Embryonic development is a self-organized process during which cells divide, interact, change fate according to a complex gene regulatory network and organize themselves in a three-dimensional (3D) space. Here, we model this complex dynamic phenomenon in the context of the acquisition of epiblast (Epi) and primitive endoderm (PrE) identities within the inner cell mass (ICM) of the preimplantation embryo in the mouse. The multiscale model describes cell division and biomechanical interactions between cells, as well as biochemical reactions inside each individual cell and in the extracellular matrix. We use the model to study the Epi and PrE lineage development and the appearance of a so-called salt-and-pepper pattern which the two lineages form.

We consider the simple random walk on two types of tilings of the hyperbolic plane. The first by 2π⁄q-angled regular polygons, and the second by the Voronoi tiling associated to a random discrete set of the hyperbolic plane, the Poisson point process. In the second case, we assume that there are on average λ points per unit area.

In both cases the random walk (almost surely) escapes to infinity with positive speed, and thus converges to a point on the circle. The distribution of this limit point is called the harmonic measure of the walk.

I will show that the Hausdorff dimension of the harmonic measure is strictly smaller than 1, for q sufficiently large in the Fuchsian case, and for λ sufficiently small in the Poisson case. In particular, the harmonic measure is singular with respect to the Lebesgue measure on the circle in these two cases.

The proof is based on a Furstenberg type formula for the speed together with an upper bound for the Hausdorff dimension by the ratio between the entropy and the speed of the walk.

This is joint work with P. Lessa and E. Paquette.

We will discuss the asymptotic behaviour of the critical two-point function for a long-range version of the n-component $|varphi|^4$ model and the weakly self-avoiding walk (WSAW) on the d-dimensional Euclidean lattice with d=1,2,3. The WSAW corresponds to the case n=0 via a supersymmetric integral representation. We choose the range of the interaction so that the upper-critical dimension of both models is $d+epsilon$. Our main result is that, for small $epsilon$ and small coupling strength, the critical two-point function exhibits mean-field decay, confirming a prediction of Fisher, Ma, and Nickel. The proof makes use of a renormalisation group method of Bauerschmidt, Brydges, and Slade, as well as a cluster expansion. This is joint work with Martin Lohmann and Gordon Slade.

Meromorphic solutions of soliton equations usually do not fit in the standard spectral transform scheme. We show, that the spectral theory for the corresponding linear problems should be formulated in terms of Pontrjagin spaces – pseudo-Hilbert spaces with a finite number of negative squares. This observation uses the following property: all eigenfucntions of these linear operators with special singularities are meromorphic for all values of spectral parameter.

 

We also discuss a two-dimensional analog of this property.

 

I will talk about loop infrared effects in de Sitter QFT. Namely about their types, physical meaning and origin and also about their resumation and physical consequences. The talk is based on arXiv:1701.07226.

In this talk I present the phase diagram of a U(N)^2 x O(D) invariant fermionic planar matrix quantum mechanics (equivalently tensor or complex SYK models) in the new large D limit dominated by melonic graphs. The Schwinger-Dyson equations can have two solutions describing either a "large" black hole phase a la SYK or a "small" black hole with trivial IR behavior. In the strongly coupled region of the mass-temperature plane, there is a line of first order phase transitions between the small and large black hole phases. This line terminates at a new critical point which can be studied numerically in detail. The critical exponents are non-mean-field and different on the two sides of the transition. If time allows, I will compare this to purely bosonic unstable and stable melonic models.

A variety of spin systems, including Heisenberg and XXZ models, can be represented in terms of random loops in d+1 dimensions. The (conjectured) phase transitions are expressed in terms of the occurrence of macroscopic loops. Moreover, the macroscopic loops are expected to obey a Poisson–Dirichlet distribution. For Zd such results are lacking. Here we present progress on mean-field models: on the complete graph and on regular tress. Based on ongoing work with Jürg Fröhlich and Daniel Ueltschi.

In this talk we will start by explaining a spectacular analogy between topological spaces and higher category theory : In [1] Michael Batanin built the globular weak higher groupoid fundamental for any topological space X, wished by Alexander Grothendieck. To do such construction he used the coendomorphism operad associated to the coglobular object of disks in Top, the category of topological spaces. He was able to built the weak higher groupoid fundamental functor thanks to the contractibility of this operad. In the other hand the author [2] built a coglobular object in the category of higher operads, such that algebras for the first object are weak infini-categories, algebras for the second object are weak infini-functors, etc. With such coglobular object we also get a coendomorphism operad, built itself with operads instead of topological spaces. We conjecture that this operad is contractible like the topological one. If this is true then the globular weak higher category of globular weak higher categories exist. Also we will explain how to build the free cubical contractible higher operad which algebras are cubical weak higher categories, and if we have time we will explain how from this we obtain the cubical weak higher groupoid fundamental functor for topological spaces.

[1] Michael Batanin, Monoidal globular categories as a natural environment for the theory of weak-n-categories, Advances in Mathematics, 1998.

 

[2] Camell Kachour, Steps toward the weak higher category of weak higher categories in the globular setting, Category and Algebraic Structures with Applications, 2015.

In the recent development of modular geometry on toric noncommutative manifolds (Connes-Moscovici 2014), metrics are parametrized by self-adjoint elements in the ambient C*-algebra, whose exponential are called Weyl factors. Local invariants, such as the Riemannian curvature, are encoded in the coefficients of certain heat kernel expansion. The new ingredient, purely due to noncommutativity, is the the inner automorphism generated by the Weyl factor, whose corresponding derivation can be viewed as a noncommutative differential.  From analytic point of view, curvature is designed to measure the commutators of covariant derivatives. In this talk, we will discuss some intriguing spectral functions which define the interplay between the inner automorphisms and the classical differentials.  I recently found that hypergeometric functions and its multivariable generalization are the building blocks. Geometric applications such as Gauss-Bonnet theorem lead to some functional relations/equations between them which are still begging for more conceptual understanding.

I will prove that the compactly supported cohomology of certain unitary or symplectic Shimura varieties at level Gamma_1(p^infty) vanishes above the middle degree. The key ingredients come from p-adic Hodge theory and studying the Bruhat decomposition on the Hodge-Tate flag variety. I will describe the steps in the proof using modular curves as a toy model. I will also mention an application to Galois representations for torsion classes in the cohomology of locally symmetric spaces for GL_n. This talk is based on joint work in preparation with D. Gulotta, C.Y. Hsu, C. Johansson, L. Mocz, E. Reineke, and S.C. Shih.

I will discuss a relation between conformal blocks, describing kinematics of a CFT, and integrable models of quantum-mechanical particles. I will show how the dependence of blocks on cross-ratios is encoded in equations of motion of a Calogero-Sutherland model and their dependence on conformal dimension and spin of the exchanged operator – in those of a relativistic Calogero-Sutherland model. Both are simultaneously controlled by an integrable connection generalizing 2d Knizhnik-Zamolodchikov equations. I will review how this connection, associated to representations of degenerate double affine Hecke algebra, comes from a q-deformed bispectrally symmetric setting.