L4

Logarithmic and orbifold structures provide two independent ways to model curves in a variety with tangency along a normal crossings divisor. The associated systems of Gromov-Witten invariants benefit from complementary techniques; this has motivated extensive interest in comparing the two approaches.

I will report on work in which we establish a complete comparison which, crucially, incorporates negative tangency orders. Negative tangency orders appear naturally in the boundary splitting formalisms of both theories. As such, our comparison opens the way for the wholesale importation of techniques from one side to the other. Work of Sam Johnston uses our comparison to give a new proof of the associativity of the Gross-Siebert intrinsic mirror ring.

Along the way, I will discuss the pathological geometry of negative tangency mapping spaces, and how this can be understood and controlled via tropical geometry. A crucial contribution of our work is the discovery of a "refined virtual class" on the logarithmic moduli space, which gives rise to a distinguished sector of the Gromov-Witten theory.

This is joint work with Luca Battistella and Dhruv Ranganathan.

I will discuss some recent progress on the Donaldson Segal programme, and in particular how calibrated cycles (coassociative submanifolds, special Lagrangians) arise from the large mass limit of $G_2$ and Calabi Yau monopoles.

S-duality is an intriguing symmetry of (twisted) N=4 supersymmetric Yang-Mills theory on a four-manifold. When the four-manifold underlies a complex projective surface, it leads to the Vafa-Witten invariants defined by Tanaka-Thomas in 2017. I will discuss some developments related to Azumaya algebras, universality, Seiberg-Witten invariants, wall-crossing for Nakajima quiver varieties, the structure of S-duality, and modular curves (including relations to the Rogers-Ramanujan continued fraction and Klein quartic).

The Bogomolov-Miyaoka-Yau inequality for minimal compact complex surfaces of general type was proved in 1977 independently by Miyaoka, using methods of algebraic geometry, and by Yau, as an outgrowth of his proof of the Calabi conjectures. In this talk, we outline our program to prove the conjecture that symplectic 4-manifolds with $b^+>1$ obey the Bogomolov-Miyaoka-Yau inequality. Our method uses Morse theory on the gauge theoretic moduli space of non-Abelian monopoles, where the Morse function is a Hamiltonian for a natural circle action and natural two-form.  We shall describe generalizations of Donaldson’s symplectic subspace criterion (1996) from finite to infinite dimensions. These generalized symplectic subspace criteria can be used to show that the natural two-form is non-degenerate and thus an almost symplectic form on the moduli space of non-Abelian monopoles. This talk is based on joint work with Tom Leness and the monographs https://arxiv.org/abs/2010.15789  (to appear in AMS Mathematical Surveys and Monographs), https://arxiv.org/abs/2206.14710 and https://arxiv.org/abs/2410.13809. 

 In the first part of the talk, after revisiting some classical models for dilute polymeric fluids, we show that thermodynamically 
consistent models for non-isothermal flows of such fluids can be derived in a very elementary manner. Our approach is based on identifying the 
energy storage mechanisms and entropy production mechanisms in the fluid of interest, which in turn leads to explicit formulae for the Cauchy 
stress tensor and for all the fluxes involved. Having identified these mechanisms, we first derive the governing system of nonlinear partial 
differential equations coupling the unsteady incompressible temperature-dependent Navier–Stokes equations with a 
temperature-dependent generalization of the classical Fokker–Planck equation and an evolution equation for the internal energy. We then 
illustrate the potential use of the thermodynamic basis on a rudimentary stability analysis—specifically, the finite-amplitude (nonlinear) 
stability of a stationary spatially homogeneous state in a thermodynamically isolated system.

In the second part of the talk, we show that sequences of smooth solutions to the initial–boundary-value problem, which satisfy the 
underlying energy/entropy estimates (and their consequences in connection with the governing system of PDEs), converge to weak 
solutions that satisfy a renormalized entropy inequality. The talk is based on joint results with Miroslav Bulíček, Mark Dostalík, Vít Průša 
and Endré Süli.

I will explain how to obtain local L^\infty estimates for optimal transport problems. Considering entropic optimal transport and optimal transport with p-cost, I will show how such estimates, in combination with a geometric linearisation argument, can be used in order to obtain ε-regularity statements. This is based on recent work in collaboration with M. Goldman (École Polytechnique) and R. Gvalani (ETH Zurich).

In the 1990s, physicists introduced an ideal way to count curves inside a Calabi-Yau 3-fold, called the Gopakumar-Vafa (GV) theory. Building on several previous attempts, Maulik-Toda recently gave a mathematical rigorous definition of the GV invariants. We expect that the GV invariants and the Gromov-Witten (GW) invariants are related by an explicit formula, but this stands as a challenging open problem. In this talk, I will explain recent mathematical developments on the GV theory, especially for local curves, including the cohomological chi-independence theorem and the GV/GW correspondence in a special case.

I will speak about my work with Helge Ruddat on how to construct explicitly log structures and morphisms. I will also discuss some motivation. I will try to stay informal and assume no prior knowledge of log structures.

Modern approaches to infinitesimal deformations of algebro-geometric objects (like varieties) use the setting of formal moduli problems, from derived geometry. It allows to prove that all kinds of deformations are governed by a tangent complex equipped with a derived Lie algebra structure. I will use this framework to study equivariant deformations of varieties with respect to the action of an algebraic group. Then, I will explain how this theory of equivariant deformations allows us to prove a dichotomous behaviour for almost all varieties that are homogeneous under a reductive group : either they deform to characteristic 0, or they admit no deformation to any ring of characteristic greater than p.