Georgia Institute of Technology College of Sciences

Taking the double branched cover of $S^3$ over a knot $K$ is natural way to associate $K$ with a 3-manifold, and to study the double branched cover, we often want a Dehn surgery description for it. The Montesinos trick gives a systematic way to get such a description. In this talk, we will go over the broad statement of this trick: that a rational tangle replacement on the knot corresponds to Dehn surgery on the double branched cover. This gives particularly nice descriptions for some satellites of $K$ as surgery on $K \mathrel\# K^r$. We will also discuss an application of the trick which characterizes the 2-bridge knots with unknotting number 1.

This series will tie together algebraic, complex analytic, symplectic, and contact geometries together in one coherent story. This will be done via the study of a series of couplets from different fields of geometry:

Algebraic manifolds:
Affine and quasi-projective varieties (non-compact models)
Projective varieties (compact models)

Complex manifolds:
Stein manifolds
Stein compactifications

Symplectic manifolds:
Liouville/ Weinstein geometry
Compact Kahler manifolds 

Depending on how long it takes to discuss these items, I will also attempt to include discussions on:

• Biran-Giroux decompositions of symplectic manifolds • Boothby-Wang bundles and contact plumbings of these • Milnor's fibration theorem for isolated singularities and connections to open book decompositions and Lefschetz fibrations • Open questions and interesting avenues of research

Most of our discussion will, as a side effect, outline the topological structure behind Type IIA String theory (the "topological A-model") which requires a 6-dimensional Calabi-Yau (Kahler) background.

We investigate how fluctuations behave in large systems of interacting particles when the interaction is given by the Biot–Savart kernel, a key model from fluid dynamics. Our main result provides the first quantitative convergence rates for these fluctuations, and remarkably, the rates are optimal. The key idea is to compare the generators of the particle system and of the limiting fluctuation process in an infinite-dimensional setting. This comparison allows us to derive a sharp error bound for the fluctuations. Beyond the Biot–Savart case, the method is versatile and can also be applied to other singular interactions, such as the repulsive Coulomb kernel.

In this talk, we consider numbers with multiple close factorizations like $99990000 = 9999 \cdot 10000 = 9090 \cdot 11000$ and $3950100 = 1881 \cdot 2100 = 1890 \cdot 2090 = 1900 \cdot 2079$. We discuss optimal bounds on how close these factors can be relative to the size of the original numbers. It is related to the study of close lattice points on smooth curves.