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 January 11 4-5pm Crawford 401 no talk Friday January 18 4-5pm Crawford 401 Dr. Kanishka Perera Florida Institute of Technology Department of Mathematical Sciences Title: Noncompact variational problems I: Basic critical point theory (accessible to undergraduates) Abstract: In this introductory talk I will present some of the basic ideas of critical point theory. Topics will include the deformation lemma, mountain pass theorem, higher minimax schemes, and applications to eigenvalue problems (time permitting). The presentation will be in a finite dimensional setting in order to be accessible to undergraduates. Friday January 25 4-5pm Crawford 401 Dr. Kanishka Perera Florida Institute of Technology Department of Mathematical Sciences Title: Noncompact variational problems II: Scalar field equations Abstract: Friday February 1 4-5pm Crawford 401 Dr. Ugur Abdulla Florida Institute of Technology Department of Mathematical Sciences SIAM Sponsored Colloquium Title: On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations – Invitation to Interdisciplinary Research Abstract: In this talk, we describe a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. This type of inverse problem arose, in particular, in a bioengineering problem on the laser ablation of biological tissues. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consists of the minimization of the sum of L2-norm declinations from the available measurement of the temperature flux on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. We prove well-posedness in Sobolev spaces framework and convergence of discrete optimal control problems to the original problem both with respect to cost functional and control. Friday February 8 4-5pm Crawford 401 Dr. Gnana Tenali Florida Institute of Technology Department of Mathematical Sciences Title: First order Functional Differential Equations with Advance Arguments Friday February 15 4-5pm Crawford 401 Dr. Jim Jones Florida Institute of Technology Department of Mathematical Sciences Title: Parallel multigrid solvers for partial differential equations in coastal ocean simulation Abstract: In this talk, I'll discuss my ongoing joint research with Dr. Steven Jachec in the Marine and Environmental Systems Department. This work is focused on speeding up the calculations in the Stanford Unstructured Nonhydrostatic Terrain-following Adaptive Navier-Stokes Simulator (SUNTANS). This code is used in coastal ocean simulations. A major computational task in this code is solving a large matrix problem, Ax=b, that arises from the discretization of the partial differential equations used to model the system. I'll discuss the need for scalable solvers. In essence, a scalable solver can keep the solution time constant for bigger and bigger problems provided that we also increase the computational resources. That is, a ten fold increase in the number of unknowns in the vector x accompanied by a ten fold increase in the number of computing processors should keep the run time constant. Techniques to attain scalable solvers touch on some mathematical topics dealing with convergence of approximate solutions and some computer science topics dealing with efficient codes for execution on large parallel computers. Friday February 22 4-5pm Crawford 401 Dr. Mark Rahmes Harris Coorporation SIAM Sponsored Colloquium Title: Near Real Time Discovery and Conversion of Open Source Information to a Reward Matrix Abstract: We describe a system model for determining decision making strategies based upon the ability to perform data mining and pattern discovery utilizing open source information to automatically predict the likelihood of reactions to specific events or situational awareness from multiple information sources. Within this paper, we discuss the development of a method for determining actionable information to efficiently propitiate manpower, equipment assets, or propaganda responses to multiple application case scenario experiments. We have connected human sentiment information to other user selectable attributes using open source information and interlink to relative probabilities for different reactions to different events based upon current information and the situation. Direct user input or modifications, additions, or deletions of attributes of interest and their associated probabilities can be modeled for a multitude of scenarios from the reward matrix. The initial step utilizes a decision tree learning method that is used for classification, weighted factors and probabilistic prediction based upon the information obtained from multiple and disparate data sources. Optimal strategies are then calculated to increase the likelihood of making the best decision available using game theory in a constant sum game for a resource allocation scenario. Data fusion and visualization techniques provide the user with a useful tool to interact with the results generating near real time decisions. Our system can be extended to other applications such as course of action planning, strategy, resource management, risk assessment, and behavioral economics, and is not tied to proprietary feeds, inputs, or outputs. Our solution can have multiple algorithms/inputs/outputs based upon user needs and requirements and allow for human interaction with the resulting decision made by the system to show changes based upon different decisions made in near real time. The goal of our system solution, called GlobalSite, is to deliver trustworthy decision making analysis systems and services that understand situations or potential impacts of decisions, while being a vital tool for continuing mission operations, analyst information or other decision making scenarios. Friday March 1 4-5pm Crawford 401 Dr. Benjamin Hutz Florida Institute of Technology Department of Mathematical Sciences Title: What is projective space? Abstract: This is an introductory "What is ______ ?" talk, where a basic area of mathematics is introduced. It is aimed at graduate students and advanced undergraduates. This talk will introduce the notion of projective space which is used in many areas of mathematics and applications including perspective drawing and computer vision. The main application in this talk will be to intersections of curves in the plane (Bezout's Theorem). Friday March 8 4-5pm