I will discuss the orderability of the fundamental groups of knot complements including known results, a useful technique using some ideas of Baumslag, and some interesting questions that have recently arisen from this study.
I will discuss how a graph theoretic construction used by Hirasawa and Murasugi can be used to show that the commutator subgroup of the knot group of a two-bridge knot is a union of an ascending chain of parafree groups. Using a theorem of Baumslag, this implies that the commutator subgroup of a two-bridge knot group is residually torsion-free nilpotent which has applications to the anti-symmetry of ribbon concordance and the bi-orderability of two-bridge knots. In 1973, E. J. Mayland gave a conference talk in which he announced this result. Notes on this talk can be found online. However, this result has never been published, and there is evidence, in later papers, that a proper proof might have eluded Mayland.
Domino tilings of finite grid regions have been studied in many contexts, revealing rich combinatorial structure. They arise in applications spanning physics, computer science and probability theory and recreational mathematics. We will look at questions such as counting and sampling from large combinatorial sets, such as the set of domino tilings, providing a small sample of some of the techniques that are used.
Branched covers are a generalization of covering spaces, and give rise to interesting questions in smooth as well as contact topology. All 3 manifolds arise as branched coverings of the 3-sphere. The talk will involve a discussion of the proof of this fact due to Montesinos, and will explore the work done towards understanding which contact 3 manifolds arise as the branched cover of the standard tight 3 sphere, and how the branch set can be regulated.
The Caffarelli contraction theorem states that the Brenier map sending the
Gaussian measure onto a uniformly log-concave probability measure is
lipschitz. In this talk, I will present a new proof, using entropic
regularization and a variational characterization of lipschitz transport
maps. Based on joint work with Nathael Gozlan and Maxime Prod'homme.
The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. Specifically, neuronal networks at multiple scales utilize their structural complexities to achieve different computational goals. In this talk, I will discuss functional implications that can be inferred from the architecture of brain networks.
The first part of the talk will focus on a generalized problem of linking structure and dynamics of the whole-brain network. By simulating large-scale brain dynamics using a data-driven network of phase oscillators, we show that complexities added to the spatially embedded brain connectome by idiosyncratic long-range connections, enable rapid transitions between local and global synchronizations. In addition to the spatial dependence, I will also discuss hierarchical structure of the brain network. Based on the data-driven layer-specific connectivity patterns, we developed an unsupervised method to find the hierarchical organization of the mouse cortical and thalamic network. The uncovered hierarchy provides insights into the direction of information flow in the mouse brain, which has been less well-defined compared to the primate brain.
Finally, I will discuss computational implications of the hierarchical organization of the brain network. I will focus on a specific type of computation – discrimination of partially occluded objects— carried out by a small cortical circuitry composed of an intermediate visual cortical area V4 and its efferent prefrontal cortex. I will explore how distinct feedforward and feedback signals promote robust encoding of visual stimuli by leveraging predictive coding, a Bayesian inference theory of cortical computation which has been proposed as a method to create efficient neural codes. We implement a predictive coding model of V4 and prefrontal cortex to investigate possible computational roles of feedback signals in the visual system and their potential significance in robust encoding of nosy visual stimuli.
In sum, our results reveal the close link between structural complexity and computational versatility found in brain networks, which may be useful for developing more efficient artificial neural networks and neuromorphic devices.
This talk concerns a naturally occurring family of Calabi-Yau manifolds that degenerates in the sense of metric geometry, algebraic geometry and nonlinear PDE. A primary tool in analyzing their behavior is the recently developed regularity theory. We will give a precise description of arising singularities and explain possible generalizations.
I will introduce an isoperimetric inequality for the Hamming cube and some of its applications. The applications include a “stability” version of Harper’s edge-isoperimetric inequality, which was first proved by Friedgut, Kalai and Naor for half cubes, and later by Ellis for subsets of any size. Our inequality also plays a key role in a recent result on the asymptotic number of maximal independent sets in the cube.
This is joint work with Jeff Kahn.
(This is a joint event of ACO Student Seminar and the Combinatorics Seminar Series)
In this talk we will prove a conjecture of Talagrand, which is a fractional version of the “expectation-threshold” conjecture of Kalai and Kahn. This easily implies various difficult results in probabilistic combinatorics, e.g. thresholds for perfect hypergraph matchings (Johansson-Kahn-Vu) and bounded-degree spanning trees (Montgomery). Our approach builds on recent breakthrough work of Alweiss, Lovett, Wu, and Zhang on the Erdős-Rado “Sunflower Conjecture.”
This is joint work with Keith Frankston, Jeff Kahn, and Bhargav Narayanan.
(This is a joint event of the Combinatorics Seminar Series and the ACO Student Seminar.)
In this talk we will prove a conjecture of Talagrand, which is a fractional version of the “expectation-threshold” conjecture of Kalai and Kahn. This easily implies various difficult results in probabilistic combinatorics, e.g. thresholds for perfect hypergraph matchings (Johansson-Kahn-Vu) and bounded-degree spanning trees (Montgomery). Our approach builds on recent breakthrough work of Alweiss, Lovett, Wu, and Zhang on the Erdos-Rado “Sunflower Conjecture.”
This is joint work with Keith Frankston, Jeff Kahn, and Bhargav Narayanan.
A group is said to be torsion-free if it has no elements of finite order. An example is the group, under composition, of self-homeomorphisms (continuous maps with continuous inverses) of the interval I = [0, 1] fixed on the boundary {0, 1}. In fact this group has the stronger property of being left-orderable, meaning that the elements of the group can be ordered in a way that is nvariant under left-multiplication. If one restricts to piecewise-linear (PL) homeomorphisms, there exists a two-sided (bi-)ordering, an even stronger property of groups.
I will discuss joint work with Danny Calegari concerning groups of homeomorphisms of the cube [0, 1]^n fixed on the boundary. In the PL category, this group is left-orderable, but not bi-orderable, for all n>1. Also I will report on recent work of James Hyde showing that left-orderability fails for n>1 in the topological category.
