Mathematics Colloquium

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

When

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

Where

Campbell Hall 443

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.

Where

Campbell Hall 443

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.

Where

Campbell Hall 443

Speaker

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.

Where

Campbell Hall 443

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.

Where

Campbell Hall 443

Speaker

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.


Brief History of the Boltzmann-Sinai Hypothesis

When

October 12, 2018 | 2:30 - 3:30 p.m.

Where

Campbell Hall 443

Speaker

Dr. Nandor Simanyi

Abstract

The Boltzmann-Sinai Hypothesis dates back to 1963 as Sinai’s modern formulation of Ludwig Boltzmann’s statistical hypothesis in physics, actually as a conjecture: Every hardball system on a flat torus is (completely hyperbolic and) ergodic (i. e. ”chaotic”, by using a nowadays fashionable, but a bit profane language) after fixing the values of the obviously invariant kinetic quantities. In the half century since its inception, quite a few people have worked on this conjecture, made substantial steps in the proof, created useful concepts and technical tools, or proved the conjecture in some special cases, sometimes under natural assumptions. Recently I was able to complete this project by putting the last, missing piece of the puzzle to its place, getting the result in full generality. In the talk, I plan to present the brief history of the proof by sketching the most important concepts and technical tools that the proof required.