A loop of Hamiltonian diffeomorphisms of a symplectic manifold $X$ defines, by clutching, a symplectic fibration over the two-sphere with fibre $X$.  We prove that the integral cohomology of the total space splits additively, answering a question of McDuff, and extending the rational cohomology analogue proved by Lalonde-McDuff-Polterovich in the late 1990’s. The proof uses a virtual fundamental class of moduli spaces of sections of the fibration in Morava K-theory. This talk reports on joint work with Mohammed Abouzaid and Mark McLean.

Bacteria represent the major component of the world’s biomass. A number of these bacteria are motile and swim with the use of flagellar filaments, which are slender helical appendages attached to a cell body by a flexible hook. Low Reynolds number hydrodynamics is the key for flagella to generate propulsion at a microscale [1]. In this talk I will discuss two projects related to swimming of a model bacterium Escherichia coli (E. coli).

E. coli has many flagellar filaments that are wrapped in a bundle and rotate in a counterclockwise fashion (if viewed from behind the cell) during the so-called ‘runs’, wherein the cell moves steadily forward. In between runs, the cell undergoes quick ‘tumble’ events, during which at least one flagellum reverses its rotation direction and separates from the bundle, resulting in erratic motion in place. Alternating between runs and tumbles allows cells to sample space by stochastically changing their propulsion direction after each tumble. In the first part of the talk, I will discuss how cells reorient during tumble and the mechanical forces at play and show the predominant role of hydrodynamics in setting the reorientation angle [2].

In the second part, I will talk about hydrodynamics of bacteria near walls in visco-elastic fluids. Flagellar motility next to surfaces in such fluids is crucial for bacterial transport and biofilm formation. In Newtonian fluids, bacteria are known to accumulate near walls where they swim in circles [3,4], while experimental results from our collaborators at the Wu Lab (Chinese University of Hong Kong) show that in polymeric liquids this accumulation is significantly reduced. We use a combination of analytical and numerical models to propose that this reduction is due to a viscoelastic lift directed away from the plane wall induced by flagellar rotation. This viscoelastic lift force weakens hydrodynamic interaction between flagellated swimmers and nearby surfaces, which results in a decrease in surface accumulation for the cells. 

References

[1] Lauga, Eric. "Bacterial hydrodynamics." Annual Review of Fluid Mechanics 48 (2016): 105-130.

[2] Dvoriashyna, Mariia, and Eric Lauga. "Hydrodynamics and direction change of tumbling bacteria." Plos one 16.7 (2021): e0254551.

[3] Berke, Allison P., et al. "Hydrodynamic attraction of swimming microorganisms by surfaces." Physical Review Letters 101.3 (2008): 038102.

[4] Lauga, Eric, et al. "Swimming in circles: motion of bacteria near solid boundaries." Biophysical journal 90.2 (2006): 400-412.

TBC

In this talk, I will explain how to obtain the space of local operators of a 4d N=2 gauge theory using the category of line operators in the Kapustin twist (holomorphic topological twist). This category is given a precise definition by Cautis-Williams, as the category of equivariant coherent sheaves on the space of Braverman-Finkelberg-Nakajima. We compute the derived endomorphism of the monoidal unit in this category, and show that it coincides with the vacuum module of the Poisson vertex algebra of Oh-Yagi and Butson. The Euler character of this space reproduces the Schur index. I will also explain how to obtain the space of local operators at the junction of minimal Wilson-t’Hooft line operators. Its Euler character can be compared to the index formula of Cordova-Gaiotto-Shao. This is based on arXiv: 2112.12164.

There have been several proposals of scale-separated AdS vacua in the literature. All known examples arise from the effective field theory of flux compactifications with low supersymmetry, and there are often doubts about their consistency as 10 or 11d backgrounds in string theory. These issues can often be tackled in the bulk theory, or by analysis of the dual CFT via holography. I will review the most common issues, and focus the analysis on the recently constructed family of 3d scale-separated AdS vacua, which is dual to a two-dimensional CFT, emphasizing the discrete symmetry structure of the model in comparison to DGKT. Finally, I will comment on the tantalizing observation of integer operator dimensions in DGKT-like vacua, and comment on possible places to look for consistency issues in these models.

Let X be a smooth projective curve over a finite field equipped with an action of a finite group G. I’ll first briefly introduce the corresponding Artin L-function and a certain equivariant Euler characteristic. The main result will be a precise relation between the epsilon constants appearing in the functional equations of Artin L-functions and that Euler characteristic if the projection X → X/G is at most weakly ramified. This generalises a theorem of Chinburg for the tamely ramified case. I’ll end with some speculations in the arbitrarily wildly ramified case. This is joint work with Helena Fischbacher-Weitz and with Adriano Marmora.

In this talk, we discuss existentially closed measure preserving actions of countable groups.  A classical result of Berenstein and Henson shows that the model companion for this class exists for the group of integers and their analysis readily extends to cover all amenable groups.  Outside of the class of amenable groups, relatively little was known until recently, when Berenstein, Henson, and Ibarlucía proved the existence of the model companion for the case of finitely generated free groups.  Their proof relies on techniques from stability theory and is particular to the case of free groups.  In this talk, we will discuss the existence of model companions for measure preserving actions for the much larger class of universally free groups (also known as fully residually free groups), that is, groups which model the universal theory of the free group.  We also give concrete axioms for the subclass of elementarily free groups, that is, those groups with the same first-order theory as the free group.  Our techniques are ergodic-theoretic and rely on the notion of a definable cocycle.  This talk represents ongoing work with Brandon Seward and Robin Tucker-Drob.