Department of Mathematical Sciences
Colloquium Series
Purpose: To discuss mathematical research and interdisciplinary problems in all areas of mathematics and applied mathematics. Talks are welcome from faculty, graduate students, and outside speakers from academia and industry. Talks should be at a level accessible to graduate students. Students and faculty at all levels and from all departments are welcome to attend.
Date/Time | Speaker | Title/Abstract |
Friday, May 20
3-4pm The MAC |
Dr. Steven J. Miller
Department of Mathematics and Statistics Williams College |
Title: Results and Research in REUs: Cookie Monster Meets the Fibonacci Numbers. Mmmmmm -- Theorems!
Abstract: After discussing some of my experiences guiding over 300 undergradutes in research projects over the past 20 years, I'll discuss some joint work with students at the Williams SMALL REU program. A beautiful theorem of Zeckendorf states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers. Once this has been shown, it's natural to ask how many Fibonacci numbers are needed. Lekkerkerker proved that the average number of such summands needed for integers in [F_n, F_{n+1}) is n / (phi^2 + 1), where phi is the golden mean. We present a combinatorial proof of this through the cookie problem and differentiating identities, and further prove that the fluctuations about the mean are normally distributed and the distribution of gaps between summands is exponentially decreasing. These techniques apply to numerous generalizations, which we'll discuss as time permits. The only background required is some elementary knowledge about cookies. |
Friday, September 23
3:00-4:30pm The MAC |
Dr. Ugur G. Abdulla
Department of Mathematical Sciences Florida Institute of Technology |
AMS Graduate Chapter at FIT: Opening Ceremony and Colloquium Title: Topological Dynamics, Sharkovski Ordering and Feigenbaum Universality in ChaosAbstract: This lecture will give an introduction to topological dynamics of continuous maps on the interval. Sharkovski's striking theorem on the coexistence of periodic orbits and its connection to celebrated Feigenbaum-Collet-Eckmann scenario of the transition from periodic to chaotic behavior of unimodal maps will be discussed. Recent advances made by my research group on Dynamical Systems and Chaos Theory on the fine classification of the periodic orbits through directed graphs and cyclic permutations, and fascinating order on the distribution of periodic windows in the chaotic regime for the nonlinear iterated maps on the interval will be reported. I will discuss some open problems in the field. Talk will be accessible to graduate and advanced undergraduate students. |
Friday, October 21
3:00-4:00pm The MAC |
Dr. Roberto Peverati
Chemistry Department Florida Institute of Technology |
Title: How Mathematics and Chemistry Can Work Together to Understand and Predict Nature
Abstract: In this seminar we are going to explore how theoretical and computational chemistry are becoming crucial tools to understand chemical molecules and reactions, and how they are linked to applied mathematics. We will discuss about the quantum mechanics foundations, the many rigorous attempts to approximate the Schrödinger equation, and about the semi-empiricisms that are necessary to apply the theories to such a complex science like chemistry. We will also see how mathematics and information science can help to shape the future of this field, and how we plan to develop a new research program based on these premises in the Chemistry Department here at FIT. This seminar aims to be the starting point for an open discussion between the two Departments, so that students, researchers, and professors can benefit from the sharing of ideas and projects, and ultimately advance their research opportunities in both fields. |
Friday, January 20
3:00-4:00pm The MAC |
Dr. Ugur G. Abdulla
Department of Mathematical Sciences Florida Institute of Technology |
Title: Recent Advances on Optimal Control of Parabolic Free Boundary Problems - Invitation to Research
Abstract: This talk presents recent advances on inverse Stefan type free boundary problems, where some of the coefficients of the PDE or some boundary data on the fixed boundary is missing and must be found along with the temperature and free boundary. I discuss both one-phase and multiphase cases. In one-phase case optimal control framework is employed where missing data and free boundary are components of the control vector. Multiphase optimal Stefan problem is reduced to optimal control problem for quasilinear PDE with discontinuous coefficient. We prove the well-posedness of the optimal control problem and the convergence of the sequence of the discrete optimal control problems to the original problem both with respect to cost functional and optimal control. We prove Frechet differentiability in Besov spaces, necessary condition for optimality, Pontryagin type maximum principle under minimal regularity assumptions on the data. I will formulate some open problems for PhD and postdoctoral research. |
Friday, January 27
3:00-4:00pm The MAC |
Dr. Martin E. Glicksman
College of Engineering Florida Institute of Technology |
Title: Pattern Formation: Uncovering Interface Perturbation Fields
Abstract: Solid-liquid microstructures support 4th-order capillary-mediated perturbation fields that locally add or withdraw small amounts of energy. These perturbations importantly self-interact with their interfaces to modulate their motion, and, ultimately, alter their shape and stimulate complex pattern formation. The details of these fields depend sensitively on interface geometry and the anisotropy of interfacial energy. The actual presence of these heretofore undetected Poisson fields was recently exposed by using entropy-functional multiphase-field methods to uncover and measure them. Their intensity distributions were determined by simulating stationary solid-liquid interfaces configured as grain boundary grooves constrained in a uniform thermal gradient. Energy source fields resident on such microstructures were exposed and measured as interface “residuals”, found by subtracting the linear component imposed on non-linearly perturbed-interface potentials. Perturbation source fields so revealed entail persistent self-induced cooling (energy withdrawal) everywhere along groove profiles, a result that confirms prior analytical predictions based on sharp-interface thermodynamics and the calculus of variations. Phase-field simulations provide independent quantitative support for the active presence of cooling fields on grooved solid-liquid interfaces. This study is directed to the long-standing question of what stimulates the formation of microstructures–stochastic or deterministic signals? It also provides a basic understanding of diffusion-limited pattern formation dynamics in nature and technology, possibly extensible to biological morphogenesis. |
