Hirzebruch proved a beautiful inequality for complex line arrangements in CP^2, giving strong bounds on the their combinatorics. In the quest for a topological proof of this inequality, Paolo Aceto and I studied odd and even line arrangements (which I will define in the talk). We proved Hirzebruch-like inequalities for these arrangements, and drew some corollaries about configurations of lines. Time (and audience) permitting, I will also discuss some more speculative ideas and generalisations of our results.

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

In this talk, I will begin by presenting some classic constructions of smooth non-orientable 4-manifolds arising from certain Brieskorn homology 3-spheres. I will then explain how to construct new examples, including infinitely many smooth fake copies of *RP4#*CP2. In addition, I will describe a method for generating a collection of Brieskorn homology 3-spheres that can be realized via integer surgery on knots in the 3-sphere. This is joint work with Jae Choon Cha and Oguz Savk.

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

Any reasonable exotic phenomena in simply-connected 4-manifolds are unstable. It is an open question if there is an universal upper bound to the number of stabilizations needed. The case of 1 stabilization was proven in works of Lin and Guth-K., but whether we need more than two stabilizations has been open because it is significantly harder. In this talk, we discuss my recent proof with Park and Taniguchi that two stabilizations are indeed not enough for exotic diffeomorphisms.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.