Mathematics

Treating Limits as Colimits and Colimits as Limits ... with Applications!

Jordan Watts (University of Colorado Bolder)

Abstract: Actually, in this talk, we will restrict ourselves to treating subspaces as quotient spaces and quotient spaces as subspaces ... with applications. To elaborate, consider a manifold. Typically it is defined to be a certain gluing of open subsets of Euclidean space (a quotient space), although we know we can embed any manifold into some large Euclidean space (a subspace). Conversely, the level set of a regular value of a smooth real-valued function (a subspace) is a manifold (a quotient space). This is all elementary, but when one starts treating singular spaces in this fashion, interesting math occurs! We will first focus on orbifolds, and review how this point-of-view leads to an essentially injective functor between orbifolds and differentiable (local) semi-algebraic varieties. As an application, we use this to prove that a symplectic reduced space of a Hamiltonian circle action is never diffeomorphic to an orbit space of a Lie group action, unless it is an orbifold. Moreover, it is only ever an orbifold if its dimension is at most 2, or if the reduction is performed at a regular value of the momentum map.

On the lack of density of Lipschitz mappings in Sobolev spaces with model filiform target

Derek Jung   [email] (UIUC Math)

Abstract: In 2014, Dejarnette, Hajlasz, Lukyanenko, and Tyson published a paper with the same title, except replacing model filiform with Heisenberg. They prove Lipschitz mappings $\mathbb{D}\to \mathbb{H}^1$ are not dense in the Sobolev space $W^{1,p}(\mathbb{D},\mathbb{H}^1)$ when $1\le p<2$. In this talk, I will present my efforts in translating their result from the Heisenberg group to the model filiform groups, which can be realized as the class of jet spaces $J^k(\mathbb{R},\mathbb{R})$. I will begin by briefly describing the structure of the Heisenberg group and discussing the main parts of their proof. After introducing the model filiform groups, I will give an example of a biLipschitz embedding of $\mathbb{S}^1$ into $J^k(\mathbb{R},\mathbb{R})$ for each $k$. I will conclude by remarking on the main obstacle to proving the non-density result: the regularity problem for the Carnot-Caratheodory metric.

On tame expansions of the group of integers

Gabriel Conant (Notre Dame Math)

Abstract: We discuss some recent work concerning expansions of the group $\mathbb{Z}$ of integers, which are tame with respect to model theoretic dp-rank. Our focus is on the ordered group of integers (also called Presburger arithmetic), which is a well-known example of a dp-minimal expansion of $\mathbb{Z}$. It was asked by Dolich et. al. whether every dp-minimal expansion of $\mathbb{Z}$ is a reduct of Presburger. We present a result in the opposite direction: there are no intermediate structures strictly between the group of integers and Presburger arithmetic. The proof of this result uses Cluckers' cell decomposition for Presburger sets, as well as work of Kadets on the geometry of convex polyhedra.

Distribution of the periodic points of the Farey map

Byron Heersink   [email] (UIUC )

Abstract: A result of Series established a cross section of the geodesic flow in the tangent space of the modular surface which provided a lucid explanation of the connection between the geodesics in the modular surface and continued fractions. Pollicott later utilized this connection to show the limiting distribution of the periodic points of the Gauss map, i.e., the periodic continued fractions, when ordered according to the length of corresponding closed geodesics. In this talk, we outline how to extend the work of Series and Pollicott to obtain results for the Farey map. In particular, we expand the cross section of Series so that the return map under the geodesic flow is a double cover of the Farey map's natural extension. We then show how to adapt the method of Pollicott, which uses the analysis of a certain nuclear operator on the disk algebra, to prove an equidistribution result for the periodic points of the Farey map.

Stability theorems for graphs without long cycles

Ruth Luo (Illinois Math)

Abstract: We show stability versions of two Turán-type theorems for graphs without long cycles. The first is a theorem of Erdős from 1962 which gives an upper bound for the number of edges in a nonhamiltonian graph with prescribed minimum degree. A sharpness example $H_{n,d}$ is provided. The second is a theorem by Erdős and Gallai from 1959 which gives an upper bound on the number of edges in a graph with circumference less than $k$ (for some fixed $k$). The strongest sharpening of the Erdős-Gallai theorem was due to Kopylov who also provided sharpness examples $H_{n,k,t}$ and $H_{n,k,a}$. We show that 1. any 2-connected nonhamiltonian graph with minimum degree at least $d$ and "close" to the maximum number edges is a subgraph of $H_{n,d}$, and 2. any 3-connected graph with circumference less than $k$ and "close" to the maximum number of edges is a subgraph of either $H_{n,k,t}$ or $H_{n,k,2}$. This is joint work with Zoltán Füredi, Alexandr Kostochka, and Jacques Verstraëte.

The integrality conjecture and the Kac positivity conjecture

Ben Davison (EPFL)

Abstract: Without assuming knowledge of any incarnation of Donaldson-Thomas theory, I'll give an introduction to the categorified version of it. I'll also explain what this upgrade of DT theory has to do with proving positivity conjectures, via my favourite example: the Kac positivity conjecture (originally proved by Hausel Letellier and Villegas), stating that polynomials counting absolutely indecomposable representations of quivers over F_q have positive coefficients.

Lecture 1. What is it about the plane?

Vaughan Jones (Vanderbilt University)

Abstract: It is natural to think that combinatorial problems become simpler if the objects involved are supposed to be planar. The isomorphism problem for graphs for instance. But in fact the restriction of planarity can make problems harder. Much harder. Take Sudoku for instance. We will give a couple of problems about contracting tensors which are unsolvable in the planar version but relatively easy without the planar restriction. In the process we will develop a general framework for manipulating planar diagrams.