Date | Speaker | Title |
---|---|---|
13/1 | Wael Bahsoun (UVic) | Chaotic dynamical systems: a probabilistic approach I |
20/1 | Wael Bahsoun (UVic) | Chaotic dynamical systems: a probabilistic approach II |
27/1 | Chris Bose (UVic) | Approximation of Invariant Measures in Ergodic Theory I |
3/2 | Chris Bose (UVic) | Approximation of Invariant Measures in Ergodic Theory II |
10/2 | Matthijs Vos (UVic Biology) | Systems of many differential equations are usually unstable Systems of many ecological species are usually stable How to resolve the paradox of diversity-stability relations in ecology |
17/2 | Robert Moody (UVic) | Dynamics in the Theory of Diffraction in Systems with Long-range Aperiodic Order |
24/2 | No seminar | No seminar |
Tuesday 28/2 | Izabella Laba (UBC)  | [Note: Different Time/Place: DSB C116 on Tuesday] Distance sets: Combinatorics and Fourier Analysis |
10/3 | Ciprian Demeter (UCLA)   | Breaking the duality in the Return Times Theorem |
17/3 | Ian Putnam (UVic) | Unique ergodicity for interval exchange transformations |
Let (X,B,m) be a measure space and T:X -> X be a measurable nonsingular transformation.
1. Does T admit an invariant probability measure which is absolutely continuous (acim for short) with respect to the ambient measure?
2. If 1. is satisfied, does the acim persist under small perturbation of T?
3. If 1. is satisfied, can we find the density of the acim?
4. If we can not find the density of the acim, can we approximate it by a computable density and find a bound on the approximation error?
To answer these questions we study the Perron-Frobenius operator associated with T.
In the first lecture, I will introduce the Perron-Frobenius operator and state its properties. Moreover, I will give specific examples where 1. and 3. are satisfied.
In the second lecture, I will state (and prove if time permits) stability of the isolated eigenvalues of transfer operators (including Perron-Frobenius) satisfying a certain inequality. This will give answers to questions 2 and 4.
One way to try to get around the problem is to derive a closed form expression for an APPROXIMATION to the invariant measure in question. Such approximations would then be used to compute ESTIMATES on the asymptotics.
Lecture 1:
Many methods for measure approximation have been suggested over the years -- I will review some of these ideas and attempt to organize them into a cohesive framework involving either
1. long orbit calculations
2. Finite-rank projection methods
3. Convex optimization methods.
Lecture 2:
I will present in some detail a case study from the convex optimization approach: maximal entropy approximations, and show how rigourous convergence results can be obtained in quite general settings.
Simple theory thus predicts that complex communities cannot exist or persist for long. However, they clearly do so in nature.
In this talk I discuss the characteristics that differentiate ecological communities from randomly assembled mathematical communities. I show how introducing ecological realism into the equations can reverse the sign of diversity-stability relations from negative to positive.
This is joint work with Michael Lacey, Terence Tao and Christoph Thiele.