Crawford 401 no talk spring break Friday March 15 4-5pm Crawford 401 Dr. Ugur Abdulla Florida Institute of Technology Department of Mathematical Sciences Title: Wiener Test at infinity for Elliptic and Parabolic PDEs,and its Measure-Theoretical, Topological, and Probabilistic Consequences Abstract: Wiener's celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results in XX century mathematics. It has shaped the boundary regularity theory for elliptic and parabolic PDEs, and has become a central result in the development of potential theory at the intersection of functional analysis, PDE, and measure theories. This talk describes the new developments which precisely characterize the regularity of the point at $\infty$ for second order elliptic and parabolic PDEs and broadly extend the role of the Wiener test in classical analysis. The Wiener test at $\infty$ arises as a global characterization of uniqueness in boundary value problems for arbitrary unbounded open sets. From a topological point of view, the Wiener test at $\infty$ arises as a thinness criteria at $\infty$ in fine topology. In a probabilistic context, the Wiener test at $\infty$ characterizes asymptotic laws for the characteristic Markov processes whose generator is the given differential operator. The counterpart of the new Wiener test at a finite boundary point leads to uniqueness in the Dirichlet problem for a class of unbounded functions growing at a certain rate near the boundary point; a criteria for the removability of singularities and/or for unique continuation at the finite boundary point. The Wiener criterion at minimal Martin boundary points is a largely unexplored issue of great importance in various disciplines, most notably in Nonlinear Potential Theory, Nonlinear Elliptic and Parabolic PDEs, Calculus of Variations, Topology, and Probability Theory. The talk will end with a description of some open problems in those directions. Friday March 22 3:30-4:30pm Crawford 220 Dr. Adrian Peter Florida Institute of Technology Department of Systems Engineering Title: Information Geometry for Shape Analysis: Probabilistic Models for Shape Matching and Data-Driven Estimation Abstract: Shape analysis is a core area of study in machine learning and computer vision. It is a fundamental building block of higher level cognitive tasks such as recognition. This talk will discuss novel approaches to basic shape analysis tasks, including shape similarity metrics and methods for matching. Our investigations into these methods will also highlight a supporting statistical tool that has general applicability outside the realm of shape analysis: a new wavelet density estimation procedure. All of the techniques are theoretically grounded in the framework of information geometry. Information geometry is an emerging discipline in mathematics that utilizes differential geometry to characterize the space of probability distributions. Within this context, our basic approach to shape analysis is: represent shapes as probability densities and then use the intrinsic geometry of the space of densities to establish geodesics between shapes. Valid intermediate densities (shapes) can be obtained by traversing along the geodesics and the length of the geodesic immediately gives us a similarity measure between shapes. These concepts are illustrated by using a wavelet expansion model to represent probability densities. Beyond shape analysis, information geometry has applications to image analysis, data fusion, document mining and a host of other areas that require distance/divergence measures between probability distributions. Time permitting, I will also highlight other research projects from the Information Characterization and Exploitation (ICE) Lab. Friday March 29 4-5pm Crawford 401 Dr. Brian Moore University of Central Florida SIAM Sponsored Colloquium Title: Bistable Waves in Discrete Inhomogeneous Media Abstract: Bistable differential-difference equations are crude models for conduction of electrical impulses in the nervous system. Employing a piecewise linear approximation of the nonlinearity, one can derive exact solutions of this system such that a portion of the medium for conduction is deteriorated, characteristic of diseases that affect the nervous system. Using Jacobi operator theory and transform techniques, wave-like solutions are constructed for problems with essentially arbitrary inhomogeneous discrete diffusion. A thorough study of the steady state solutions provides necessary and sufficient conditions for traveling waves to fail to propagate due to inhomogeneities in the medium. Solutions with nonzero wave speed are also derived, and we learn how the wave speed and wave form are affected by the inhomogeneities. Friday April 5 4-5pm Crawford 401 Dr. Ugur Abdulla Florida Institute of Technology Department of Mathematical Sciences Title: Sharkovski's Theorem and Universality in Chaos Abstract: Recently, a fascinating order was revealed for the distribution of periodic windows in the chaotic regime for discrete dynamical systems. In this talk I will analyze this phenomenon through classification of directed graphs and cyclic permutations of minimal orbits. Friday April 12 4-5pm Crawford 401 Friday April 19 4-5pm Crawford 401 Friday April 26 Refreshments: 4pm Talk 5-6pm Olin Engineering 118 Dr. Ken Golden Department of Mathematics University of Utah First Annual SIAM Public Lecture Title: Mathematics and the Melting Polar Ice Caps Abstract: In September of 2012, the area of the Arctic Ocean covered by sea ice reached its lowest level ever recorded in more than three decades of satellite measurements. In fact, compared to the 1980's and 1990's, this represents a loss of more than half of the summer Arctic sea ice pack. While global climate models generally predict sea ice declines over the 21st century, the precipitous losses observed so far have significantly outpaced most projections. Dr. Golden will discuss how mathematical models of composite materials and statistical physics are being used to study key sea ice processes and advance how sea ice is represented in climate models. This work is helping to improve projections of the fate of Earth's ice packs, and the response of polar ecosystems. In addition, a video from a 2012 Antarctic expedition where sea ice properties were measured will be shown.

Past Semesters