Summer school on Involutions in gauge theory and algebraic geometry: abstracts
May 30–June 2, 2023
University of Nantes

Schedule: All lectures will be held at the Campus Sciences of the University of Nantes, in Building 26 (Bâtiment 26), Room 112 (Amphitéâtre 112). Lunch will be at the university cafeteria. The conference dinner will be on Wednesday evening.

	Tuesday	Wednesday	Thursday	Friday
9:30-10:45	Manzaroli	Manzaroli	Scaduto	Scaduto
10:45-11:15	Coffee break	Coffee break	Coffee break	Coffee break
11:15-12:30	Scaduto	Scaduto	Manzaroli	Manzaroli
12:30-14:00	Lunch	Lunch	Lunch	Lunch
14:00-15:00	Yan	Framba	Exercises M	Saint-Criq
15:00-15:30			Coffee break
15:30-16:30			Exercises S

Mini-courses:

Matilde Manzaroli (University of Tübingen)
Topology of real algebraic curves
This course aims to dive into different aspects of the study of the topology of real algebraic curves embedded in real surfaces which may admit different anti-holomorphic involutions. This involves questions related to the 16th Hilbert problem: we look at obstructions of topological types for real algebraic curves and at the realisation of topological types via different construction techniques.

Christopher Scaduto (University of Miami)
Instanton Floer homology, knots, and involutions
The study of involutions on manifolds of dimensions 3 and 4 is closely related to the study of knots and embedded surfaces. The first lecture of this mini-course will be an introduction to fundamental aspects of this story. In the remaining lectures, I will describe a powerful tool, called (singular) instanton Floer homology, which is a theory specifically adapted to involutions on 3-manifolds and cobordisms between them. In the final lecture I will also discuss some recent developments on detecting 4-dimensional corks, and survey some related techniques.

Short talks:

Giovanni Framba (University of Pisa)
A new invariant of equivariant concordance and two-bridge knots
We are interested in the group of equivariant concordance classes of strongly invertible knots. We introduce the moth link, i.e. a two-component link associated to a given strongly invertible knot. Kojima&endash;Yamasaki's eta-function of the moth link is an equivariant concordance invariant. Using this invariant we prove that the equivariant concordance order of two-bridge knots is always infinite.

Anthony Saint-Criq (University of Toulouse III)
Bounding the number of non-empty ovals of an odd degree curve
Topologically-speaking, real plane algebraic curves are a collection of embedded circles in RP(2). The (still open) Hilbert sixteenth problem asks about classifying the possible mutual positions of those circles. A powerful tool in studying even-degree curves is taking the double branched cover of CP(2) ramified along the complexification. We will review how one can tweak the ambiant manifold and allow for double covering over an odd-degree curve, and derive an upper bound on the number of non-empty components of the real part. This method generalizes to curves on a quartic, as well as to a new notion of non-orientable flexible curves.

Jiajun Yan (University of Virginia)
A new gauge-theoretic construction of 4-dimensional hyperkähler ALE spaces
Non-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of non-compact hyperkähler spaces. They are in one-to-one correspondence with the finite subgroups of SU(2) and have interesting connections with representation theory and singularity theory, captured by the McKay Correspondence. In this talk, we first introduce the finite-dimensional construction of 4-dimensional hyperkähler ALE spaces given by Peter Kronheimer in his PhD thesis. Then we give a new gauge theoretic construction of these spaces. Time permitting, I will talk about some future directions.

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