We describe an extension of Pitman's theorem on Brownian motion and the three dimensional Bessel process to several dimensions. We show how this extension is suggested by considering a random walk on a noncommutative space, and is connected with crystals and Littelmann paths.

Einstein's field equations in general relativity admit a variety of solutions with spacetime singularities. Numerical relativity has recently revealed the properties of somewhat generic spacetime singularities. It has been found that in a variety of systems self-similar solutions can describe asymptotic or intermediate behaviour of more general solutions in an approach to singularities. The typical example is the convergence to an attractor self-similar solution in gravitational collapse. This is closely related to the cosmic censorship violation in the spherically symmetric collapse of a perfect fluid. The self-similar solution also plays an important role in critical phenomena in gravitational collapse. The critical phenomena are understood as the intermediate behaviour around a critical self-similar solution. We see that the convergence and critical phenomena are understood in a unified manner in terms of attractors of codimension zero and one, respectively, in renormalisation group flow.

This is a survey article on the recent results of the “horospherical geometry” in the hyperbolic space. Detailed arguments for the results have been appeared or will appear in several different articles.

We compute topological one-point functions of the chiral operator $\mathrm{Tr}\ \varphi^k$ in the maximally confining phase of $U(N)$ supersymmetric gauge theory. These chiral one-point functions are of particular interest from gauge/string theory correspondence, since they are related to the equivariant Gromov–Witten theory of $\mathbf{P}^1$. By considering the power sums of Jucys–Murphy elements in the class algebra of the symmetric group we can derive a combinatorial identity that leads the relations among chiral one-point functions. Using the operator formalism of free fermions, we also compute the vacuum expectation value of the loop operator $\langle \mathrm{Tr}\ e^{it\varphi}\rangle$ which gives the generating function of the one-point functions.

We present an operator algebraic approach to superconformal field theory and a classification result in this framework. This is based on a joint work with S. Carpi and R. Longo

Each elliptic curve can be embedded uniquely in the projective plane, up to projective equivalence. The hessian curve of the embedding is generically a new elliptic curve, whose isomorphism type depends only on that of the initial elliptic curve. One gets like this a rational map from the moduli space of elliptic curves to itself. We call it the hessian dynamical system. We compute it in terms of the $j$-invariant of elliptic curves. We deduce that, seen as a map from a projective line to itself, it has 3 critical values, which correspond to the point at infinity of the moduli space and to the two elliptic curves with special symmetries. Moreover, it sends the set of critical values into itself, which shows that all its iterates have the same set of critical values. One gets like this a sequence of dessins d'enfants. We describe an algorithm allowing to construct this sequence.

The goal of this text is to motivate for an effective version of the quantum Birkhoff normal form, which gives precise spectral asymptotics even in the presence of resonances, in the semiclassical limit. Microlocal analysis, via pseudo-differential operators and Toeplitz operators is used in order to achieve this.

We will discuss a relation between a special value of Ruelle–Selberg L-function of a unitary local system on a hyperbolic threefold of finite volume and Alexander invariant. A philosophy of our results are based on Iwasawa Main Conjecture in number theory.

The purpose of this note is to describe recent development on the study of the local structure of singular spaces with curvature bounded above, due to Lychak and Nagano[16], [17] for the characterization of topological manifolds, and due to Kleiner, Nagano, Shioya and Yamaguchi [15] for the general characterization of 2-dimensional spaces.

This paper provides a survey of some recent results (mostly in [68, 71, 72, 73, 74]) concerning high-order representation and processing of singular data. We present these results from a certain general point of view which we call a “Model-Net” approach: this is a method of representation and processing of various types of mathematical data, based on the explicit recovery of the hierarchy of data singularities. As an example we use a description of singularities and normal forms of level surfaces of “product functions” recently obtained in [68, 34] and on this base describe in detail the structure of the Model-net representation of such surfaces.

Then we discuss a “Taylor-net” representation of smooth functions consisting of a net of Taylor polynomials of a prescribed degree $k$ (or $k$-jets) of this function stored at a certain grid in its domain. Following [72, 74] we present results on the stability of Hermite fitting, which is the main tool in acquisition of Taylor-net data.

Next we present (following [71, 73, 74]) a method for numerical solving PDE's based on Taylor-net representation of the unknown function. We extend this method also to the case of the Burgers equation near a formed shock-wave.

Finally, we shortly discuss (following [28, 56]) the problem of a non-linear acquisition of Model-nets from measurements, as well as some additional implementations of the Model-net approach.

We study an analogue of Serre fibrations in the setting of $C^*$-algebra bundles. We derive in this framework a Leray–Serre type spectral sequence. We investigate a class of examples which generalise on one hand principal bundles with a $n$-torus as structural group and on the other hand non-commutative tori.

