Cube Complexes with Coupled Links (CLCC) are a special family of non-positively curved cube complexes that are constructed by specifying what the links of their vertices should be. In this talk I will introduce the construction of CLCCs and try to motivate it by explaining how one can use information about the local geometry of a cube complex to deduce global properties of its fundamental group (e.g. hyperbolicity and cohomological dimension). On the way, I will also explain what fly maps are and how to use them to deduce that a CAT(0) cube complex is hyperbolic.

Equivariant cohomology is adapted from ordinary cohomology to better capture the action of a group on a topological space. In Floer theory, given an autonomous Hamiltonian, there is a natural action of the circle on 1-periodic flowlines given by time translation. Combining these two ideas leads to the definition of  $S^1$-equivariant symplectic cohomology. In this talk, I will introduce these ideas and explain how they are related. I will not assume prior knowledge of Floer theory.

Symplectic cohomology is a Floer cohomology invariant of compact symplectic manifolds 
with contact type boundary, or of open symplectic manifolds with a certain geometry 
at the infinity. It is a graded unital K-algebra related to quantum cohomology, 
and for cotangent bundle, it recovers the homology of a loop space. During the talk 
I will define symplectic cohomology and show some of the results on its (non) vanishing. 
Time permitting, I will also mention natural TQFT algebraic structure on it.

The Grojnowski-Nakajima theorem states that the direct sum of the homologies of the Hilbert schemes on n points on an algebraic surface is an irreducible highest weight representation of an infinite-dimensional Heisenberg superalgebra. We present an idea to rederive the Grojnowski-Nakajima theorem using Halpern-Leistner's categorical Kirwan surjectivity theorem and Joyce's theorem that the homology of a moduli space of sheaves is a vertex algebra. We compute the homology of the moduli stack of perfect complexes of coherent sheaves on a smooth quasi-projective variety X, identify it as a (modified) lattice vertex algebra on the Lawson homology of X, and explain its relevance to the aforementioned problem.

We consider the discrete Bilaplacian on a cube in two and three dimensions with zero boundary data and prove estimates for its Green's function that are sharp up to the boundary. The main tools in the proof are Caccioppoli estimates and a compactness argument which allows one to transfer estimate for continuous PDEs to the discrete setting. One application of these estimates is to understand the so-called membrane model from statistical physics, and we will outline how these estimates can be applied to understand the phenomenon of entropic repulsion. We will also describe some connections to numerical analysis, in particular another approach to these estimates based on convergence estimates for finite difference schemes.

We study Brownian motion and stochastic parallel transport on Perelman's almost Ricci flat manifold,  whose dimension depends on a parameter $N$ unbounded from above. By taking suitable projections we construct sequences of space-time Brownian motion and stochastic parallel transport whose limit as $N\to \infty$ are the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on Perelman’s manifold and for the horizontal Laplacian on the corresponding orthonormal frame bundle.

As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman's manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and Haslhofer.

This is joint work with Robert Haslhofer.

Lagrangian Floer cohomology can be enriched by using local coefficients to record some homotopy data about the boundaries of the holomorphic disks counted by the theory. In this talk I will explain how one can do this under the monotonicity assumption and when the Lagrangians are equipped with local systems of rank higher than one. The presence of holomorphic discs of Maslov index 2 poses a potential obstruction to such an extension. However, for an appropriate choice of local systems the obstruction might vanish and, if not,
one can always restrict to some natural unobstructed subcomplexes. I will showcase these constructions with some explicit calculations for the Chiang Lagrangian in CP^3 showing that it cannot be disjoined from RP^3 by a Hamiltonian isotopy, answering a question of Evans-Lekili. Time permitting, I will also discuss some work-in-progress on the topology of monotone Lagrangians in CP^3, part of which follows from more general joint work with Jack Smith on the topology of monotone Lagrangians of maximal Maslov number in
projective spaces.

Stability conditions on derived categories of algebraic varieties and their wall-crossings have given algebraic geometers a number of new tools to study the geometry of moduli spaces of stable sheaves. In work in progress with Macri, Lahoz, Nuer, Perry and Stellari, we are extending this toolkit to a the "relative" setting, i.e. for a family of varieties. Our construction comes with relative moduli spaces of stable objects; this gives additional ways of constructing new families of varieties from a given family, thereby potentially relating different moduli spaces of varieties.

In this talk we discuss the recent proof for the existence of $C^{1,1}$ isometric immersions of several classes of negatively curved surfaces into $\R^3$, including the Lobachevsky plane, metrics of helicoid type and a one-parameter family of perturbations of the Enneper surface. Our method, following Chen--Slemrod--Wang and Cao--Huang--Wang, is to transform the Gauss--Codazzi equations into a system of hyperbolic balance laws, and prove the existence of weak solutions by finding the invariant regions. In addition, we provide further characterisation of the $C^{1,1}$ isometrically immersed generalised helicoids/catenoids established in the literature.

We present some recent results on the regularity criteria for weak solutions to the incompressible Navier--Stokes equations (NSE) in 3 dimensions. By the work of Constantin--Fefferman, it is known that the alignment of vorticity directions is crucial to the regularity of NSE in $\R^3$.  In this talk we show a boundary regularity theorem for NSE on curvilinear domains with oblique derivative boundary conditions. As an application, the boundary regularity of incompressible flows on balls, cylinders and half-spaces with Navier boundary condition is established, provided that the vorticity is coherently aligned up to the boundary. The effects of  vorticity alignment on the $L^q$, $1<q<\infty$ norm of the vorticity will also be discussed.