arXiv research

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

Noise causes learning plateaus in neural networks.

We compare existence and equivariance phenomena for weak moment maps and homotopy moment maps in multisymplectic geometry.

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

The paper defines strong emergence in field theories and proves it exists between certain theories.

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…

Complexity science offers new insights into macroeconomics and finance.

In this paper we show how the existence of a certain stable cylinder determines (locally) the ambient manifold where it is immersed. This cylinder has to verify a {\it bifurcation phenomena}, we make this explicit in the introduction. In particular, the existence of such a stable cylinder implies that the ambient manif…

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

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…

We survey the known existence and non-existence results for $G_2$-instantons on non-compact cohomogeneity-1 $G_2$-manifolds and their consequences, including an explicit example of a family of $G_2$-instantons where bubbling, removable singularities and conservation of energy phenomena occur. We also describe several o…

Study on limits and cut-off phenomena in deep neural networks.

New bounds for KRR condition number reveal overfitting phenomena.

In this paper we study the Palais-Smale sequences of the conformal Dirac-Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the existence of infinitely many solutions to the problem provided that the underlying mani…

NNs accurately predict energy eigenvalues and other physical phenomena in 1D quantum mechanics.

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…

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

Differentiable NAS frameworks grow networks wider and deeper, revealing biases in wiring evolution.

Framework for interpreting ML models to reveal properties of real-world 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…

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

We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert-Einstein functional on such conformal classes, near homogeneous metrics. Both bifurcation and local…

Paper studies rigidity phenomena for elliptic systems on Riemannian manifolds.

The paper finds multiple ways a special curvature can blow up in high dimensions.

We establish the existence of new rigidity and rationality phenomena in the theory of nonabelian group actions on the circle, and introduce tools to translate questions about the existence of actions with prescribed dynamics into finite combinatorics. A special case of our theory gives a very short new proof of Naimi's…

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.

Survey of rigidity and gap phenomena in sphere-ball submanifolds.

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…

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 …

In this paper we consider the functional whose critical points are solutions of the fractional CR Yamabe type equation on the sphere. We firstly study the behavior of the Palais-Smale sequences characterizing the bubbling phenomena and therefore we prove a multiplicity type result by showing the existence of infinitely…

A new sampler tackles critical phenomena by leveraging scale invariance.

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

We initiate the systematic study of $G_2$-instantons with $SU(2)^2$-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on $\mathbb{R}^4\times S^3$ with its two explicitly known distinct holonomy $G_2$ metrics, whic…

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 introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geome…

Analytic K-semistability connects curvature to metric existence.

The fractional Yamabe problem, proposed by González-Qing (2013, Anal. PDE) is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the classical Yamabe problem and the boundary Yamabe problem to the realm of nonlocal co…

We describe the induced geometry on several classes of Kodaira moduli spaces of rational curves in twistor spaces. By constructing connections and frames on the moduli spaces we build and review twistor theories pertaining to relativistic and non-relativistic geometries. Focussing on the cases of three- and five-dimens…

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.

The systolic ratio of a contact form on a closed three-manifold is the quotient of the square of the shortest period of closed Reeb orbits by the contact volume. We show that every co-orientable contact structure on any closed three-manifold is defined by a contact form with arbitrarily large systolic ratio. This shows…