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We introduce the notion of coupled embeddability, defined for maps on products of topological spaces. We use known results for nonsingular biskew and bilinear maps to generate simple examples and nonexamples of coupled embeddings. We study…
We study a topological method to calculate the first Milnor fiber cohomology of a defining polynomial of a reduced projective hyperplane arrangement $X$ of degree $d$. We can show the vanishing of a monodromy eigenspace of the first Milnor…
A cyclic quotient singularity of type $p^2/pq-1$ ($0<q<p, (p,q)=1$) has a smoothing whose Milnor fibre is a $\mathbb Q$HD, or rational homology disk (i.e., the Milnor number is $0$) ([9], 5.9.1). In the 1980's, we discovered additional…
For a smooth cubic fourfold Y, we study the moduli space M of semistable objects of Mukai vector $2\lambda_1+2\lambda_2$ in the Kuznetsov component of Y. We show that with a certain choice of stability conditions, M admits a symplectic…
We compute symplectic cohomology for Milnor fibres of certain compound Du Val singularities that admit small resolution by using homological mirror symmetry. Our computations suggest a new conjecture that the existence of a small resolution…
The analysis of manifold valued data using embedding based methods is linked to the problem of finding suitable embeddings. In this paper we are interested in embeddings of quotient manifolds $\mathrm{SO}(3)/\mathcal{S}$ of the rotation…
A systematic and comprehensive study of p-adic refinement equations and subdivision scheme associated with a finitely supported refinement mask are carried out in this paper. The Lq -convergence of the subdivision scheme is characterized in…
This article is a summary of the author's unpublished Ph.D thesis. Its purpose is to generalise a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the simple singularities of type $A_r$, $D_r$, $E_6$, $E_7$ and…
Fractional Sobolev spaces, also known as Besov or Slobodetzki spaces, arise in many areas of analysis, stochastic analysis in particular. We prove an embedding into certain q-variation spaces and discuss a few applications. First we show…
Mixed boundary conditions are introduced to finite element exterior calculus. We construct smoothed projections from Sobolev de Rham complexes onto finite element de Rham complexes which commute with the exterior derivative, preserve…
The Mellin transform of fibre integral is calculated for certain isolated singularities of quasihomogeneous complete intersections (especially the unimodal singualrities of the list by Giusti and Wall). We show the property of symmetry…
We introduce a theory of finite polynomial cohomology with coefficients in this paper. We prove several basic properties and introduce an Abel-Jacobi map with coefficients. As applications, we use such a cohomology theory to study…
The Borel-Weil-Bott theorem can be used to decompose the cohomology of twisted sheaves of holomorphic forms on the complex Grassmannian into irreducible representations of the general linear group. By analyzing this decomposition, we…
This article studies the symplectic cohomology of affine algebraic surfaces that admit a compactification by a normal crossings anticanonical divisor. Using a toroidal structure near the compactification divisor, we describe the complex…
Convenient mixed functions with strongly non-degenerate Newton boundaries have Milnor fibrations, as the isolatedness of the singularity follows from the convenience. In this paper, we consider the Milnor fibration for non-convenient mixed…
We consider two linear reductive algebraic groups $ G $ and $ G' $ over $ C $. Take a finite dimensional rational representation $ W $ of $ G \times G' $. Let $ Y = W // G := Spec C[W]^G $ and $ X = W // G' := \Spec C[W]^{G'} $ be the…
The rational homology of unordered configuration spaces of points on any surface was studied by Drummond-Cole and Knudsen. We compute the rational cohomology of configuration spaces on a closed orientable surface, keeping track of the mixed…
On a smooth algebraic curve X with genus greater than 1 we consider a flat principal bundle with a reductive structure group S and a vector bundle associated with it. To this set of information we put in correspondence a pro-algebraic group…
Let $\mathcal{L}$ be the noncrossing partition lattice associated to a finite Coxeter group $W$. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of $\mathcal{L}$. We define a…
We study generalisations to totally real fields of methods originating with Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction.…