Colloquium
Reducibility of the Fermi surface for periodic quantum-graph operators
When
February 15, 2019 | 2:30 - 3.30 p.m.
Where
Campbell Hall 443
Speaker
Dr. Stephen Shipman, LSU
Abstract
The Fermi, or Floquet, surface for a periodic operator at a given energy level is an algebraic variety that describes all complex wave vectors admissible by the periodic operator at that energy. Its reducibility is intimately related to the construction of embedded eigenvalues supported by local defects. The rarity of reducibility is reflected in the fact that a generic polynomial in several variables cannot be factored. The "easy" mechanism for reducibility is symmetry. However, reducibility ensues in much more general and interesting situations. This work constructs a class of non-symmetric periodic Schrödinger operators on metric graphs (quantum graphs) whose Floquet surface is reducible. The graphs in this study are obtained by coupling two identical copies of a periodic quantum graph by edges to form a bilayer graph. Reducibility of the Floquet surface for all energies ensues when the coupling edges have potentials belonging to the same asymmetry class, that is, when their "spectral A-functions" are identical. If the potentials of the connecting edges belong to different asymmetry classes, then typically the Floquet surface is not reducible. Bilayer graphene is a notable exception—its Floquet surface is always reducible.
Supergeometry: introduction and some applications
When
February 8, 2019 | 2:30 - 3.30 p.m.
Where
Campbell Hall 443
Speaker
Dr. Theodore Voronov, University of Manchester
Abstract
Supergeometry owes its name to supersymmetry, which is a (theoretical) symmetry mixing bosons and fermions in physics. Supersymmetric models appeared in 1970s, but "super" ideas have much deeper roots both in mathematics and physics e.g. in the works of Hermann Graßmann and Élie Cartan. The pioneer of supermathematics was Felix Alexandrovich Berezin. His discovery of what we now call "Berezin integral" was used by Faddeev and Popov for quantization of gauge fields even before the official birth of supermathematics.
Supergeometry provides powerful tools for "ordinary" mathematics and mathematical physics, and has led to spectacular applications such as analytic proof of the Atiyah-Singer index theorem based on supermanifold quantization. It possesses a unifying power, making it possible to see e.g. Clifford algebra and differential operators as basically the same thing. Recent applications of supergeometry are related with symmetry structures "up to homotopy." This includes the proof of the existence of deformation quantization of Poisson manifolds due to Kontsevich.
I will give an introduction to the main ideas of supergeometry with examples (including my own work). No prior knowledge will be assumed. I hope to touch on some of my recent works, such as volumes of classical supermanifolds (originated from my counterexample to a conjecture by Witten), differential operators on the superline, and (time permitting) an entirely new idea of "microformal geometry."
An overview of numerical algorithms for the Poisson-Boltzmann equation in biomolecular electrostatics
When
February 1, 2019 | 2:30 - 3.30 p.m.
Where
Campbell Hall 443
Speaker
Dr. Shan Zhao, University of Alabama
Abstract
The Poisson-Boltzmann Equation (PBE) is a widely used implicit solvent model for the electrostatic analysis of solvated biomolecules. The numerical solution of the PBE is known to be challenging, due to the consideration of discontinuous coefficients, complex geometry of protein structures, singular source terms, and strong nonlinearity. In this talk, I will offer a brief overview of recent studies in the literature as well as new developments in our group for resolving the PB numerical difficulties.
- For treating dielectric interface and complex geometry, both finite element methods and Cartesian grid finite difference methods have been developed for delivering a second order accuracy in space.
- In the framework of pseudo-time integration, we have constructed an analytical treatment to suppress the nonlinear instability.
- For treating charge singularity in solvated biomolecules, we have introduced a new regularization approach, which combines the efficiency of two-component schemes with the accuracy of the three-component methods. Finally, numerical experiments of several benchmark examples and free energy calculations of protein systems are presented to demonstrate the stability, accuracy, and efficiency of the new algorithms.
Tanglegrams and tanglegram crossing numbers
When
January 25, 2019 | 2:30 - 3.30 p.m.
Where
Campbell Hall 443
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.
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.
Decidability of Thurston equivalence
When
October 5, 2018 | 2:30 - 3:30 p.m.
Where
Campbell Hall 443
Speaker
Dr. Nikita Selinger, UAB Math. Department
Abstract
In the early 1980’s Thurston proved a prominent theorem in the field of Complex Dynamics. His characterization theorem provides a topological criterion of whether a given Thurston map (i.e. a topological map with finite combinatorics) can be realized by a rational map. In a joint work with M. Yampolsky and K. Rafi, we produce an algorithm of checking whether two Thurston maps are equivalent.
Using Computer Animations to Help Teach Mathematics
When
September 28, 2018 | 2:30 - 3:30 p.m.
Where
Campbell Hall 443
Speaker
Mr. Michael Pogwizd
Abstract
In this presentation, I share a collection of online images, animations, and videos designed to help students better understand mathematical concepts, ranging from high-school algebra to real analysis. Depending on their level of difficulty, chalk-board explanations of these concepts can require 10-20 minutes.
