The lower central series of a group $G$ is defined by $\gamma_1=G$ and $\gamma_n = [G,\gamma_{n-1}]$. The "dimension series", introduced by Magnus, is defined using the group algebra over the integers: $\delta_n = \{g: g-1\text{ belongs to the $n$-th 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 $\delta_n\ge\gamma_n$, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with $\delta_4/\gamma_4$ cyclic of order 2. On the positive side, Sjogren showed that $\delta_n/\gamma_n$ is always a torsion group, of exponent bounded by a function of $n$. Furthermore, it was believed (and falsely proven by Gupta) that only $2$-torsion may occur.
In joint work with Roman Mikhailov, we prove however that for every prime $p$ there is a group with $p$-torsion in some quotient $\delta_n/\gamma_n$.
Even more interestingly, I will show that the dimension quotient $\delta_n/gamma_n$ is related to the difference between homotopy and homology: our construction is fundamentally based on the order-$p$ element in the homotopy group $\pi_{2p}(S^2)$ due to Serre.
 

 In the late seventies, Casson and Gordon developed several knot invariants that obstruct a knot from being slice, i.e. from bounding a disc in the 4-ball. In this talk, we use twisted Blanchfield pairings to define twisted generalisations of the Levine-Tristram signature function, and describe their relation to the Casson-Gordon invariants. If time permits, we will present some obstructions to algebraic knots being slice. This is joint work with Maciej Borodzik and Wojciech Politarczyk.

NSOP1 is a class of first order theories containing simple theories, which contains many natural examples that somehow slip-out of the simple context.

As in simple theories, NSOP1 theories admit a natural notion of independence dubbed Kim-independence, which generalizes non-forking in simple theories and satisfies many of its properties.

In this talk I will explain all these notions, and in particular talk about recent progress (joint with Nick Ramsey) in the study of Kim-independence, showing transitivity and several consequences.

Here Z is Zermelo’s set theory of 1908, as later formulated: full separation, but no replacement or collection among its axioms. PROVI was presented in lectures in Cambridge in 2010 and later published with improvements by Nathan Bowler, and is, I claim, the weakest subsystem of ZF to support a recognisable theory of set forcing: PROV is PROVI shorn of its axiom of infinity. The provident sets are the transitive non-empty models of PROV. The talk will begin with a presentation of PROV, and then discuss more recent applications and problems: in particular an answer in the system Z + PROV to a question posed by Eugene Wesley in 1972 will be sketched, and two proofs (fallacious, I hope) of 0 = 1 will be given, one using my slim models of Z and the other applying the Spector–Gandy theorem to certain models of PROVI. These “proofs”, when re-interpreted, supply some arguments of Reverse Mathematics.

Starting with a countable transitive model of V=L, we show that by 
adding a single Cohen real, c, most intermediate models do no satisfy choice. In 
fact, most intermediate models to L[c] are not even definable.

The key part of the proof is the Bristol model, which is intermediate to L[c], 
but is not constructible from a set. We will give a broad explanation of the 
construction of the Bristol model within the constraints of time.

 The talk is about the normal modal logics of elementary classes defined by first-order formulas of the form
 'for all x_0 there exist x_1, ..., x_n phi(x_0, x_1, ... x_n)' with phi being a conjunction of binary atoms.
 I'll show that many properties of these logics, such as finite axiomatisability,
 elementarity,  axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula,
 together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.
 

Polynomials have been at the heart of mathematics for a millennium, yet when it comes to applying them, there are many puzzles and surprises. Among others, our tour will visit Newton, Lagrange, Gauss, Galois, Runge, Bernstein, Clenshaw and Chebfun (with a computer demo).