arXiv research

Survey of recent developments in racks and quandles.

Recent work connects Thompson's groups to knot theory.

Recent developments in Seiberg-Witten theory and relations with Complex Geometry.

Survey on minimizing a volume function for singularities.

Survey on singularity theory and its relation to minimal model program.

Survey of recent austere submanifold results.

Machine learning explores symmetries in field theory and algebra.

This article presents a survey of some recent results in the theory of spatial graphs. In particular, we highlight results related to intrinsic knotting and linking and results about symmetries of spatial graphs. In both cases we consider spatial graphs in $S^3$ as well as in other $3$-manifolds.

This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by Kálmán, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extend…

Recent years have seen noteworthy progress in the mathematical formulation of quantum field theory and perturbative string theory. We give a brief survey of these developments. It serves as an introduction to the more detailed collection "Mathematical Foundations of Quantum Field Theory and Perturbative String Theory".

In a pair of recent papers (one to appear and one forthcoming), the author develops a general version of small cancellation theory applicable in higher dimensions, and then applies this theory to the Burnside groups of sufficiently large exponent. The present article gives a brief introduction to the methods and techni…

We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory. We also describe related work and recent progress.

Survey recent constructions of cyclic cocycles for Lie groups.

Recent advances in discrete subgroups of SL2(C) influenced by Thurston's work.

Study on Hodge theory for almost complex manifolds.

New theory extends Hamilton-Jacobi for contact systems, ensuring integrability.

Survey on recent developments in isometric immersions using PDE techniques.

The asymptotic dimension theory was founded by Gromov in the early 90s. In this paper we give a survey of its recent history where we emphasize two of its features: an analogy with the dimension theory of compact metric spaces and applications to the theory of discrete groups.

We summarize the main results of our recent investigation of bundles of real Clifford modules and briefly touch on some applications to string theory and supergravity.

We survey some recent results concerning Stanilov-Tsankov-Videv theory, conformal Osserman geometry, and Walker geometry which relate algebraic properties of the curvature operator to the underlying geometry of the manifold.

Study one-dimensional topological theories with linear generating functions.

Survey on broken ray transforms and open problems.

Paper shows all groups are quasi-sofic.

Quantum cellular automata form a homology theory.

Explains the history and challenges of minimal surfaces.

In this paper, we study the properties of the colored HOMFLY polynomials via HOMFLY skein theory. We prove some limit behaviors and symmetries of the colored HOMFLY polynomial predicted in some physicists' recent works.

Machine learning knot invariants with physics applications.

Surveying recent developments in K-stability and metric geometry.

Unified view of spectral networks linking geometry and gauge theory.

Recent work extends Turaev's modular categories to non-semisimple settings.

Observable structures of a topological field theory of AKSZ type are analyzed. From a double (or multiple) complex structure of observable algebras, new topological invariants are constructed. Especially, Donaldson polynomial invariants and their generalizations are constructed from a topological field theory of AKSZ t…

Surveying stability of klt singularities with new solutions.

Survey on statistical learning theory for control, focusing on linear systems.

Unified approach to invariants in equivariant geometry.

Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.

Overparameterized models generalize well despite fitting noisy data.

In this paper we show how to describe the general theory of a linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the algebra of Clifford bivectors is isomorphic to the Lie algebra of Sl(2,C). In that way the pullback of the …

It is well-known that an n-dimensional Poincaré complex $X^n$, $n \ge 5$, has the homotopy type of a compact topological $n$-manifold if the total surgery obstruction $s(X^n)$ vanishes. The present paper discusses recent attempts to prove analogous result in dimension 4. We begin by reviewing the necessary algebraic an…

Extended equivariant BV formalism to manifolds with boundaries.

New theory connects geometry without relying on connections.

Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in K- and L-theories. An important consequence of these algebraic conjectures is the topological rigidity of compact aspherical manifolds. Our goal is to strip the basi…

Recent results on ergodic theory for Riemann surface laminations and foliations.

Abstract notes on robust statistical learning theory.

In recent years, we have established the iteration theory of the index for symplectic matrix paths and applied it to periodic solution problems of nonlinear Hamiltonian systems. This paper is a survey on these results.

Researchers find new $G_2$-conifolds in $M$-theory with potential field theory duals.

Homology theories for associative algebraic structures are well established and have been studied for a long time. More recently, homology theories for self-distributive algebraic structures motivated by knot theory, such as quandles and their relatives, have been developed and investigated. In this paper, we study ass…

The purpose of this contribution is to point out connections between recent ideas about gerbes and gerbal actions (as higher categorical extension of representation theory) and old discussion in quantum field theory on commutator anomalies, gauge group extensions, and 3-cocycles. The unifying concept is the classical o…

Algebraic homology and cohomology theories for quandles have been studied extensively in recent years. With a given quandle 2(3)-cocycle one can define a state-sum invariant for knotted curves(surfaces). In this paper we introduce another version of quandle (co)homology theory, say positive quandle (co)homology. Some p…