Mathematics Department - Special Seminar - Spring 2017

# Special Seminar - Spring 2017

## Wednesday, February 1st

### "Symmetric intersecting families"

Time: 12:15 PM
Location: Hill 705
Abstract: A family of sets is said to be intersecting if any two sets in the family have nonempty intersection. Families of sets subject to various intersection conditions have been studied over the last fifty years and a common feature of many of the results in the area is that the extremal families are often quite asymmetric. Motivated by this, P{'e}ter Frankl conjectured in 1981 that symmetric intersecting families must be very small; more precisely, Frankl conjectured that a family of subsets of ${1, 2, dots, n}$ where any three sets intersect must have size $o(2^n)$ if its automorphism group is transitive. In this talk, I shall prove this conjecture. Based on joint work with David Ellis.

## Monday, January 23rd

### "Arithmetic hyperbolic manifold groups contain nonseparable subgroups"

Time: 3:00 PM
Location: Hill 525
Abstract: Given a group, whether all finitely generated subgroups are separable is an interesting group theoretical property, and it is closely related with low-dimensional topology, e.g. the virtual Haken conjecture (solved by Agol). I will show that, for almost all arithmetic hyperbolic manifolds with dimension at least 4, their fundamental groups contain nonseparable subgroups. The main ingredient is a study of certain graph of groups with hyperbolic 3-manifold groups as vertices, and the fact that hyperbolic 3-manifolds have a lot of virtual fibering structures. The proof also implies that, for a compact irreducible 3-manifold with empty or tori boundary, it supports one of eight Thurston's geometries if and only if its fundamental group is subgroup separable.

## Thursday, January 19th

### "Incorporation of Geometry into Learning Algorithms and Medicine"

Time: 3:00 PM
Location: Hill 705
Abstract: This talk focuses on two instances in which scientific fields outside mathematics benefit from incorporating the geometry of the data. In each instance, the applications area motivates the need for new mathematical approaches and algorithms, and leads to interesting new questions. (1) A method to determine and predict drug treatment effectiveness for patients based off their baseline information. This motivates building a function adapted diffusion operator high dimensional data X when the function F can only be evaluated on large subsets of X, and defining a localized filtration of F and estimation values of F at a finer scale than it is reliable naively. (2) The current empirical success of deep learning in imaging and medical applications, in which theory and understanding is lagging far behind. By assuming the data lies near low dimensional manifolds and building local wavelet frames, we improve on existing theory that breaks down when the ambient dimension is large (the regime in which deep learning has seen the most success).

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