In this talk we discuss modern mathematical models arising from modeling of tumor-immune system interactions. They are described as systems of ordinary differential equations of generalized Lotka-Volterra type. The corresponding vector fields are rational functions of dimension 2 or 3 containing many free parameters. The elementary qualitative theory can be applied to investigate the solutions of these equations. In particular, we derive conditions for global stability of certain equilibrium points using the Lyapunov functions method.

I will discuss the physical background of compressible Euler equations, its mathematical difficulties, and some important research problems.

I will discuss a method, based on p-adic height pairings, for determining the integral solutions of certain hyperelliptic equations.

The method produces, for a hyperelliptic curve over the rational numbers whose rank equals its genus, an explicit function on the p-adic points on the curve that takes finitely many explicitly determined values on the integral points. The proof goes via p-adic Arakelov theory.

Time permitting I will explain how one can use the generated function to effectively find the integral points and suggest some potential applications.

Associated to a closed hyperbolic surface is its length spectrum, the set of the lengths of all of its closed geodesics. Two surfaces are said to be isospectral if they share the same length spectrum.

The talk will be about the following questions and how they relate:

How many questions do you need to ask a length spectrum to determine it?
How many different surfaces can be isospectral to a surface of a given genus?

The approach to these questions will include finding adapted coordinate sets for moduli spaces and exploring McShane type identities.

Thurston conjectured that quasi-Fuchsian manifolds are uniquely determined by the induced hyperbolic metrics on the boundary of their convex core and Mess extended this conjecture to the context of globally hyperbolic anti de-Sitter spacetimes. In this talk I will discuss a universal version of Thurston and Mess' conjectures: any quasisymmetric homeomorphism from the circle to itself is obtained on the convex hull of a quasicircle in the boundary at infinity of the 3-dimensional hyperbolic (resp. anti-de Sitter) space. We will also discuss a similar result for convex domains bounded by surfaces of constant curvature K in (-1,0) in the hyperbolic setting and of curvature K in (-∞,-1) in the anti de-Sitter setting with a quasicircle as their asymptotic boundary.

(This is joint work in progress with F. Bonsante, J. Danciger and J.-M. Schlenker.)

During pre-implantation development, the mammalian embryo forms the blastocyst, which will implant into the uterus. The architecture of the blastocyst is essential to the specification of the first mammalian lineages and to the implantation of the embryo. Consisting of an epithelium enveloping a fluid-filled cavity and the inner cell mass, the blastocyst is sculpted by a succession of morphogenetic events. These deformations result from the changes in the forces and mechanical properties of the tissue composing the embryo. Using microaspiration, live-imaging, genetics and theoretical modelling, we study the biophysical and cellular changes leading to the formation of the blastocyst. In particular, we uncover the crucial role of acto-myosin contractility, which generates periodic waves of contractions, compacts the embryo, controls the position of cells within the embryo and influences fate specification.

I will talk about the "cohesive module" introduced by Jonathan Block. Concepturally it is a complex of vector bundles with a superconnection. I will describe the geometric nature of this object. In particular, on complex manifolds, the cohesive module gives a DG enhancement of the bounded derived category of coherent sheaves. Nevertheless, it applies to many other places.

The famous Hironaka's theorem asserts that any integral algebraic variety X of characteristic zero can be modified to a smooth variety Xres by a sequence of blowings up. Later it was shown that one can make this compatible with smooth morphisms Y → X in the sense that Yres → Y is the pullback of Xres → X.
In a joint project with D. Abramovich and J. Wlodarczyk, we construct a new algorithm which is compatible with all log smooth morphisms (e.g. covers ramified along exceptional divisors). We expect that this algorithm will naturally extend to an algorithm of resolution of morphisms to log smooth ones. In particular, this should lead to functorial semistable reduction theorems.
In my talk I will tell about main ideas of the classical algorithm and will then discuss logarithmic and stack-theoretic modifications we had to make in the new algorithm.

I review a class of nonlocally modified gravity models which were proposed to explain the current phase of cosmic acceleration without dark energy. Among the topics considered are deriving causal and conserved field equations, adjusting the model to make it support a given expansion history, why these models do not require an elaborate screening mechanism to evade solar system tests, degrees of freedom and kinetic stability, and the negative verdict of structure formation. Although these simple models are not consistent with data on the growth of cosmic structures many of their features are likely to carry over to more complicated models which are in better agreement with the data. The talk is be based on 1401.0254.

It is generally stated that the determination of the light travel time up to the order G2 is sufficient for modelling space missions in project — such as LATOR, ASTROD, SAGAS, ODYSSEY or GAME — designed to reach an accuracy of 10-7 to 5 × 10-9 in measuring the post-Newtonian parameter γ. However, this statement has to be re-examined in light of our recent demonstration that a so-called `enhanced' term of order G3 in the time delay may become comparable to some of the terms of order G2 when the emitter and the receiver are almost on opposite sides of the central body — a configuration of crucial importance in experimental gravitation.
This talk yields an overview of the new methods enabling us to carry out the calculation of the light travel time as a function of the positions of the emitter and the receiver (time transfer function) at any order of approximation for a large class of static, spherically symmetric metrics generalizing the Schwarzschild solution. The time transfer function is explicitly determined up to order G3. The enhanced terms are inferred from this solution and their significance for some tests of general relativity in the solar system is analyzed.
To finish, the appearance of enhanced terms in the direction of light propagation is discussed.

I will explain a recent work by Hidetoshi Oyama on a reformulation of the mod p analogue of the Simpson correspondence by Ogus and Vologodsky in terms of crystals on certain sites. He follows a similar work of the speaker on the p-adic Simpson correspondence by Faltings.
In the new formulation, the correspondence between D-modules and Higgs modules and the comparison between de Rham cohomology and Higgs cohomology are both given by the direct and inverse image functors of a certain morphism of topos.

We constructed a family of Calabi-Yau varieties over the l-line P1 Z[1/2]
{0, 1, ¥}, which is a projective smooth model of the affine scheme

[ w2
= x1·s xn(1-x1)·s(1-xn)(1 – l x1·s xn), ]

such that the generalized hypergeometric series n+1Fn(1/2, ·s,1/2; 1, ·s, 1; l) appear in the middle cohomology as a period function. In this talk we recall the construction of the family and how to calculate various cohomologies (Betti, de Rham, etale, and crystalline), discuss torsion freeness, up to 2-torsions, of integral cohomologies, and prove the integral version of degeneration of the Hodge to de Rham spectral sequence.