Diophantine approximation deals with quantitative and qualitative aspects of approximating numbers by rationals. A major breakthrough by Kleinbock and Margulis in 1998 was to study Diophantine approximations for manifolds using homogeneous dynamics. After giving an overview of recent developments in this subject, I will talk about Diophantine approximation in the S-arithmetic set-up, where S is a finite set of valuations of
Projective modules over rings are the algebraic analogues of vector bundles; more precisely, they are direct summands of free modules. Some rings have non-free projective modules. For instance, the ideals of a number ring are projective, and for some number rings they need not be free. Even for rings like ℤ, over which all finitely generated projective modules are free, the category of such modules contains a wealth of interesting information. In this talk I will introduce algebraic K-theory, which encodes this information. I will also explain why one would try to use some kind of trace from K-theory to simpler invariants, outline the cyclotomic trace and briefly show how it is used in calculations.
The talk will consist of a long historical introduction to the topic of deviation of ergodic averages for locally Hamiltonian flows on compact surafces as well as some current results obtained in collaboration with Corinna Ulcigrai and Minsung Kim. New developments include a better understanding of the asymptotic of so-called error term (in non-degenerate regime) and the appearance of new exponents in the deviation spectrum (in degenerate regime).
The lower central series of a group G is defined by γ1=G and γn = [G,γn-1]. The 'dimension series', introduced by Magnus, is defined using the group algebra over the integers:
δn = { g : g-1 belongs to the nth power of the augmentation ideal }.
It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has δn ≥ γn, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with δ4 / γ4 cyclic of order 2. On the positive side, Sjogren showed that δn / γn is always a torsion group, of exponent bounded by a function of n. Furthermore, it was believed (and falsely proved by Gupta) that only 2-torsion may occur.
In joint work with Roman Mikhailov, we prove however that every torsion abelian group may occur as a quotient δn / γn; this proves that Sjogren's result is essentially optimal.
Even more interestingly, we show that this problem is intimately connected to the homotopy groups πn(Sm) of spheres; more precisely, the quotient δn / γn is related to the difference between homotopy and homology. We may explicitly produce p-torsion elements starting from the order-p element in the homotopy group π2p(S2) due to Serre.
This talk is concerned with connections between arithmetic dynamics and complex dynamics. The first aim of the talk is to discuss several open problems from arithmetic dynamics and to explain how these problems are related to complex dynamical tool: bifurcation measures. If time allows, I will give a strategy to tackle several of those problems at the same time.
Given a topological dynamical system (X,T), a bounded sequence (an) and f ∈C(X) we are interested in the asymptotic behaviour of
1/(∑n≤N |an|) ∑n≤N an f(Tnx).
This will be a survey talk about recent progress on pointwise convergence problems for multiple ergodic averages along polynomial orbits and their relations with the Furstenberg-Bergelson-Leibman conjecture.
In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of bo ⋀ bo and BP⟨1⟩ ⋀ BP⟨1⟩. These splittings helped make it feasible to do computations using the bo- and BP⟨1⟩-based Adams spectral sequences. In this talk, we will discuss an analogous splitting for BP⟨2⟩ ⋀ BP⟨2⟩ at primes larger than 3.
I'll talk about the tmf-based Adams spectral sequence, and how it detects most of the v2-periodic elements in the known range of the 2-primary stable stems. Parts of the material I will discuss are joint with Dominic Culver, Prasit Bhattacharya, JD Quigley, and Mark Mahowald.
Sheaves sit at an interface of algebra and geometry. Equivariant sheaves offer even more structure, allowing for different group actions at different stalks. We are interested in the case where both the base space and group of equivariance are profinite (that is, compact, Hausdorff and totally disconnected). This combination provides many useful consequences, such as a good notion of equivariant presheaves and an explicit construction of infinite products.
The 2019 PhD thesis of Sugrue used equivariant sheaves to give an algebraic model for rational G-equivariant stable homotopy theory, where G is profinite. In this talk I will explain the model and related results, such as the equivalence between equivariant sheaves and rational Mackey functors (for profinite G).
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