better understand how cell behave and interact, especially during degenerative processes (i.e.

tumorogenesis and metastasis), it has become necessary to combine both numerical and experimental

approaches. Particularly, numerical models allow determining those parameters that are still very

difficult to experimentally measure such as strains and stresses.

During the last few years, I have developed new finite element models to simulate morphogenetic

movements in Drosophila embryo, limb morphogenesis, bone remodeling as well as single and

collective cell migration. The common feature of these models is the multiplicative decomposition of

the deformation gradient which has been used to take into account both the active and the passive

deformations undergone by the cells. I will show how this mechanical approach, firstly used in the

seventies by Lee and Mandel to describe large viscoelastic deformations, can actually be very

powerful in modeling the biological phenomena mentioned above.