The visuals introduced in this presentation will do three things: 1), they will greatly reduce the amount of time needed to explain concepts; 2) they will increase the students’ understanding of the concepts; and 3) they will make learning math entertaining – not just an end unto itself, but a great way to improve retention.
My goal is to demonstrate to members of the Math Department the advantages of using these visuals in teaching and tutoring. All animations are available to use for free and can be found on my UAB website.
Spectral Theory for Systems of Ordinary Differential Equations with Distributional Coefficients
When
September 14, 2018 | 2:30 - 3:30 p.m.
Where
Campbell Hall 443
Speaker
Rudi Weikard, Chair and Professor, Department of Mathematics, University of Alabama at Birmingham
Abstract
We discuss the spectral theory of the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. We do not require the definiteness condition customarily made on the coefficients of the equation.
Specifically, we construct associated minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we construct Green's function and prove the existence of a spectral (or generalized Fourier) transformation.
Classical Matrix Inequalities and their Extensions
When
May 2, 2018 | 2:30 - 3:30 p.m.
Where
Campbell Hall 443
Speaker
Tin-Yau Tam, Chair and Professor, Department of Mathematics and Statistics, Auburn University
Abstract
We will discuss some classical matrix inequalities and their extensions including Schur-Horn inequalities, Sing-Thompson’s inequalities, Weyl-Horn’s inequalities, Bhatia’s inequality etc. Most of them are related to my new book Matrix Inequalities and Their Extensions to Lie Groups.
Floating Mats and Sloping Beaches: Steklov Problem on Domains with Corners
When
April 13, 2018 | 2:30 - 3:30 p.m.
Where
Campbell Hall 443
Speaker
Leonid Parnovski, University College London
Abstract
I will discuss recent results on the asymptotic behaviour of eigenvalues of Steklov operators on domains with corners. These results are rather surprising: the asymptotics depends on the arithmetic properties of the corners.
Some Sylvester-type Matrix Equations and Tensor Equations over the Quaternion Algebra
When
April 6, 2018 | 2:30 - 3:30 p.m.
Where
Campbell Hall 443
Speaker
Zhuo-Heng He, Auburn University
Abstract
Sylvester-type equations have many applications in neural network, robust control, output feedback control, the almost noninteracting control by measurement feedback problem, graph theory, and so on. In this talk, we consider some Sylvester-type matrix equations and tensor equations over the quaternion algebra. We present some necessary and sufficient conditions for the solvability to these Sylvester-type matrix equations and tensor equations over the quaternion algebra. Moreover, the general solutions to these quaternion matrix equations and tensor equations are explicitly given when they are solvable. We also provide some numerical examples to illustrate our results.
Elementary and Advanced Perspectives of Measurement and Ratio
When
March 30, 2018 | 2:30 - 3:30 p.m.
Where
Campbell Hall 443
Speaker
James Madden, Louisiana State University
Abstract
Measurement, ratio, and proportion are topics in elementary school mathematics, yet there are profound connections to current research in algebra and analysis, e.g., the theorem of Hölder on archimedean totally-ordered groups, the Yosida Representation Theorem for archimedean vector lattices, and my own work interpreting the Yosida Theorem in point-free topology. In this talk, I will trace the history of ratio from Eudoxus to "point-free Yosida", with stops along the way to examine interactions between academic mathematics and the mathematics taught in school.
Decomposition Towers and their Forcing
When
March 23, 2018 | 2:30 - 3:30 p.m.
Where
Campbell Hall 443
Speaker
Alexander Blokh, UAB and Michal Misiurewicz, IUPUI, Indianapolis
Abstract
We define the decomposition tower, a new characteristic of cyclic permutations. A cyclic permutation π of the set N = {1,…,n} has a block structure if N can be divided into consecutive blocks permuted by π. The set N might be partitioned into blocks in a few ways; then those partitions get finer and finer. Decomposition towers reflect the variety of sizes of blocks of such partitions. Set
4 >> 6 >> 3 >> … >> 4n >> 4n + 2 >> … >> 2 >> 1,
define the lexicographic extension of >> onto towers, and denote it >> too. We prove that if N >> M and an interval map f has a cycle with decomposition tower N then f must have a cycle with decomposition tower M. The results are joint with Michal Misiurewicz (IUPUI, Indianapolis), inspired by the Sharkovsky Theorem, and based upon our (M – B) recent results.
Optimal Quantization
When
February 9, 2018 | 2:30 - 3:30 p.m.
Where
Campbell Hall 443
Speaker
Mrinal Kanti Roychowdhury, The University of Texas Rio Grande Valley
Abstract
The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Though the term 'quantization' is known to electrical engineers for the last several decades, it is still a new area of research to the mathematical community. In my presentation, first I will give the basic definitions that one needs to know to work in this area. Then, I will give some examples, and talk about the quantization on mixed distributions. Mixed distributions are an exciting new area for optimal quantization. I will also tell some open problems relating to mixed distributions.