We briefly review properties of the $\mathcal{N} = 2$ $U(N)$ gauge model composed of $\mathcal{N} = 1$ superfields. This model can be regarded as a low-energy effective action of $\mathcal{N} = 2$ Yang–Mills theory equipped with electric and magnetic Fayet–Iliopoulos terms. In this model, the $\mathcal{N} = 2$ supersymmetry is spontaneously broken to $\mathcal{N}= 1$, and the Nambu–Goldstone fermion comes from the overall $U(1)$ part of $U(N)$ gauge group. We also give $\mathcal{N} = 1$ supermultiplets appearing in the vacua. In addition, we give a manifestly $\mathcal{N} = 2$ symmetric formulation of the model by employing the unconstrained $\mathcal{N} = 2$ superfields in harmonic superspace. Finally, we study a decoupling limit of the Nambu–Goldstone fermion and identify the origin of the fermionic shift symmetry with the second, spontaneously broken supersymmetry.

Based on projective representations of smooth Deligne cohomology groups, we introduce an analogue of the space of conformal blocks to compact oriented $(4k + 2)$-dimensional Riemannian manifolds with boundary. For the standard $(4k + 2)$-dimensional disk, we compute the space concretely to prove that its dimension is finite.

We construct special solutions of the quantum Knizhnik–Zamolodchikov equation on the tensor product of the vector representation of the quantum algebra of type $A_{N-1}$. They are constructed from non-symmetric Macdonald polynomials through the action of the affine Hecke algebra.

We compute local Gromov–Witten invariants of cubic surfaces via nef toric degeneration.

We enumerate all the $\theta$-curves and handcuff graphs with up to seven crossings by using algebraic tangles and prime basic $\theta$-polyhedra. Here, a $\theta$-polyhedron is a connected graph embedded in a 2-sphere, whose two vertices are 3-valent, and the rest are 4-valent. There exist twenty-four prime basic $\theta$-polyhedra with up to seven 4-valent vertices. We can obtain a $\theta$-curve and handcuff graph diagram from a prime basic $\theta$-polyhedron by substituting algebraic tangles for their 4-valent vertices.

The sixth Painlevé equation has the affine Weyl group symmetry of type $D_{4}^{(1)}$ as a group of Bäcklund transformations and is written as a Hamiltonian system. We propose a quantization of the sixth Painlevé equation with the extended affine Weyl group symmetry of type $D_{4}^{(1)}$.

It is known that holomorphic sections of an ample line bundle $L$ (and its tensor power $L^k$) on an Abelian variety $A$ are given by theta functions. Moreover, a natural basis of the space of holomorphic sections is related to a certain Lagrangian fibration of $A$. We study projective embeddings of $A$ given by the basis for $L^k$, and show that moment maps of toric actions on the ambient projective spaces, restricted to $A$, approximate the Lagrangian fibration of $A$ for large $k$. The case of Kummer variety is also discussed.

We define a generalization of Schur's Q-function for an arbitrary pair of strict partitions, which is called the generalized Q-function. We prove that all the generalized Q-functions solve a series of non-linear differential equations called the UC hierarchy of B-type (BUC hierarchy). We furthermore investigate the BUC hierarchy from the viewpoint of representation theory. We consider the Fock representation of the algebra of neutral fermions and establish the boson-fermion correspondence. Using this, we discuss the relationship between the BUC hierarchy and a certain infinite dimensional Lie algebra.

I report on recent work of Sylvain Barré and myself on the space of triangle buildings.

From a set-theoretic point of view the space of triangle buildings is the family of all triangle buildings (also called Bruhat–Tits buildings of type $\tilde{A}_2$) considered up to isomorphism. This is a continuum. We shall see that it provides new tools and a general framework for studying triangle buildings, which connects notably to foliation and lamination theory, quasi-periodicity of metric spaces, and noncommutative geometry.

This text is a general presentation of the subject and explains some of these connections. Several open problems are mentioned. The last sections set up the basis for an approach via $K$-theory.

We prove that metric measure spaces with nonnegative Ricci curvature have at most two ends.

If a group acts geometrically (i.e., properly discontinuously, cocompactly and isometrically) on two geodesic spaces $X$ and $X'$, then an automorphism of the group induces a quasi-isometry $X \to X'$. We find a geometric action of a Coxeter group $W$ on a CAT(0) space $X$ and an automorphism $\phi$ of $W$ such that the quasi-isometry $X \to X$ arising from $\phi$ can not induce a homeomorphism on the boundary of $X$ as in the case of Gromov-hyperbolic spaces.

This is an expository article on manifolds which are locally modeled on the standard representation of a torus and their classifications.