arXiv research

The paper explores higher property T in lattices and its connections to geometric phenomena.

We present examples of foliations with infinite dimensional basic symplectic and com- plex cohomologies, along with a general sufficient condition for such phenomena. This puts re- strictions on possible generalizations of several finiteness results from Riemannian foliations to any broader class. The examples are also…

Study cohomological equation for robotic screw motions on SE(3).

This is an introduction to some of the analytic (or integrable systems) aspects of quantum cohomology which have attracted much attention during the last few years. The small quantum cohomology algebra, regarded as an example of a Frobenius manifold, is described in the original naive manner, without going into the tec…

Analyzes complex structure deformations using cohomology contraction methods.

Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.

We give explicit formulae for fringe lengths of the Calegari-Walker Ziggurats -- i.e. graphs of extremal rotation numbers associated to positive words in free groups. These formulae reveal (partial) integral projective self-similarity in ziggurat fringes, which are low-dimensional projections of characteristic polyhedr…

Floer theory constructs filtrations on quantum cohomology for symplectic manifolds.

We continue the study of a general class of spaces of 0-cycles on a manifold defined and begun by Farb-Wolfson-Wood. Using work of Gadish on linear subspace arrangements, we obtain representation stability for the cohomology of the ordered version of these spaces. We establish subexponential bounds on the growth of uns…

Study rational homology of moduli space via Morse functions, proving stability phenomena.

Analytic K-semistability connects curvature to metric existence.

Formula connects surgeries to Seiberg-Witten invariants.

Floer homotopy theory applies to Lagrangians, overcoming curvature issues.

String structures in degree four are associated with cancellation of anomalies of string theory in ten dimensions. Fivebrane structures in degree eight have recently been shown to be associated with cancellation of anomalies associated to the NS5-brane in string theory as well as the M5-brane in M-theory. We introduce …

The paper argues against the inefficiency of explaining deep learning phenomena.

We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical t…

In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative $1$-Lipschitz functions with Dirichlet boundary condition around zero, which is called boundary concentration phenomena. We first examine relation between boundary concen…

This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative aspects. The unifying framework for this study is the stratification of the character …

Sharp changes in time series representing market dynamics are studied by means of the self--similar analysis suggested earlier by the authors. These sharp changes are market booms and crashes. Such crises phenomena in markets are analogous to critical phenomena in physics. A simple classification of the market crisis p…

Study on Lane-Emden equation on curved spaces, revealing new existence and non-existence phenomena.

Paper connects Stokes phenomena to quantum groups and Poisson-Lie groups.

In this paper we observe that 2-dimensional 0-surgery occurs in natural processes, such as tornado formation and other phenomena reminiscent of hole drilling. Inspired by such phenomena, we introduce new theoretical concepts which enhance the formal definition of 2-dimensional 0-surgery with the observed dynamics. To d…

We give a survey of some known results and of the many open questions in the study of generic phenomena in geometrically interesting groups.

Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.

We consider the action of symplectic monodromy on chain-level enhancements of quantum cohomology. First, we construct a family version of $A_\infty$-structure on quantum cohomology (this should morally correspond to Hochschild cohomology of a "family of Fukaya categories over the circle"). Following Kaledin, we look at…

Analyzes magnetic Laplacian on hyperbolic surfaces, highlighting key quantum phenomena.

Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where a sphere of dimension 0 or 1 is selected, forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromo…

New framework for diffusion geometry simplifies complex calculations.

Study exotic phenomena in small 4-manifolds using diffeomorphisms.

The paper studies hyperbolic phenomena on closed surfaces using bicorn curves.

Knowing and modelling the migration phenomena and especially the social and economic consequences have a theoretical and practical importance, being related to their consequences for development, economic progress (or as appropriate, regression), environmental influences etc. One of the causes of migration, especially …

This paper extends Lusternik-Schnirelmann category to non-compact manifolds.

The paper explains emergent phenomena in deep learning using entropic forces.

Mathematical structures link Gromov-Witten to Donaldson-Thomas invariants.

We connect topological changes that can occur in $3$-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena, including relationships between quantum entanglement and wormhole formation. By considering the initial manifold as the $3$-dimensional spatial sectio…

In this paper we try to establish a connection between a three-dimensional Lotka--Volterra dynamical system and two-dimensional topological surgery. There are many physical phenomena exhibiting two-dimensional topological surgery through a `hole drilling' process. By our connection, such phenomena may be modelled mathe…

We investigate different concentration-compactness phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in $R^{2m}$, then that of a closed manifold and, finally, the particular case of the sphere $S^{2m}$. In all cases we allow the sign of the Q-curvature to vary, …

Study non-squeezing phenomena in contact geometry using specific capacities.

New neural networks model complex phenomena with fewer parameters.

This article demonstrates the possibility of constructing indicators of critical and crisis phenomena in the volatile market of cryptocurrency. For this purpose, the methods of the theory of complex systems such as recurrent analysis of dynamic systems and the calculation of permutation entropy are used. It is shown th…

We expect manifolds obtained by Dehn filling to inherit properties from the knot manifold. To what extent does that hold true for the Heegaard structure? We study four changes to the Heegaard structure that may occur after filling: (1) Heegaard genus decreases, (2) a new Heegaard surface is created, (3) a non-stabilize…

Form a pure mathematical point of view, common functional forms representing different physical phenomena can be defined. For example, rates of chemical reactions, diffusion and heat transfer are all governed by exponential-type expressions. If machine learning is used for physical problems, inferred from domain knowle…

Noise causes learning plateaus in neural networks.

The paper explores gaps in curvature-related metrics and rigidity.

We review the nature of some well-known phenomena such as volatility smiles, convexity adjustments and parallel derivative markets. We propose that the market is incomplete and postulate the existence of intrinsic risks in every contingent claim as a basis for understanding these phenomena. In a continuous time framewo…

Extends cohomology theory for infinite volume transformation groups.

Topological surgery occurs in natural phenomena where two points are selected and attracting or repelling forces are applied. The two points are connected via an invisible `thread'. In order to model topologically such phenomena we introduce dynamics in 1-, 2- and 3-dimensional topological surgery, by means of attracti…

In this short note we define a new cohomology for a Lie algebroid $\mathcal{A}$, that we call the \emph{twisted cohomology} of $\mathcal{A}$ by an odd cocycle $θ$ in the Lie algebroid cohomology of $\mathcal{A}$. We proof that this cohomology only depends on the Lie algebroid cohomology class $[θ]$ of the odd cocycle $…