Three Hopf algebras from number theory, physics and topology, and their common operadic, simplicial and categorical background
Imma IMMA GALVEZ-CARRILLO, Ralph M. KAUFMANN, Andy TONKS - 2018-06-18 (M/18/08)
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. The primary examples are the Hopf algebras of Goncharov for multiple zeta values, that of Connes--Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework.
On the Ramanujan conjecture for automorphic forms over function fields I. Geometry
Will SAWIN, Nicolas TEMPLIER - 2018-05-31 (M/18/07)

Global bifurcations of limit cycles in the Leslie-Gover model with the Allee effect
Valery A. GAIKO, Jean-Marc GINOUX, Cornelis VUIK - 2018-05-23 (M/18/06)
In this paper, we complete the global qualitative analysis of the Leslie-Gover system with the Allee effect which models the dynamics of the populations of predators and their prey in a given ecological or biomedical system. In particular, studying global bifurcations, we prove that such a system can have at most two limit cycles surrounding one singular point. We also conduct some numerical experiments to illustrate the obtained results.
Multi-parameter planar polynomial dynamical systems
Valery GAIKO, Cornelis VUIK - 2018-04-13 (M/18/05)
In this paper, we study multi-parameter planar dynamical systems and carry out the global bifurcation analysis of such systems. To control the global bifurcations of limit cycle in these systems, it is necessary to know the properties and combine the effects of all their field rotation parameters. It can be done by means of the development of our bifurcational geometric method based on the Wintner-Perko termination principle and application of canonical systems with field rotation parameters. Using this method, we solve, e.g., Hilbert's Sixteenth Problem on the maximum number of limit cycles and their distribution for the general Li\'{e}nard polynomial system and a Holling-type quartic dynamical system. We also conduct some numerical experiments to illustrate the obtained results.
Pre-Calabi-Yau algebras as noncommutative Poisson structures
Natalia IYUDU, Maxim KONTSEVICH - 2018-03-27 (M/18/04)
We show how the double Poisson algebra introduced in \cite{VdB} appear as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on $Ac part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structure. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson bracket on representation spaces $({\rm Rep}_n A)^{Gl_n}$ for any associative algebra $A$.
Introduction to Higher Cubical Operads. Second Part: The Functor of Fundamental Cubical Weak $\infty$-Groupoids for Spaces
Camell KACHOUR - 2018-01-16 (M/18/03)

Some Aspects of Dynamical Topology: Dynamical Compactness and Slovak Spaces
Sergiy KOLYADA - 2018-01-08 (M/18/02)
The area of dynamical systems where one investigates dynamical properties that can be described in topological terms is "Topological Dynamics". Investigating the topological properties of spaces and maps that can be described in dynamical terms is in a sense the opposite idea. This area is recently called as "Dynamical Topology". For (discrete) dynamical systems given by compact metric spaces and continuous (surjective) self-maps we (mostly) survey some results on two new notions: "Slovak Space" and "Dynamical Compactness". Slovak space is a dynamical analogue of the rigid space: a nontrivial compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Dynamical compactness is a new concept of chaotic dynamics. The omega-limit set of a point is a basic notion in theory of dynamical systems and means the collection of states which "attract" this point while going forward in time. It is always nonempty when the phase space is compact. By changing the time we introduced the notion of the omega-limit set of a point with respect to a Furstenberg family. A dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this omega-limit set is nonempty. A nice property of dynamical compactness: all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.
From Euler's play with infinite series to the anomalous magnetic moment
Ivan TODOROV - 2018-01-05 (P/18/01)