In this talk, as a continuation of my talk in the Members' Colloquium but with a specialized audience in mind, I will discuss in more detail some of the general geometric and dynamical structures underlying the theoretical aspects of the restricted 3-body problem, and outline new research directions.
We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, 𝒜t(G) of finite injective dimension, a homology theory π∗𝒜t taking values in 𝒜t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence
Ext𝒜t(G)∗ , ∗ (π∗𝒜t(X), π∗𝒜t(Y)) → [X,Y]∗G
for rational G-spectra X and Y.
This approach should be viewed as an analogue of the Cousin complex in algebraic geometry. It is expected that a similar method will apply to other tensor triangulated categories with finite-dimensional Noetherian Balmer spectra.
The honest answer to the question is that I actually do not know. I will therefore rather talk about several famous examples that are widely called 'h-principle results' and try to explain some of the ideas behind the ones I am most familiar with.
I am going to talk about the group-theoretic aspects of the Andrews-Curtis conjecture, some recent results, and some old. From my viewpoint the Andrews-Curtis conjecture is not just a hard stand-alone question, coming from topology, but a host of very interesting problems in group theory.
Mackey functors play a central role in equivariant homotopy theory, where homotopy groups are replaced by homotopy Mackey functors. In this talk, I will discuss joint work with Dan Dugger and Christy Hazel classifying perfect chain complexes of constant Mackey functors for G=ℤ/2. Our decomposition leads to a computation of the Balmer spectrum of the derived category. We extend these results to classify all finite modules over the equivariant Eilenberg-MacLane spectrum Hℤ/2.
Factorization systems (both weak and strong) are commonly defined as consisting of two classes of maps satisfying a certain orthogonality relation and a factorization axiom. The standard definition of algebraic weak factorization system, involving comonads and monads, is rather different. The goal of this talk will be to describe an equivalent definition of algebraic weak factorization system emphasising orthogonality and factorization.
Quandles are algebraic structures designed to mesh with the Reidemeister moves of knot theory. Joyce and Matveev showed that quandles give rise to a complete invariant of oriented knots. Since the Yang-Baxter equation resembles the third Reidemeister move, it is not surprising that quandles also form a class of set-theoretic solutions of the Yang-Baxter equation. In this talk I will explain how quandles and connected quandles can be enumerated up to isomorphism and list a few open problems. I will also present two additional classes (involutive and idempotent) of set-theoretic solutions of the Yang-Baxter equation with rich algebraic theory.
We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of ∞-categories, and exploit this to prove many classical theorems about monads in the ∞-categorical setting. Among other things, we prove that the category of algebras for an accessible monad on a locally presentable ∞ category is locally presentable. We also apply the result to construct examples of ∞-categorical monads from pretheories.
In this series of lectures we will introduce the 2-matrix model and the issue of mixed traces, then we shall give the answers as formulas. Some formulas will be proved during the lectures, but the main goal is to explain how to use the formulas for practical computations. We shall largely follow the Chapter 8 of the book Counting surfaces, B. Eynard, Birkhäuser 2016.
Usual free probability theory was introduced by Voiculescu in the context of operator algebras. It turned out that there exists also a relation to random matrices, namely it describes the leading order of expectation values of the trace for multi-matrix models. Higher order versions of free probability were later introduced by Collins, Mingo, Sniady, Speicher in order to capture in the same way the leading order of correlations of several traces. A prominent role in free probability theory is played by “free cumulants” and “moment-cumulant formulas”, and the underlying combinatorial objects are “non-crossing partitions” and, for the higher order versions, “partitioned permutations”. I will give in my talks an introduction to free probability theory, with special emphasis on the higher order versions, and an eye towards possible relations to topological recursion. In particular, it seems that the problem of symplectic invariance in topological recursion has, at least in the planar sector, something to do with the transition between moments and free cumulants.
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