In the third part, we extend this method of equivariant degree to the realm of coupled cell networks, which are (generally non-symmetric) networks of dynamical systems, where the time evolution of one system is influenced by others. Under the general framework of coupled cell networks, we apply the newly defined ``lattice equivariant degree'' to studying synchrony-breaking bifurcations, a local bifurcation through which a fully synchronous equilibrium loses its stability and bifurcates to states of less synchrony in space.

In the second part of this course, we give a standard degree-approach to studying equivariant bifurcations in equivariant dynamical systems, as well as systems with constant delay. By means of equivariant degree we learned from the first part, results of existence, multiplicity and symmetric properties can be obtained and classified for bifurcating branches of solutions. Several models will be looked at in population dynamics.

A topological degree, in its simplest form, may be thought as a generalization of the {\it winding number} of a continuous circle map, which counts how many times the image of the map has traveled counterclockwise around the origin. This count remains unchanged if the map is perturbed slightly. Also, the addition of winding numbers corresponds to the conjunction of maps, and the negation of winding numbers can be realized by rewinding the direction of maps. The topological degree is thus usually referred as ``an algebraic count of the zeros of a continuous map''.
Equivariant degree theory is a topological degree theory that is concerned with {\it equivariant maps}, that is, maps that commute with the actions of a group on their space of domain and image. A main objective of the equivariant degree theory is to attain the topological structure of the zeros of an equivariant map and their algebraic properties induced by the equivariance.
In the first part of this course, we give a definition of equivariant degree, using a list of its most important properties such as existence and homotopy invariance. We also give examples of computations and among others, computational formulas for basic maps, which will be used frequently in application.

I will give an account of old and new results about quantitative isoperimetry. The issue of quantifying the stability in the isoperimetric inequality has attracted the interest of mathematicians since the beginning of the last century. Even though several problems still remain unsolved, there have been some remarkable achievements in the very last years, obtained via symmetrization techniques and optimal transport.
Here I will present a new, variational method called "Selection Principle", which can be successfully applied to the analysis of quantitative forms of the isoperimetric inequality. The method is based on a penalization argument combined with the regularity theory for quasiminimizers of the perimeter. Some applications of the Selection Principle will be presented: first, a new proof of the sharp quantitative isoperimetric inequality in R^n, with an improvement on the estimate of the optimal asymptotic constant appearing in the inequality; second, the proof of a conjecture due to Hall (1992) about the precise value of the above-mentioned constant in dimension 2. Finally, I will briefly discuss other applications of the method (in particular, the quantitative stability for the double bubble) as well as some open problems.

I will give an account of old and new results about quantitative isoperimetry. The issue of quantifying the stability in the isoperimetric inequality has attracted the interest of mathematicians since the beginning of the last century. Even though several problems still remain unsolved, there have been some remarkable achievements in the very last years, obtained via symmetrization techniques and optimal transport.
Here I will present a new, variational method called "Selection Principle", which can be successfully applied to the analysis of quantitative forms of the isoperimetric inequality. The method is based on a penalization argument combined with the regularity theory for quasiminimizers of the perimeter. Some applications of the Selection Principle will be presented: first, a new proof of the sharp quantitative isoperimetric inequality in R^n, with an improvement on the estimate of the optimal asymptotic constant appearing in the inequality; second, the proof of a conjecture due to Hall (1992) about the precise value of the above-mentioned constant in dimension 2. Finally, I will briefly discuss other applications of the method (in particular, the quantitative stability for the double bubble) as well as some open problems.

I will give an account of old and new results about quantitative isoperimetry. The issue of quantifying the stability in the isoperimetric inequality has attracted the interest of mathematicians since the beginning of the last century. Even though several problems still remain unsolved, there have been some remarkable achievements in the very last years, obtained via symmetrization techniques and optimal transport.
Here I will present a new, variational method called "Selection Principle", which can be successfully applied to the analysis of quantitative forms of the isoperimetric inequality. The method is based on a penalization argument combined with the regularity theory for quasiminimizers of the perimeter. Some applications of the Selection Principle will be presented: first, a new proof of the sharp quantitative isoperimetric inequality in R^n, with an improvement on the estimate of the optimal asymptotic constant appearing in the inequality; second, the proof of a conjecture due to Hall (1992) about the precise value of the above-mentioned constant in dimension 2. Finally, I will briefly discuss other applications of the method (in particular, the quantitative stability for the double bubble) as well as some open problems.

We intend in the first two lectures to review the main introductory topics on stochastic analysis: martingales, Markov process, Brownian motion, Itô formula, stochastic differential equation and stochastic flow. In the last two lectures we show applications of stochastic calculus in geometry and dynamical systems, including stochastic processes in Lie groups, isometric decomposition of stochastic flows in Riemannian manifolds, and others.

We intend in the first two lectures to review the main introductory topics on stochastic analysis: martingales, Markov process, Brownian motion, Itô formula, stochastic differential equation and stochastic flow. In the last two lectures we show applications of stochastic calculus in geometry and dynamical systems, including stochastic processes in Lie groups, isometric decomposition of stochastic flows in Riemannian manifolds, and others.

We intend in the first two lectures to review the main introductory topics on stochastic analysis: martingales, Markov process, Brownian motion, Itô formula, stochastic differential equation and stochastic flow. In the last two lectures we show applications of stochastic calculus in geometry and dynamical systems, including stochastic processes in Lie groups, isometric decomposition of stochastic flows in Riemannian manifolds, and others.

We intend in the first two lectures to review the main introductory topics on stochastic analysis: martingales, Markov process, Brownian motion, Itô formula, stochastic differential equation and stochastic flow. In the last two lectures we show applications of stochastic calculus in geometry and dynamical systems, including stochastic processes in Lie groups, isometric decomposition of stochastic flows in Riemannian manifolds, and others.