I will discuss results relating different partially wrapped Fukaya categories. These include a K\"unneth formula, a `stop removal' result relating partially wrapped Fukaya categories relative to different stops, and a gluing formula for wrapped Fukaya categories. The techniques also lead to generation results for Weinstein manifolds and for Lefschetz fibrations. The methods are mainly geometric, and the key underlying Floer theoretic fact is an exact triangle in the Fukaya category associated to Lagrangian surgery along a short Reeb chord at infinity. This is joint work with Sheel Ganatra and Vivek Shende.

# Past Geometry and Analysis Seminar

In this talk, we will explain how to construct embedded closed Lagrangian submanifolds in mirror quintic threefolds using tropical curves and the toric degeneration technique. As an example, we will illustrate the construction for tropical curves that contribute to the Gromov–Witten invariant of the line class of the quintic threefold. The construction will in turn provide many homologous and non-Hamiltonian isotopic Lagrangian

rational homology spheres, and a geometric interpretation of the multiplicity of a tropical curve as the weight of a Lagrangian. This is a joint work with Helge Ruddat.

In this talk, we are going to talk about the Type I singularity on 4-dimensional manifolds foliated by homogeneous S3 evolving under the Ricci

flow. We review the study on rotationally symmetric manifolds done by Angenent and Isenberg as well as by Isenberg, Knopf and Sesum. In the latter, a global frame for the tangent bundle, called the Milnor frame, was used to set up the problem. We shall look at the symmetries of the manifold, derived from Lie groups and its ansatz metrics, and this global tangent bundle frame developed by Milnor and Bianchi. Numerical simulations of the Ricci flow on these manifolds are done, following the work by Garfinkle and Isenberg, providing insight and conjectures for the main problem. Some analytic results will be proven for the manifolds S1×S3 and S4 using maximum principles from parabolic PDE theory and some sufficiency conditions for a neckpinch singularity will be provided. Finally, a problem from general relativity with similar metric symmetries but endowed on a manifold with differenttopology, the Taub-Bolt and Taub-NUT metrics, will be discussed.

We consider the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. We construct smooth solutions with concentrated vorticities around $k$ points which evolve according to the Hamiltonian system for the Kirkhoff-Routh energy, using an outer-inner solution gluing approach. The asymptotically singular profile around each point resembles a scaled finite mass solution of Liouville's equation.

We also discuss the {\em vortex filament conjecture} for the three-dimensional case. This is joint work with Juan D\'avila, Monica Musso and Juncheng Wei.

After discussing a general excision technique for constructing canonical orientations for moduli spaces that derive from an elliptic equation, I shall

explain how to carry out this program in the case of G2-instantons and the 7-dimensional real Dirac operator. In many ways our approach can

be regarded as a categorification of the Atiyah-Singer index theorem. (Based on joint work with Dominic Joyce.)

This talk is based on joint work with Dominic Joyce and Markus Upmeier. Issues we'd like to talk about are a) the orientability of moduli spaces that

appear in various gauge-theoretic problems; and b) how to orient those moduli spaces if they are orientable. We begin with briefly mentioning backgrounds and motivation, and recall basics in gauge theory such as the Atiyah-Hitchin-Singer complex and the Kuranishi model by taking the anti-self-dual instanton moduli space as an example. We then describe the orientability and canonical orientations of the anti-self-dual instanton moduli space, and other

gauge-theoretic moduli spaces which turn up in current research interests.

Einstein metrics are fixed points (up to scaling) of Hamilton's Ricci flow. A natural question to ask is whether a given metric is stable in the sense that the flow returns to the Einstein metric under a small perturbation. I'll give a brief survey of this area focussing on the case when the Einstein constant is positive. An interesting class of metrics where this question is not completely resolved are the compact symmetric spaces. I'll report on some recent progress with Tommy Murphy and James Waldron where we have been able to use a criterion due to Kroencke to show the Kaehler-Einstein metric on some Grassmannians and the bi-invariant metric on the Lie group G_2 are unstable.

Amplituhedra are mathematical objects generalising the notion of polytopes into the Grassmannian. Proposed as a geometric construction encoding scattering amplitudes in the four-dimensional maximally supersymmetric Yang-Mills theory, they are mathematically interesting objects on their own. In my talk I strengthen the relation between scattering amplitudes and geometry by linking the amplituhedron to the Jeffrey-Kirwan residue, a powerful concept in symplectic and algebraic geometry. I focus on a particular class of amplituhedra in any dimension, namely cyclic polytopes, and their even-dimensional

conjugates. I show how the Jeffrey-Kirwan residue prescription allows to extract the correct amplituhedron canonical differential form in all these cases. Notably, this also naturally exposes the rich combinatorial structures of amplituhedra, such as their regular triangulations

The goal of the talk will be to introduce a class of functions that answer a question in topology, can be computed via analytic methods more common in the theory of dynamical systems, and in the end turn out to enjoy beautiful modular properties of the type first observed by Ramanujan. If time permits, we will discuss connections with vertex algebras and physics of BPS states which play an important role, but will be hidden "under the hood" in much of the talk.

Introduced by Konno, hyperpolygon spaces are examples of Nakajima quiver varieties. The simplest of these is a noncompact complex surface admitting the structure of a gravitational instanton, and therefore fits nicely into the Kronheimer-Nakajima classification of complete ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU(2). For more general hyperpolygon spaces, we can speculate on how

this classification might be extended by studying the geometry of hyperpolygons at "infinity". This is ongoing work with Hartmut Weiss.