2007 Abel prize has been awarded to S R S Varadhan for creating a unified theory of large deviations. We attempt to give a flavour of this branch of probability theory, highlighting the role of Varadhan.

For an abelian group 𝐺, the Davenport constant $D(G)$ is defined to be the smallest natural number 𝑘 such that any sequence of 𝑘 elements in 𝐺 has a non-empty subsequence whose sum is zero (the identity element). Motivated by some recent developments around the notion of Davenport constant with weights, we study them in some basic cases. We also define a new combinatorial invariant related to $(\mathbb{Z}/n\mathbb{Z})^d$, more in the spirit of some constants considered by Harborth and others and obtain its exact value in the case of $(\mathbb{Z}/n\mathbb{Z})^2$ where 𝑛 is an odd integer.

For the alternating group $A_6$ of degree 6, Zassenhaus’ conjecture about rational conjugacy of torsion units in integral group rings is confirmed.

Let 𝑅 be a complete semi-local ring with respect to the topology defined by its Jacobson radical, $\mathfrak{a}$ an ideal of 𝑅, and 𝑀 a finitely generated 𝑅-module. Let $D_R(-):=\mathrm{Hom}_R(-,E)$, where 𝐸 is the injective hull of the direct sum of all simple 𝑅-modules. If 𝑛 is a positive integer such that $\mathrm{Ext}^j_R(R/\mathfrak{a},D_R(H^t_{\mathfrak{a}}(M)))$ is finitely generated for all $t>n$ and all $j\geq 0$, then we show that $\mathrm{Hom}_R(R/\mathfrak{a},D_R(H^n_{\mathfrak{a}}(M)))$ is also finitely generated. Specially, the set of prime ideals in $\mathrm{Coass}_R(H^n_{\mathfrak{a}}(M))$ which contains $\mathfrak{a}$ is finite.

Next, assume that $(R,\mathfrak{m})$ is a complete local ring. We study the finiteness properties of $D_R(H^r_{\mathfrak{a}}(R))$ where 𝑟 is the least integer 𝑖 such that $H^i_{\mathfrak{a}}(R)$ is not Artinian.

We define the local fundamental group scheme and study its properties under base change of the base field.

Given two irreducible representations $\mu,\upsilon$ of the symmetric group $S_d$, the Kronecker problem is to find an explicit rule, giving the multiplicity of an irreducible representation, 𝜆, of $S_d$, in the tensor product of 𝜇 and 𝜐. We propose a geometric approach to investigate this problem. We demonstrate its effectiveness by obtaining explicit formulas for the tensor product multiplicities, when the irreducible representations are parameterized by partitions with at most two rows.

Let 𝑀 be a compact 3-manifold with a triangulation 𝜏. We give an inequality relating the Euler characteristic of a surface 𝐹 normally embedded in 𝑀 with the number of normal quadrilaterals in 𝐹. This gives a relation between a topological invariant of the surface and a quantity derived from its combinatorial description. Secondly, we obtain an inequality relating the number of normal triangles and normal quadrilaterals of 𝐹, that depends on the maximum number of tetrahedrons that share a vertex in 𝜏.

Let $\mathfrak{A}$ be a Banach algebra and let 𝑋 be a Banach $\mathfrak{A}$-bimodule. In studying $\mathcal{H}^1(\mathfrak{A},X)$ it is often useful to extend a given derivation $D:\mathfrak{A}\to X$ to a Banach algebra $\mathfrak{B}$ containing $\mathfrak{A}$ as an ideal, thereby exploiting (or establishing) hereditary properties. This is usually done using (bounded/unbounded) approximate identities to obtain the extension as a limit of operators $b\mapsto D(ba)-b.D(a), a\in\mathfrak{A}$ in an appropriate operator topology, the main point in the proof being to show that the limit map is in fact a derivation. In this paper we make clear which part of this approach is analytic and which algebraic by presenting an algebraic scheme that gives derivations in all situations at the cost of enlarging the module. We use our construction to give improvements and shorter proofs of some results from the literature and to give a necessary and sufficient condition that biprojectivity and biflatness is inherited to ideals.

A quasi-normed cone is a pair $(X, p)$ such that 𝑋 is a (not necessarily cancellative) cone and 𝑞 is a quasi-norm on 𝑋. The aim of this paper is to prove a closed graph and an open mapping type theorem for quasi-normed cones. This is done with the help of appropriate notions of completeness, continuity and openness that arise in a natural way from the setting of bitopological spaces.

We shall investigate the use of Abel transform on $PSL_2(\mathbb{R})$ as a tool beyond 𝐾-biinvariant setup, discuss its properties and show some applications.

First, we prove the decomposition theorem for the regularities of multifractal Hausdorff measure and packing measure in $\mathbb{R}^d$. This decomposition theorem enables us to split a set into regular and irregular parts, so that we can analyze each separately, and recombine them without affecting density properties. Next, we give some properties related to multifractal Hausdorff and packing densities. Finally, we extend the density theorem in [6] to any measurable set.

Every quantum Lévy process with a bounded stochastic generator is shown to arise as a strong limit of a family of suitably scaled quantum random walks.

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.

This paper investigates $2m-\mathrm{th}(m\geq 2)$ order singular 𝑝-Laplacian boundary value problems, and obtains the necessary and sufficient conditions for existence of positive solutions for sublinear $2m$-th order singular 𝑝-Laplacian BVPs on closed interval.

This paper is devoted to the study of boundary value problem of third-order functional differential equations. We obtain some existence results for the problem at resonance under the condition that the nonlinear terms is bounded or generally unbounded. In this paper we mainly use the topological degree theory.