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We give an informal survey, emphasizing examples and open problems, of two interconnected research programs in moduli of curves: the systematic classification of modular compactifications of $M_{g,n}$, and the study of Mori chamber…

We perform a general study of the structure of locally compact modules over compactly generated abelian groups. We obtain a devissage result for such modules of the form "compact-by-sheer-by-discrete", and then study more specifically the…

To any Hamiltonian action of a reductive algebraic group $G$ on a smooth irreducible symplectic variety $X$ we associate certain combinatorial invariants: Cartan space, Weyl group, weight and root lattices. For cotangent bundles our…

In this series of papers, we propose a theory of enumerative invariants counting self-dual objects in self-dual categories. Ordinary enumerative invariants in abelian categories can be seen as invariants for the structure group $\mathrm{GL}…

Let $G$ be a (non compact) connected simply connected locally compact second countable Lie group, either abelian or unimodular of type I, and $\rho$ an irreducible unitary representation of $G$. Then, we define the analytic torsion of $G$…

Given a finite-dimensional inner product space $V$ and a group $G$ of isometries, we consider the problem of embedding the orbit space $V/G$ into a Hilbert space in a way that preserves the quotient metric as well as possible. This inquiry…

We construct the moduli space of r-jets at a point of Riemannian metrics on a smooth manifold. The construction is closely related to the problem of classification of jet metrics via differential invariants. The moduli space is proved to be…

According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism G^ --> D^ of the dual groups is a topological isomorphism. We introduce four conditions on D…

The moduli space of twisted holomorphic 1-forms on Riemann surfaces, equivalently dilation surfaces with scaling, admits a stratification and GL(2,R)-action as in the case of moduli spaces of translation surfaces. We produce an analogue of…

Let $G$ be the identity component of the isometry group for an arbitrary curved two-point homogeneous space $M$. We consider algebras of $G$-invariant differential operators on bundles of unit spheres over $M$. The generators of this…

By a recent result of Juh\'{a}sz and van Mill, a locally compact topological group whose dense subspaces are all separable is metrizable. In this note we investigate the following question: is every locally compact group having all dense…

In this paper, we give a approximation characterization, embedding properties and the duality of matrix weighted modulation spaces.

We show that a bi-invariant metric on a compact connected Lie group $G$ is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric $g_0$ on $G$ there is a positive integer $N$ such…

In this paper we characterize spaces of $L^\infty$-functions on a compact Hausdorff space that are invariant under a transitive and continuous group action. This work generalizes the author's 2021 results concerning the specific case of…

We investigate the problem of the superuniversality of the phase transition between different quantum Hall plateaus. We construct a set of models which give a qualitative description of this transition in a pure system of interacting…

In two-dimensional conformal field theory, we analyze conformally invariant boundary conditions which break part of the bulk symmetries. When the subalgebra that is preserved by the boundary conditions is the fixed algebra under the action…

This survey deals with two closely connected topics: first, the stability of Einstein metrics under the Einstein-Hilbert functional, and second, their deformation theory and the study of the moduli space of Einstein metrics on a compact…

We describe a large-scale project in applied automated deduction concerned with the following problem of considerable interest in loop theory: If $Q$ is a loop with commuting inner mappings, does it follow that $Q$ modulo its center is a…

We study various compactifications of moduli space of Newton maps. Mainly, we focus on GIT compactifiaction and Deligne-Mumford compactification. Then we explore the relations among these compactifications.

We investigate the dependence of nonabelian T-duality on various identification of the group of target space isometries of nonlinear sigma models with its orbits, i.e. with respect to the location of the group unit on manifolds invariant…