Related papers: A counterexample to Thiagarajan's conjecture on re…
Precedence constraints are inequalities used to model time dependencies. In 1958, Gallai proved that a finite system of precedence constraints admits solutions if and only if the corresponding precedence graph does not contain…
This is the second in a series of papers in which we investigate ideal triangulations of the interiors of compact 3-manifolds with tori or Klein bottle boundaries. Such triangulations have been used with great effect, following the…
We introduce weaves, which are random sets of non-crossing c\`{a}dl\`{a}g paths that cover space-time $\overline{\mathbb{R}}\times\overline{\mathbb{R}}$. The Brownian web is one example of a weave, but a key feature of our work is that we…
For a random graph subject to a topological constraint, the microcanonical ensemble requires the constraint to be met by every realisation of the graph (`hard constraint'), while the canonical ensemble requires the constraint to be met only…
We study what happens when coherence fails. Categories with a tensor product and a natural associativity isomorphism that does not necessarily satisfy the pentagon coherence requirements (called associative categories) are considered.…
Causal nets (CNs) are Petri nets where causal dependencies are modelled via inhibitor arcs. They play the role of occurrence nets when representing the behaviour of a concurrent and distributed system, even when reversibility is considered.…
We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to…
We study categories for reversible computing, focussing on reversible forms of event structures. Event structures are a well-established model of true concurrency. There exist a number of forms of event structures, including prime event…
We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…
We introduce a new categorical and constructive foundation for analytic approximation based on a Contextual Choice Principle (CCP), which enforces locality and compatibility in the construction of mathematical objects. Central to our…
Using meromorphic differentials with real periods, we prove Arbarello's conjecture: any compact complex cycle of dimension $g-n$ in the moduli space $\M_g$ of smooth genus $g$ algebraic curves must intersect the locus of curves having a…
We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we…
In 2004, Kim and Vu conjectured that, when $d=\omega(\log n)$, the random $d$-regular graph $G_d(n)$ can be sandwiched with high probability between two random binomial graphs $G(n,p)$ with edge probabilities asymptotically equal to…
We establish the existence of the universal type structure in presence of conditioning events without any topological assumption, namely, a type structure that is terminal, belief-complete, and non-redundant, by performing a construction…
The long-standing topological Tverberg conjecture claimed, for any continuous map from the boundary of an $N(q,d):=(q-1)(d+1)$-simplex to $d$-dimensional Euclidian space, the existence of $q$ pairwise disjoint subfaces whose images have…
Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these…
The well-known Conley's theorem states that the complement of chain recurrent set equals the union of all connecting orbits of the flow $\phi$ on the compact metric space $X$, i.e. $X-\mathcal{CR}(\phi)=\bigcup [B(A)-A]$, where…
We give exact relations for certain types of the hierarchic fractal structures. In the blatant distinction from regular networks of the "small world" (SW) topology [1], regular fractal networks manifests the logarithmic dependence of the…
We introduce a new notion of "regularity structure" that provides an algebraic framework allowing to describe functions and / or distributions via a kind of "jet" or local Taylor expansion around each point. The main novel idea is to…
Ne\v{s}et\v{r}il and Ossona de Mendez recently proposed a new definition of graph convergence called structural convergence. The structural convergence framework is based on the probability of satisfaction of logical formulas from a fixed…