# Mathematics Colloquium

## Tanglegrams and tanglegram crossing numbers

### When

January 25, 2019 | 2:30 - 3.30 p.m.

### Speaker

Dr. Laszlo Szekely, University of South Carolina

### Abstract

A tanglegram is a graph, consisting of two rooted binary trees of the same size and a perfect matching joining their leaves. A tanglegram layout is a special straight line drawing of this graph, where the leaves of the trees are placed on two parallel lines, the two trees are drawn as plane trees on either side of the strip created by these lines, and the matching is drawn inside the strip. It is desirable to draw a tanglegram with the least possible number of pairs of crossing edges in the strip. This is the Tanglegram Layout Problem. The minimum number of the crossings obtained in this way is the tanglegram crossing number of the tanglegram. The tanglegram is planar, if it has a layout without crossings. The Tanglegram Layout Problem is NP-hard and is well studied in computer science.

Tanglegrams play a major role in phylogenetics, especially in the theory of coevolution. The first binary tree is the phylogenetic tree of hosts, while the second binary tree is the phylogenetic tree of their parasites, e.g. gopher and louse. The matching connects the host with its parasite. The tanglegram crossing number has been related to the number of times parasites switched hosts.

I present a Kuratowski type characterization of planar tanglegrams and a conjecture for a similar finite characterization of some more general tanglegram crossing number problems. This is a joint work with Eva Czabarka and Stephan Wagner.

## Subintervals of (Random Subsets of) Intervals and Entanglement in Quantum Spin Chains

### When

January 18, 2019 | 2:30 - 3.30 p.m.

### Speaker

Dr. Gunter Stolz, UAB

### Abstract

The entanglement entropy of eigenstates provides a measure which can be used to study if an interacting quantum system is localized. In particular, guided by the well known phenomenon of Anderson localization for non-interacting systems, one expects that the exposure of the system to disorder will increase its tendency towards localization. We discuss this in the example of the quantum Ising chain. This simple model allows for a very explicit description of how eigenstate entanglement is reduced by the introduction of disorder. This is part of a joint project with Houssam Abdul-Rahman and Christoph Fischbacher.

## Deep Learning: Is it the Answer to Artificial Intelligence?

### When

November 30, 2018 | 2:30 - 3:30 p.m.

### Speaker

Prof. Nidhal Bouaynaya, Rowan University

### Abstract

Within the field of machine learning, deep learning approaches have resulted in state-of-the-art accuracy in visual object detection, speech recognition, and many other domains including genomics. Deep learning techniques hold the promise of emerging technologies, such as autonomous unmanned vehicles, smart cities infrastructure, personalized treatment in medicine, and cybersecurity. However, deep learning models are deterministic, and as a result are unable to understand or assess their uncertainty, a critical part of any predictive system’s output. This can have disastrous consequences, especially when the output of such models is fed into higher-level decision making procedures, such as medical diagnosis or autonomous vehicles. This talk is divided into two parts. First, we provide intuitive insights into deep learning models and show their applications in healthcare and aviation. We then introduce Bayesian deep learning to assess the model’s confidence in its prediction and show preliminary results on robustness to noise and artifacts in the data as well as resilience to adversarial attacks.

## Isoperimetric Type Inequalities and Geometric Evolution Equations on Riemannian Manifolds

### When

November 9, 2018 | 2:30 - 3:30 p.m.

Dr. Junfang Li

### Abstract

In this talk, we will survey some sharp isoperimetric type inequality and its generalization to quermassintegrals on hypersurfaces in Riemannian manifolds. Most of the talk will be about my joint work with Pengfei Guan (McGill University) over the past 10 years. We will focus on two main aspects of this research line: geometric inequalities as the goal and fully nonlinear PDE techniques as the tool. There have been a lot of active researches on related topics recently. One main idea of recent development is to extend these classical inequalities from Euclidean space to space forms, then to manifolds with warped product structures, and also to space like hypersurfaces in the black hole models from General Relativity. I will also list some of the open problems along this line.

Graduate and undergraduate students are welcome to attend. Most of the talk will be non-technical.

## A Birman-Krein-Vishik-Grubb (BKVG) Theory for Sectorial Operators

### When

November 2, 2018 | 2:30 - 3:30 p.m.

### Speaker

Dr. Christoph Fischbacher

### Abstract

This talk is split in two parts. Firstly, we will review and discuss classical results by Birman, Krein, Vishik, and Grubb on the theory of non-negative selfadjoint extensions of a strictly positive symmetric operator $S$ on a complex Hilbert space. In particular, we will see that any such extension $\widehat{S}$ satisfies $S_K\leq\widehat{S}\leq S_F$, where $S_F$ and $S_K$ are the exceptional Friedrichs and Krein-von Neumann extensions of $S$, respectively. After this, we will discuss what happens if $S$ is perturbed by a skew-symmetric operator of the form $iV$, where $V$ is symmetric such that $A=S+iV$ is sectorial and derive a modified BKVG theory for this case.

## Fixed-point-free Mappings of Tree-like Continua

### When

October 19, 2018 | 2:30 - 3:30 p.m.

Dr. Logan Hoehn

### Abstract

A topological space has the "fixed-point property" if every continuous function of the space to itself has at least one point which is mapped to itself. The well-known Brouwer fixed-point theorem states that for each n, the closed n-dimensional ball in Euclidean space has the fixed-point property. I will survey some further results and questions on the fixed-point property in the theory of compact connected metric spaces.