We will introduce the multi-parameter flag Leibniz rules. The formulation can be perceived as compositions of differential operators or partial differential operators and the complexity reflects the number of compositions. Nonetheless, the nontrivial range of boundedness exponents cannot be derived by iterative applications of Leibniz rules of lower complexities. We will then focus on a particular example of flag Leibniz rules to illustrate main ideas in the proof. This is joint work with Cristina Benea.

Larvae (or maggots) collected at crime scenes contribute important pieces of information to police investigations. Their hatching time provides a lower bound for the post-mortem interval, i.e. the interval between death and the discovery of the body. A functional data analysis approach is described here to model the local growth rate from the experimental data on larval development, where larvae have been exposed to a small number of constant temperature profiles. This allows us to reconstruct varying temperature growth profiles and use them to estimate the hatching time for a sample of larvae from the crime scene.

All oriented links $L$ have special diagrams. Based on such a diagram we construct a sutured handlebody $M$ which embeds in the branched double cover of $L$. From the sutured Floer homology of $M$ we recover the Alexander polynomial $\Delta$ of $L$ via a simple forgetful map. More surprisingly, in cases when the diagram is also positive (so that $L$ is a special alternating link), $\mathrm{SFH}(M)$ can be used to compute those coefficients of the HOMFLY polynomial of $L$ whose sum is the leading coefficient of $\Delta$. To extract this information algebraically, we develop the notion of the interior polynomial of a bipartite graph. The talk involves joint results with A. Juh\'asz, H. Murakami, A. Postnikov, J. Rasmussen, and D. Thurston.

The prototypical question in metric systolic geometry is to bound the length of a shortest closed geodesic on a closed Riemannian manifold by the volume of the manifold. This question has been extensively studied for non simply connected manifolds, but in the recent years there has been some progress also for simply connected manifolds, on which closed geodesics cannot be found simply by minimizing the length. This progress involves extending systolic questions to Reeb flows, a class of dynamical systems generalising geodesic flows. On the one hand, this extension and the use of symplectic techniques provide some answers to classical questions within metric systolic geometry.

In a 4-manifold, the Whitney trick seeks to remove a pair of oppositely signed intersection points between immersed surfaces. I will discuss recent joint work with Chris Davis and JungHwan Park where we describe a relative Whitney trick. The relative Whitney trick seeks to remove a single double point in a properly immersed surface, at the expense of changing the boundary of the surface by a homotopy. Our main application is to prove that any link in a homology 3-sphere is homotopic to a link that bounds a collection of locally flatly embedded discs in a contractible topological 4-manifold. In other words, every link in a homology sphere is homotopic to a topologically slice link.

I will explain a geometric construction of an Enriques surface fibration over P^1 of even index. This answers a question of Colliot-Thelene and Voisin, and provides new counterexamples to the Integral Hodge conjecture. This is joint work with Fumiaki Suzuki.

We consider complex affine surfaces S{p,q} given in C^4 by {yx^d=z-p(x), wz^e=x-q(z)}, where p,q are polynomials of degrees d,e; p(0)=q(0)=1. Using these surfaces as a simple example, we introduce various notions in algebraic geometry and topology. First, we compute their standard boundaries, showing that S{p,q} is isomorphic to S{p',q'} if and only if {p,q}={p',q'}. Next, applying calculus of graph manifolds to tubular neighborhoods of these boundaries, we show that S{p,q} is homeomorphic to S{p',q'} if and only if {d,e}={d',e'}. In fact, we will show a topological construction of S{p,q} via a 0-surgery on a 2-bridge knot.

Using Heegaard Floer theory we find obstructions for configurations of singular points on general curves in CP^2, generalizing previous results of Bodnár, Borodzik, Celoria, Golla, Hedden and Livingston. We show concrete examples, when "genus cannot be traded for double points". This is a joint work with Beibei Liu and Ian Zemke.

Given a closed 4-manifold X with an indefinite intersection form, we consider smoothly embedded surfaces in X-int(B^4), with boundary a given knot K in the 3-sphere. We give several methods to bound the genus of such surfaces in a fixed homology class. Our techniques include adjunction inequalities from Heegaard Floer homology and the Bauer-Furuta invariants, and the 10/8 theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disc) in a 4-manifold and show that the set of H-slice knots can detect exotic smooth structures on closed 4-manifolds. This is joint work with Ciprian Manolescu and Lisa Piccirillo.