Related papers: Mixed characteristic homological theorems in low d…
In the representation theory of real reductive Lie groups, many objects have finiteness properties. For example, the lengths of Verma modules and principal series representations are finite, and more precisely, they are bounded. In this…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially…
Attractor-repeller decompositions of isolated invariant sets give rise to so-called connecting homomorphisms. These homomorphisms reveal information on the existence and structure of connecting trajectories of the underlying dynamical…
We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the…
Topologies on algebraic and equational theories are used to define germ determined, near-point determined, and point determined rings of smooth functions, without requiring them to be finitely generated. It is proved, that any commutative…
Let $G$ be a unimodular locally compact group. We define a property of irreducible unitary $G$-representations $V$ which we call c-temperedness, and which for the trivial $V$ boils down to F{\o}lner's condition (equivalent to the trivial…
All components of complements of discriminant varieties of simple real function singularities are explicitly listed. New invariants of such components (for not necessarily simple singularities) are introduced. A combinatorial algorithm…
In this note we give a simple unifying proof of the undecidability of several diagrammatic properties of term rewriting systems that include: local confluence, strong confluence, diamond property, subcommutative property, and the existence…
In this work we attempt to generalize our result in [6] [7] for real rings (not just von Neumann regular real rings). In other words we attempt to characterize and construct real closure * of commutative unitary rings that are real. We also…
We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for…
Representing graphs by their homomorphism counts has led to the beautiful theory of homomorphism indistinguishability in recent years. Moreover, homomorphism counts have promising applications in database theory and machine learning, where…
We study the almost complete intersection ring $R$ defined by $n+1$ general quadrics in a polynomial ring in $n$ variables over a field $\sf{k}$ and a corresponding linked Gorenstein ring $A$. The overarching theme is that, while not Koszul…
Topological data analysis is a relatively new branch of machine learning that excels in studying high dimensional data, and is theoretically known to be robust against noise. Meanwhile, data objects with mixed numeric and categorical…
We prove new results concerning the topology and Hodge theory of singular varieties. A common theme is that concrete conditions on the complexity of the singularities, from a number of different perspectives, are closely related to the…
A theorem of Hunter ensures that the complete homogeneous symmetric polynomials of even degree are positive definite functions. A probabilistic interpretation of Hunter's theorem suggests a broad generalization: the construction of…
Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric…
We build free, bigraded bidifferential algebra models for the forms on a complex manifold, with respect to a strong notion of quasi-isomorphism and compatible with the conjugation symmetry. This answers a question of Sullivan. The resulting…
We study the algebraic $K$-theory of rings of the form $R[x]/x^e$. We do this via trace methods and filtrations on topological Hochschild homology and related theories by quasisyntomic sheaves. We produce computations for $R$ a perfectoid…
In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of $n$ definable sets belonging to some fixed definable family of sets in an o-minimal structure. This generalizes the…