2nd Simons Math Summer Workshop: Moduli – July 1 – 19, 2024

Week 1 (July 1-5)

Theme: Moduli in differential and symplectic topology

There have been many instances in which moduli theory has played a definitive role in the study of manifolds. This week will focus on two such examples: K-stability and pseudoholomorphic curves.

Celebrated work of Chen, Donaldson,and Song resolved a conjecture of Donaldson and Tian by showing that on a Fano manifold, the condition of K-stability, formulated in the language of algebraic geometry, is a sharp obstruction to the existence of a Kaehler-Einstein metric. Further developments have revealed deep connections between concepts in birational geometry and differential geometry, and that K-stability plays an essential role in the moduli theory of higher dimensional Fano varieties.

Gromov introduced pseudoholomorphic curves as a tool in symplectic topology in 1986, and since then the study of these and related moduli problems has blossomed in many different directions. Moduli spaces of pseudoholomorphic curves are used to define enumerative invariants of symplectic manifolds, such as Gromov-Witten invariants, and, as well as richer invariants like the Fukaya category and its generalizations. At the same time, this theory has led to very fruitful interactions between symplectic topology and other disciplines, including mathematical physics, algebraic geometry, and gauge theory.

Despite its long history of development, many new developments in the subject hinge on ever-more sophisticated understanding of the structure of these moduli spaces. Examples include the construction of Floer-theoretic invariants defined over the integers and other spectra, the study of Gopakumar-Vafa invariants, and the introduction of “virtual” techniques inspired by derived geometry.

Week 2 (July 8-12)

Theme: Moduli of Higgs bundles and the Langlands program

The geometry of the moduli of Higgs bundles on an algebraic curve is studied from different perspectives in many different disciplines. Under the Hitchin fibration, moduli spaces of semistable Higgs bundles are important examples of algebraic completely integrable systems, and are studied in mathematical physics from this perspective. These moduli spaces also serve as local models for Lagrangian fibrations arising in the study of compact hyperkaehler manifolds.

The nonabelian Hodge correspondence identifies moduli spaces of Higgs bundles with character varieties, moduli spaces of representations of the fundamental group of the curve. This diffeomorphism of moduli spaces has revealed surprising connections, still being explored, between the topology of these integral systems and the topology and Hodge theory of character varieties.

The moduli of Higgs bundles also makes contact with number theory and representation theory via the Langlands program. Building on ideas in the proof of the Langlands correspondence for GL_n over function fields in finite characteristic, Beilinson and Drinfeld proposed the geometric langlands program, which primarily focuses on a quantization of the moduli space of Higgs bundles. Fully developing the latter program has been an inspiration for many developments in derived algebraic geometry.

The moduli theoretic perspective has also influenced the classical Langlands program in important ways, such as the use of Hitchin fibrations in finite characteristic in Ngo’s proof of the fundamental lemma. The interaction between geometric and arithmetic Langlands has grown even closer over the past few years, with the work of Fargues-Scholze formulating the local Langlands correspondence in a geometric context, and the work of Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky reformulating and generalizing geometric Langlands to include the Langlands correspondence for function fields in finite characteristic as a special case.

Special Event: Tuesday, July 9th, 2024 –  Moduli Journal Launch Celebration

The Foundation Compositio Mathematica is supporting this special day dedicated to the launch of the Moduli Journal during the 2nd SCGP Math Summer workshop.

The day will include special talks from diverse mathematicians, followed by a Jazz Concert from the 2024 SCGP Summer Concert Series, and conclude with a Special Banquet Dinner at 6:15PM in the SCGP Cafe.

Week 3 (July 15-19)

Theme: Compactifications of moduli spaces

Many moduli spaces are inherently non-compact. Studying different ways to compactify them is interesting in its own right, and at the same time provides important tools to understand the fine structure of the moduli spaces themselves better. If individual points in a moduli space parameterize a certain kind of geometric structure, then points in the boundary correspond to interesting ways in which that structure can degenerate.

The constructions of compactifications involve very different techniques, ranging from classical algebraic, geometric, or analytical tools into tropical geometry.

Well known compactifications of the moduli space of Riemann surfaces, like the Deligne-Mumford compactification, inspire new compactifications of other, more complicated moduli spaces. Tropical geometry plays an important role in the construction of compactifications. In recent years ideas from physics have led to new constructions of compactifications of the Higgs bundle moduli space, and the introduction of the real spectrum compactification of representation varieties has unified many previous approaches.

The Workshop will conclude after lunch on the 19th.