Brown Bag Seminar
General Elementary Direct Proof of Fermat’s Last Theorem
For centuries, the Fermat’s last theorem (FLT) has eluded mathematicians and challenged mathematical theory. It was until 1995 a successful proof by contradiction was published by Andrew Wiles using many modern mathematical techniques developed since Fermat. However, only a few mathematicians are knowledgeable and able to comprehend this proof. Another question has arisen whether there is a simpler proof for FLT, especially one with the knowledge at the time of Fermat. Most attempted FLT proofs and Wiles’ proof have relied on the proof by contradiction. However, this proof presents a general elementary direct proof of FLT for all positive n integers. Zoom: https://arizona.zoom.us/j/810942010
Quantum Open Systems
Via concrete models I will present the procedures necessary to obtain the dynamics of an open quantum system. Under appropriate assumptions one can arrive at a Markovian equation which encapsulates non-Unitary dynamics responsible for dissipation, decoherence or both. This need not always be the case however, in general the equation encapsulating the dynamics of a quantum open system will be a non-Markovian equation, we talk a bit about this and the hurdles. Zoom: https://arizona.zoom.us/j/810942010
Solving Maxwell's equations using FDTD and the Piola transformation
FDTD has proved to be a robust way for numerically solving Maxwell's equations since it was introduced by Kane Yee in 1966. Approximating non rectangular domains with rectangular meshes causes staircasing and as a consequence the accuracy of the numerical solution deteriorates. Madsen and Ziolkowski proposed a way to implement FDTD for general 3D domains that suffers from late time instability. We describe a potential way to repair this instability. For this purpose we employ the Piola transformation which allows us to map cells from the physical space to the reference cell in a manner that preserves fluxes and circulations, a useful property regarding Maxwell's equations. We briefly describe the Piola transformation and also the issue of orienting the edges of the mesh in the 2D case. Zoom: https://arizona.zoom.us/j/810942010
Measuring Similarities in Data using Reeb Graphs and the Interleaving Distance
Suppose you are the observer of a single hilly landscape. If you were asked to determine which features were deemed "important" or "necessary", you might be inclined to point out positions of high speaks or low troughs (perhaps even bridges between them) while ignoring the placement of individual rocks on the hillside or specific details of the diameters of the peaks. How can we summarize this "important" information? We can first imagine that the landscape is being flooded with water. As the water level rises, we are able to see that low peaks and bridges become submerged quickly and soon only the highest peaks are left. It might become impossible to move from one peak to another without traversing water. What this allows us to observe is how the connectedness of the landscape changes while ignoring all the minute features. The Reeb Graph is the quotient space where two values are equivalent if they lie on the same level set and are in the same connected component. It contracts each connected component of level sets to single points which ultimately tracks how the connectedness of our flooded landscape changes while ignoring any volumetric data of these peaks. This data structure provides a massive decrease in the amount of information that is stored while still representing the important features of the data. Now how can we measure the similarity between these Reeb Graphs and what does it tell us about the landscapes that they emerge from? The interleaving distance provides a rigorous mathematical definition of "approximate isomorphisms" between Reeb Graphs by categorifying the structures. Thankfully, the interleaving distance also has a nice geometric realization which can be thought of as the amount of "thickening" of the Reeb Graphs needed to be done to create isomorphisms. Zoom: https://arizona.zoom.us/j/810942010
Development of COVID-19 exposure models for first responder and healthcare scenarios
Microbial exposure models describing transfer of microbes to and from surfaces during hand-to-surface contacts have been used in quantitative microbial risk assessment (QMRA) to understand how interactions with everyday surfaces affect infection risk. This approach requires multidisciplinary dialogue between engineers, public health researchers, mathematicians, and microbiologists. In response to COVID-19, specifically, our multidisciplinary teams comprised of researchers from University of Leeds, England, UK; the Ohio State University, Columbus, OH, USA; and the University of Arizona are currently developing microbial exposure models estimating coronavirus exposures for first responders and for healthcare workers, accounting for procedure-specific sequences of human behaviors, contacts with surfaces in patient rooms, and virus deposition on surfaces. The goals of these models are to gain mechanistic insight into how the virus may be spread in healthcare scenarios, to inform surface disinfection protocols, and to inform strategies for maximizing personal protective equipment (PPE) use, such as when to change out gloves, respirators, or gowns. These models can then be used to estimate infection burden and inform epidemiological models on the population level with estimates on larger time scales. These models can also serve as an education tool, where in training videos we can communicate the importance of interventions in quantitative, risk mitigation terms. Future collaboration with applied mathematicians is needed to advance current exposure modeling approaches, integrate microbial exposure models with epidemiological model frameworks, and explore novel approaches for handling uncertainty in exposure mechanisms. Place: Zoom Session https://arizona.zoom.us/j/810942010
Multi-Level Graph Spanners
I am Reyan Ahmed, 4th year Ph.D. student of the Department of Computer Science. My advisor is Professor Stephen Kobourov. We are working on graph spanners. In this problem, we approximately preserve the distance between every pairs of vertices of a graph to get space and computational efficiency. We have generalized this problem to multi-level graph spanners (MLGS).
Wiener filtering and data-driven model reduction
Microlocally correct photoacoustic reconstructions from spatially reduced data
We investigate the inverse source problem for the wave equation arising in photoacoustic and thermoacoustic tomography. If one assumes that the speed of sound is a known constant, then there are many previously developed inversion formulas for the solution of this problem. These formulas require that the data is measured on a closed surface completely surrounding the object or on certain unbounded surfaces. However, in many practical applications, data can only be measured on a finite open surface. In this talk, we will present a non-iterative approach to this problem that yields an approximate solution under certain geometric restrictions on finite open surfaces. This solution coincides with the true solution microlocally--that is, the error is a smooth function. In practical applications, such reconstructions result in qualitatively correct images. For the case of a circular acquisition surface, our method can be implemented with explicit, asymptotically fast methods. We demonstrate the work of these algorithms in a series of numerical simulations.
New analytic chain soliton and self-similar solutions to the Kadomtsev-Petviashvili equation
The KP (Kadomtsev-Petviashvili) equation is probably the most studied (2+1)-dimensional integrable wave equation of the known (2+1)-dimensional integrable models. The KP equation is often used with positive dispersion (denoted as KP-2), to describe weakly nonlinear shallow water waves with one dominant wave vector, and the KP equation with negative dispersion (denoted as KP-1) gives a one parameter group of time dependent potentials corresponding to the Schrodinger equation.
In this work, the exact nonlinear instability of the KP-1 equation is described where an unstable soliton decays into a slow soliton and fast ‘chain’ soliton. Certain aspects of the behaviour of the solutions are explored, such as, branching of the periodic chain solitons. And finally, the self-similar solution to the KP equation is constructed and the resulting ODE is related to the cylindrical KDV equation as well as the Painleve integrability criteria.
Scheduling of Ticket Inspectors in Deutsche Bahn Inter-City Express Trains
We study the underdeveloped Inspector Scheduling Problem. The Deutsche Bahn want to maximise the number of passenger tickets inspected given already set train schedules with their limited resources, the passenger ticket inspectors. We formulate this problem as a network flow and implement this as a Mixed Integer program. We account for lack of data by estimating Origin-Destination Matrices and discuss the complexity of the problem. We obtain empirical results of run time of the algorithm as a function of number of inspectors and conjecture that this problem is NP-hard. We also propose heuristic methods to solve this problem