Georgia Institute of Technology College of Sciences

The L-space conjecture has been in the news a lot lately. It predicts a surprising relationship between the algebraic, geometric, and Floer-homological properties of a 3--manifold Y. In particular, it predicts exactly which 3-manifolds admit a ``taut foliation". In this talk, I'll discuss some of my past and forthcoming work investigating these connections. In particular, I'll discuss a strategy for building taut foliations manifolds obtained by Dehn surgery along knots realized as closures of ``positive braids". As an application, I will show how taut foliations can be used to obstruct positivity for cable knots. All are welcome; no background in foliation or Floer homology theories will be assumed.

https://bccte.zoom.us/j/91883463721

Meeting ID: 918 8346 3721

Over recent years there has been much interest in both Turán and Ramsey properties of vertex ordered graphs (i.e., graphs equipped with an ordering of their vertex set). In a recent paper, József Balogh, Lina Li and I initiated the study of embedding spanning structures into vertex ordered graphs. In particular, we introduced a general framework for approaching the problem of determining the minimum degree threshold for forcing a perfect $H$-tiling in an ordered graph. In this talk I will discuss this work, in particular emphasizing how we adapt the regularity and absorbing methods to be applicable in the ordered setting.

A knot in the 3-sphere is slice if it bounds a smooth disc in the 4-ball. A knot is ribbon if it bounds a self-intersecting disc with only singularities that are closed arcs consisting of intersection points of the disc with itself. Every ribbon knot is a slice knot; the converse is a famous unsolved conjecture of Fox. This talk will show some recent interesting progress around the slice-ribbon conjecture.

In the last years, connections between graphs and Riemann surfaces have been
discovered on several different levels. In particular, graphs are closely related
to singular Riemann surfaces and the boundary in the Deligne–Mumford com-
pactification of moduli spaces. Moreover, in both settings there is a notion of a
canonical measure (the Arakelov–Bergman and Zhang measures) which reflects
crucial geometric information.
In this talk, we focus on the following question: what is the limit of the canon-
ical measures along a family of Riemann surfaces? Combining the canonical
measures on Riemann surfaces and metric graphs, we obtain a full description
and a new compactification of the moduli space of Riemann surfaces in terms
of hybrid curves.

Based on joint work with Omid Amini (École polytechnique).

BlueJeans link: https://bluejeans.com/476849994

In the past several decades, scale invariant stochastic processes have been used in a wide range of applications including internet traffic modeling and hydrology.  However, by comparison to univariate scale invariance, far less attention has been paid to characteristically multivariate models that display aspects of scaling behavior the limit theory arguably suggests is most natural.
 
In this talk, I will introduce a new scale invariance model called operator fractional Lévy motion and discuss some of its interesting features, as well as some aspects of wavelet-based estimation of its scaling exponents. This is related to joint work with Gustavo Didier (Tulane University), Herwig Wendt (CNRS, IRIT Univ. of Toulouse) and Patrice Abry (CNRS, ENS-Lyon).

For a manifold M, the (generalized) Nielsen realization problem asks if the surjection Diff(M) → π_0 Diff(M) is split, where Diff(M) is the diffeomorphism group. When M is a surface, this question was posed by Thurston in Kirby's problem list and was addressed by Morita. I will discuss some more recent work on Nielsen realization problems with connections to flat fiber bundles, K3 surfaces, and smooth structures on hyperbolic manifolds.

It is a classical fact that for any c > 0, a random permutation of length n = (1+c)k^2/4 typically contains a monotone subsequence of length k. As a far-reaching generalization, Alon conjectured that for this same n, a typical n-permutation is k-universal, meaning that it simultaneously contains every k-pattern. He also gave a simple proof for the fact that if n is increased to Ck^2 log k, then a typical n-permutation is k-universal. Our main result is that the same statement holds for n = Ck^2 log log k, getting almost all of the way to Alon's conjecture.

In this talk we give an overview of the structure-vs-randomness paradigm which is a key ingredient in the proof, and a sketch of the other main ideas. Based on joint work with Matthew Kwan.