In a recent paper Friedl, Zentner and Livingston asked when a sum of torus knots is concordant to an alternating knot. After a brief analysis of the problem in its full generality, I will describe some effective obstructions based on Floer type theories.

We develop a structure  theory for del Pezzo surfaces that are regular but geometrically non-normal, based on work of Reid, but now independence on the p-degree of the ground field. This leads to existence results, as well as non-existence results for ground fields  of p-degree one. In turn, we  settle questions arising from Koll'ar's analysis on the structure of Mori fiber spaces in dimension three. This is joint work with Andrea Fanelli.

3d N=4 supersymmetric gauge theories provide a method for constructing HyperK\”ahler singularities, known as the Coulomb branch.
This method is complementary to the more traditional way of construction using HyperK\”ahler quotients, known in physics as the “Higgs branch”.
Out of all possible gauge theories there is an interesting subclass of quiver varieties, where the Coulomb branch has been studied in some detail.
Some examples are moduli spaces of classical and exceptional instantons and closures of nilpotent orbits. An interesting feature of Coulomb and Higgs branches is the phenomenon of "3d mirror symmetry” where for a pair of gauge theories, the Higgs branch and Coulomb branch exchange.
There is a large class of “mirror pairs” which I will discuss in some detail.

A topic of recent interest is the notion of implosions. I will argue that there is a simple operation on the quiver which leads to implosion. In other words, given a quiver such that its Coulomb branch is moduli space A, a simple operation of the quiver (making a bouquet) provides the implosion of A.
This has been tested on closures of nilpotent orbits of A type and on nilpotent cones of orthogonal groups and found to agree with the expected results.
If time permits, I will discuss isometries of Coulomb branches

For many moduli problems, in order to construct a moduli space as a geometric invariant theory quotient, one needs to impose a notion of (semi)stability. Using recent results in non-reductive geometric invariant theory, we explain how to stratify certain moduli stacks in such a way that each stratum admits a coarse moduli space which is constructed as a geometric quotient of an action of a linear algebraic group with internally graded unipotent radical. As many stacks are
naturally filtered by quotient stacks, this involves describing how to stratify certain quotient stacks. Even for quotient stacks for reductive group actions, we see that non-reductive GIT is required to construct the coarse moduli spaces of the higher strata. We illustrate this point by studying the example of the moduli stack of coherent sheaves over a projective scheme. This is joint work with G. Berczi, J. Jackson and F. Kirwan.

I will talk about recent work, joint with M. Gröchenig and D. Wyss, on two related results involving the cohomology of moduli spaces of Higgs bundles. The first is a positive answer to a conjecture of Hausel and Thaddeus which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The second is a new proof of Ngô's geometric stabilization theorem which appears in the proof of the fundamental lemma. I will give an introduction to these theorems and outline our argument, which, inspired by work of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration.

In real dimension two, the symplectic mapping class group of a surface agrees with its `classical' mapping class group, whose properties are well-understood. To what extend do these generalise to higher-dimensions? We consider specific pairs of symplectic manifolds (S, M), where S is a surface, together with collections of Lagrangian spheres in S and in M, say v_1, ...,v_k and V_1, ...,V_k, that have analogous intersection patterns, in a sense that we will make precise. Our main theorem is that any relation between the Dehn twists in the V_i must also hold between Dehn twists in the v_i. Time allowing, we will give some corollaries, such as embeddings of certain interesting groups into auto-equivalence groups of Fukaya categories.

First, we will discuss sequences of closed minimal hypersurfaces (in closed Riemannian manifolds of dimension up to 7) that have uniformly bounded index and area. In particular, we explain a bubbling result which yields a bound on the total curvature along the sequence and, as a consequence, topological control in terms of index and area. We then specialise to minimal surfaces in ambient manifolds of dimension 3, where we use the bubbling analysis to obtain smooth multiplicity-one convergence under bounds on the index and genus. This is joint work with Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp

 The basic algebra-geometry dictionary for finitely generated k-algebras is one of the triumphs of 19th and early 20th century mathematics.  However, classes of related rings, such as their k-subalgebras, lack clean general properties or organizing principles, even when they arise naturally in problems of smooth projective geometry.  “Stabilization” in smooth topology and symplectic geometry, achieved by products with Euclidean space, substantially simplifies many
problems.  We discuss an analog in the more rigid setting of algebraic and arithmetic geometry, which, among other things (e.g., applications to counting rational points), gives some structure to the study of k-subalgebras.  We focus on the case of the moduli space of stable rational n-pointed curves to illustrate.