AnàlisiOn the behavior of harmonic measure density ratios at points of vanishing codimension-one density
Resum: In this talk, we study the behavior of the density ratios of harmonic measure at points of vanishing codimension-one density. In particular, we show that at $\omega$-almost every point with vanishing density, the density ratios exhibit arbitrarily large oscillations as the radius tends to $0$. As a consequence, the density ratio function fails to be almost monotone. This result holds for general open sets $\Omega\subset \mathbb{R}^{n+1}$.
GeometriaInvariantes de tipo finito y filtración de Johnson.
Resum: En esta charla presentaré resultados recientes obtenidos en colaboración con Ricard Riba (UdG) sobre la interacción entre la teoría de invariantes finitos, la filtración de Johnson del mapping class group de una variedad y la filtración inducida sobre el conjunto de las esferas de homología entera. En particular demostraré que la esfera de Poincaré no puede ser construida, mediante escición de Heegaard, por un elemento arbitrariamente lejos en la filtración de Johnson.
Geometria TropicalHyper-Operations and the Field with One Element
Resum: We will explore the algebraic structures over the "field with one element" in the sense of Connes and Consani, where ordinary operations become hyper-operations with multiple possible outcomes. We then discuss its geometric implications and formulate a base change mechanism from F1 to Z.
Sistemes DinàmicsHopf-like bifurcation: hysteresis and time-delay
Resum: We investigate planar piecewise-analytic vector fields, focusing on the merged focus and other monodromic tangential singularities when hysteresis or time-delay is incorporated into the switching condition. If the singularity is an asymptotically stable solution of the system with an instantaneous switch, then the introduction of hysteresis or time-delay causes an attracting limit cycle to be formed locally. We derive asymptotic expressions for the size and period of the limit cycle, allowing any degrees of tangency and any order for the smallest non-zero Lyapunov coefficient. We find that the growth rate of the limit cycle differs for hysteresis and time-delay, and differs to that of the related pseudo-Hopf bifurcation. Joint work with Douglas Novaes and David Simpson.
The geography of integral curves in plane quadratic systems
Resum: We picture the space of all plane quadratic systems and all cofactors as an ocean and those that admit an algebraic integral curve as dry land. To get some idea about the geography of this land we use 3 approaches: (1) A random search over a finite field. This gives an heuristic overview. (2) Constructing systems in char 0 from singular curves using classification results from algebraic geometry and singularity theory and the solution of the inverse problem. This confirms the large features of the heuristic map. (3) For the smallest isolated islands of the map (those where neither the system nor the cofactor nor the curve nor a combination of them can be deformed non-trivially) we sample all systems of a particular normalization over a finite field and lift the solutions to normalized systems with rational coefficients using Hensel lifting (if possible). This yields several 100 systems over $Q$ each of which has a completely different geometry. Among them we rediscover Colin Christopher's famous system with a degree 12 curve. This is a work in progress, in particular we have run the experiments only up to degree 12 (characteristic 13) and even the data of these runs has not been completely analysed in all details.
From Generalized Controlled Nonlinear Oscillators to Lorenz-like Systems
Resum: Damped and driven oscillators are generally modeled with a nonautonomous second-order nonlinear ordinary differential equation including a sinusoidal driving forcing term, such as the forced Duffing equation and the forced Holmes-Rand equation. These equations have been extensively studied during the last century and the last two decades. In the early 1990s, Abarbanel, Rabinovich and Sushchik proposed replacing the sinusoidal forcing term with a "force controlled by the movements of the oscillator itself", i.e. by the product of two variables: the first being the solution of the oscillator itself, while the second is the solution of a first-order nonlinear ordinary differential equation. They referred to the resulting autonomous dynamical system of two coupled nonlinear ordinary differential equations as a "controlled nonlinear oscillator". To that end, they introduced a change of variables and parameters to transform the "controlled nonlinear oscillator" that corresponds to a particular case of the forced Duffing equation into the Lorenz system. The aim of this work is to show that their idea can be further generalized and applied to many other dynamical systems, including the forced Holmes-Rand equation, Chua's cubic circuit, Chen's system and the forced Helmholtz oscillator. It is proved that a certain class of three-dimensional dynamical systems can be rewritten into the form of "generalized controlled nonlinear oscillators", which can then be transformed into various Lorenz-like systems. Such a transformation could be very useful for the study of intermittent chaos.
Teoria d'AnellsMultiplicative Ideal Factorization Theory of Hereditary Noetherian Prime Rings
Resum: Hereditary Noetherian prime (HNP) rings are a noncommutative generalization of Dedekind domains. In this talk, we will present a one-sided multiplicative ideal theory for HNP rings: To a one-sided ideal we assign a divisor, i.e., an integer combination of simple modules, and describe how multiplication of ideals affects the corresponding divisor.
TopologiaSigns in objective linear algebra
Resum: Standard objective linear algebra works with slice categories instead of vector spaces and with colimit-preserving functors instead of linear maps. (Such functors are represented by spans, so that matrix multiplication becomes pullback composition of spans.) One serious limitation has been the absence of negatives. In this talk, I will explain how this can be overcome, outlining an objective theory of signs in linear algebra. It turns out one can maintain a nice topos flavour by not having the signs directly on the objects but rather on `states' (for a monoidal structure which is not the cartesian product). By using groupoid coefficients instead of set coefficients, the signs can be encoded as little homotopies. I will illustrate some of the features of the theory with an objective treatment of exterior powers and determinants.
Multiplicative aspects of global algebraic K-theory
Resum: Global algebraic K-theory is a global equivariant refinement, due to Schwede, of the algebraic K-theory for symmetric monoidal categories developed by Segal and other authors in the 1970s. Schwede's construction takes as input a parsummable category and produces a symmetric spectrum, which represents a global equivariant (with respect to all finite groups) stable homotopy type. The equivariant homotopy groups of this spectrum carry very precise information about the source parsummable category. In the talk, I will explain this construction with a focus on multiplicative enhancements.
Pre-monoidal $\infty$-operads
Resum: Pre-monoidal $\infty$-operads were introduced by Karlsson-Scheimbauer-Walde (under the name pre-coCartesian operads) to describe a class of $\infty$-operads that appears in the study of constructible factorisation algebras. A motivating example is the following: given an object x of a symmetric monoidal category C, one cannot expect the slice category C/x to be symmetric monoidal. However, if the unit is initial, there is always a natural pre-monoidal operad structure on C/x.
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