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We introduce a moduli space of ``complete quasimaps'' to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$. The construction, following previous work for curves on projective spaces, essentially proceeds by blowing up Ciocan-Fontanine--Kim's space…
This will is an expository description of quadratic rational maps. Sections 2 through 6 are concerned with the geometry and topology of such maps. Sections 7--10 survey of some topics from the dynamics of quadratic rational maps. There are…
For given closed orientable 3-manifolds $M$ and $N$ let $\c{D}(M,N)$ be the set of mapping degrees from $M$ to $N$. We address the problem: For which $N$, $\c{D}(M,N)$ is finite for all $M$? The answer is known in Thurston's picture of…
We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers…
The degree of a map between orientable manifolds is a fundamental concept in topology that aids in understanding the structure and properties of the manifolds and the maps between them. Numerous studies have been conducted on the degree of…
In this paper I survey some recent results on finite determination, convergence, and approximation of formal mappings between real submanifolds in complex spaces. A number of conjectures are also given.
We prove some asymptotic results for the radius and the profile of large random rooted planar maps with faces of arbitrary degrees. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted planar maps and certain four-type…
Let X, Y be the universal covers of two compact Riemannian manifolds (with dimension not equal to 4) with negative sectional curvature. Then every quasiisometry between them lies at a finite distance from a bilipschitz homeomorphism.
In their 2008 paper Gau and Wu conjectured that the numerical range of a 4-by-4 nilpotent matrix has at most two flat portions on its boundary. We prove this conjecture, establishing along the way some additional facts of independent…
We generalize recent construction of four-dimensional $\mathcal{N}=1$ SCFT from wrapping six-dimensional $\mathcal{N}=(2,0)$ theory on a Riemann surface to the case of $D$-type with outer-automorphism twists. This construction allows us to…
Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence…
We discuss non-relativistic variants of four-dimensional ${\cal N}$=4 super-Yang-Mills theory obtained from generalised Newton-Cartan geometric limits of D3-branes in ten-dimensional spacetime. We argue that the natural interpretation of…
A quasiplane $f(V)$ is the image of an $n$-dimensional Euclidean subspace $V$ of ${\Bbb R}^N$ ($1\leq n\leq N-1$) under a quasiconformal map $f:{\Bbb R}^N\to{\Bbb R}^N$ . We give sufficient conditions in terms of the weak quasisymmetry…
Quasiprobability representations are well-established tools in quantum information science, with applications ranging from the classical simulability of quantum computation to quantum process tomography, quantum error correction, and…
In this paper, we use the language of noncommutative differential geometry to formalise discrete differential calculus. We begin with a brief review of inverse limit of posets as an approximation of topological spaces. We then show how to…
We consider the problem of classifying those graphs that arise as an undirected square of an oriented graph by generalising the notion of quasi-transitive directed graphs to mixed graphs. We fully classify those graphs of maximum degree…
We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed.…
We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose…
Four-dimensional N = 2 superconformal quantum field theories contain a subsector carrying the structure of a chiral algebra. Using localization techniques, we show for the free hypermultiplet that this structure can be accessed directly…
We study the space of link maps, which are smooth maps from the disjoint union of manifolds P and Q to a manifold N such that the images of P and Q are disjoint. We give a range of dimensions, interpreted as the connectivity of a certain…