Friday, March 3
3:00-4:00pm The MAC |
Dr. Stanley Snelson
University of Chicago |
Title: Regularity theory for the inhomogeneous Landau equation
Abstract: We will begin by introducing the Landau equation, an integro-differential kinetic model from plasma physics that describes the evolution of a particle density in phase space, in a regime where grazing collisions predominate. We will then discuss recent progress on existence and regularity theory for the spatially inhomogeneous Landau equation, and how this relates to the corresponding theory for the Boltzmann equation and its variants. Finally, we will present recent results, obtained in collaboration with Cameron-Silvestre, on global a priori estimates for weak solutions. Our pointwise upper bounds and Holder estimates, which improve polynomially as the velocity increases, are an important step toward proving the conjecture that solutions remain bounded and smooth as long as the hydrodynamic quantities (mass, energy, and entropy densities) are under control. |
Friday, March 31
3:00-4:00pm The MAC |
Dr. Carlos Martino
Biomedical Engineering Department Florida Institute of Technology |
Title: Optimal Control In Quantum Biology: Experiments and Theory
Abstract: This study presents experimental data and related theoretical calculations for the effects of static and radio frequency magnetic fields in the biological production of reactive oxygen species (ROS). In the first part of the talk, we discuss a mechanism of spin-biochemistry providing a fundamental relation between bioenergetics and ROS product channeling. We present evidence on a fundamentally important domain of Quantum Biology: the role and control of electron spin in the biological production of ROS and ATP. Theoretical analysis considers radio frequency magnetic field effects in a one-proton radical pair model. In the second part of this talk, we pose the statement of the optimal control problem in quantum biology of ROS, namely finding the optimal condition of the intensity and directionality of the external magnetic field for the system to evolve to a desired final state. Future directions will be discussed. |
Tuesday, April 11
2:00-3:00pm OEC 118 |
Tony Robbin
Hosted by SIAM and AMS Student Chapters |
Title: Quasicrystals for Sculpture, Architecture, Physics, and Whatever
Abstract: Quasicrystals are an endlessly fascinating visual mathematics: neither a crazy quilt of randomly placed parts, nor a rigid crystalline structure of predetermined repeats, but something in between. Quasicrystal geometry could be most fruitfully applied to architecture, where without moving parts, structures appear to change from 5-fold, to 3-fold, to 2-fold symmetry as one looks around. There are several open questions about quasicrystals: For physical quasis, What is the theory of their structural stability? For mathematical quasis, How are the different dimensional quasis related to one another? And,What are the hypercubic unit cells of the 4-dimensional quasicrystal? Finally, Can the de Bruijn algorithm be a model of quantum information at a distance? |
Friday, April 14
3:00-4:00pm The MAC |
Dr. Steven Shaw
Department of Mechanical and Aerospace Engineering Florida Institute of Technology |
Title: Designing Vibration Absorbers with Nonlinear Dynamics (using bifurcation theory to improve automotive fuel economy)
Abstract: This presentation will describe the application of nonlinear dynamics in the design of mechanical vibration absorbers. These absorbers consist of vibrational degrees of freedom that are added to a system of interest and tuned such that they passively counteract oscillatory dynamic loads, thereby reducing vibrations of the primary system. The focus here will be on a class of absorbers developed to attenuate torsional (twisting) vibrations in automotive engines, which allow for operation in previously inaccessible torque-speed ranges that offer improved fuel economy. The absorber model consists of a set of nonlinear, identical, globally coupled oscillators that are driven near resonance by engine torque pulses. Depending on absorber and input parameters, this system can exhibit different types of instabilities and bifurcations, many of which stem from the inherent symmetry of the system, and all of which are detrimental to operation. Design of these absorbers for optimal performance requires an understanding of this rich behavior, which will be the main focus of the seminar. Results from mathematical modeling, analysis using perturbation and symmetry techniques, and controlled physical experiments will be presented. |
Friday, April 28
3:00-4:00pm The MAC |
Naveed Iqbal
Department of Mathematical Sciences Florida Institute of Technology |
Title: On the Classification of the Second Minimal Orbits of the Continuous Endomorphisms on the Real Line and Universality in Chaos
Abstract: This talk presents full classification of second minimal odd periodic orbits of a continuous endomorphisms on the real line. A (2k+1)-periodic orbit is called second minimal for the map f, if 2k-1is a minimal period of f in the Sharkovskii ordering. We prove that there are 4k-3 types of second minimal (2k+1)-orbits, each characterized with unique cyclic permutation and directed graph of transitions with accuracy up to inverses. The result is applied to the problem on the distribution of periodic windows within the chaotic regime of the bifurcation diagram of the one-parameter family of unimodal maps. It is revealed that by fixing the maximum number of appearances of the periodic windows there is a universal pattern of distribution. In particular, the first appearance of all the orbits is always a minimal orbit, while the second appearance is a second minimal orbit. It is observed that the second appearance of 2k+1-orbit is a second minimal 2k+1-orbit with Type 1 digraph. The reason for the relevance of the Type 1 second minimal orbit is the fact that the topological structure of the unimodal map with single maximum is equivalent to the structure of the Type 1 piecewise monotonic endomorphism associated with the second minimal 2k+1-orbit. Yet another important report of this talk is the revelation of the universal pattern dynamics with respect to increased number of appearances. |
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