On-orbit Inspection and Aiding Visual Navigation around Non-cooperative Satellites

Thursday, March 21, 4:00 - 4:50 PM, Crawford Bldg, Room 402

Abstract:

This seminar focuses on research by the NEural TransmissionS (NETS) Lab in deep learning-based computer vision for inspecting satellites on orbit and visual navigation for rendezvous and proximity operations. First is a discussion of past work developing an end-to-end visual navigation and guidance system for autonomous swarms of chaser and observer satellites for use on low-powered on-board computers. Then, ongoing efforts to develop computer vision solutions to improve 3D scene understanding with only a single observer are considered. We use neural radiance fields and accelerated 3D Gaussian splatting to learn scene representations from data captured during an inspection orbit, which enables 3D reconstruction. The 3D model is then exploited for downstream tasks, including improved satellite component detection and tracking using our recently developed algorithm SatSplatYOLO that ensembles object detection predictions over synthetically rendered views of the target satellite. We end with planned next steps and barriers to use.

Structured Modeling of High-Dimensional Vector Autoregressions

Xianqi Li, Ph. D | Department of Mathematics & Systems Engineering, Florida Institute of Technology

Thursday, March 7, 4:00 - 4:50 PM, Crawford Bldg, Room 402

Abstract:

Network modeling of high-dimensional time series data is a crucial learning task across various application areas, including macroeconomics, finance, and healthcare. While the problem of sparse modeling based on vector autoregressive models (VAR) has been investigated in depth in the literature, more complex network structures that involve low rank and group sparse components have received considerably less attention, despite their presence in data. Failure to account for low-rank structures results in spurious connectivity among the observed time series, which may lead practitioners to draw incorrect conclusions about critical scientific questions. In order to accurately estimate a network of Granger causal interactions after accounting for latent effects, I will present a novel approach for estimating low-rank and structured sparse high-dimensional VAR models by introducing a regularized framework that involves a combination of nuclear norm and lasso (or group lasso) penalties. Then, I will discuss the obtained nonasymptotic probabilistic upper bounds on the estimation error rates of the low-rank and the structured sparse components. Meanwhile, I will introduce a fast algorithm for the formulated optimization problems.  Finally, I will demonstrate the performance of the proposed modeling framework through numerical experiments. 

Stochastics in medicine: Delaying menopause and missing drug doses

Sean Lawley, Ph.D |  Mathematics Department, University of Utah

Thursday, February 15, 4:00 - 4:50 PM, Crawford Bldg, Room 402

Abstract:

Measure Tracing in Systems Engineering

Karen Holness, Ph.D | Department of Mathematics & Systems Engineering, Florida Institute of Technology

Thursday, February 8, 4:00 - 4:50 PM, Crawford Bldg, Room 402

Abstract:

This presentation provides an overview of systems engineering (SE) and discusses the importance of system performance measures to the successful execution of an SE project. The process of formulating, evaluating, and tracing specific measures across the various lifecycle phases of a system of interest will be discussed. The variety of data collection and analysis methods used to evaluate these measures will be highlighted.   

Surjections between Euclidean spaces, changing variable formula and Brouwer fixed point theorem

Shibo Liu, Ph.D | Department of Mathematics & Systems Engineering, Florida Institute of Technology

Thursday, January 25, 4:00 - 4:50 PM, Crawford Bldg, Room 402

Abstract:

The talk will be accessible to students who have taken multivariable calculus and linear algebra. We briefly review differential calculus and the inverse function theorem. Aiming at giving another calculus proof of the fundamental theorem of algebra, we present our result on the question: when will a map between Euclidean spaces (of different dimension) be surjective. Then, using divergence theorem and some well-known results in linear algebra (including the Cauchy-Binet formula) we prove the changing variable formula for multiple integrals without cutting the domain into small pieces and estimate their volume. As an easy byproduct of our proof, we obtain the Brouwer fixed point theorem.

Recent progress on the well-posedness theory of the Boltzmann equation

Stanley Snelson, Ph.D | Department of Mathematics & Systems Engineering, Florida Institute of Technology

Thursday, January 11, 4:00 - 4:50 PM, Crawford Bldg, Room 402

Abstract:

The non-cutoff Boltzmann equation is a fundamental kinetic model in thermal and statistical physics. Mathematically, it features the interaction of transport with nonlocal, nonlinear diffusion. In this talk, after introducing the model, we will discuss some current research trends that are partly motivated by the difficult open problem of global existence for large initial data. The second half of the talk will focus on a recent result (obtained in collaboration with C. Henderson and A. Tarfulea) that constructs local classical solutions with initial data that is very general in terms of regularity, decay, and the presence of vacuum regions.

Teaching Math Best Practices

Xiomara Gezuragam, Ph.D | Basque Department of Education

Wednesday, November 29, 1:00-1:50 pm, Crawford Bldg, Room 210

The Basque Department of Education and its staff are committed to preparing and implementing changes to make the education system more active, meaningful, and motivating for students. The difficulties that mathematics present to students are many, such as excessive abstraction. However, I think we need to look for and explore ways to make mathematics teaching and learning more real, meaningful, and above all motivating. The reason for my visit to Florida Tech this Fall has been to explore strategies (good practices) to help accomplish this.

This presentation will begin with a comparison of the national standards for 5th grade mathematics of the United States and the state of Florida with the Basque standards, and then explore both teaching methods and methods for assessment and grading.

Systems Engineering and Analysis of Electro-Optic and Infrared Systems: A Systems Engineering Perspective on Unconventional Imaging Research

William W. Arrasmith, Ph.D | Department of Mathematics & Systems Engineering, Florida Institute of Technology

Monday, November 20, 1:00-1:50 pm, Crawford Bldg, Room 210

Abstract:

With the rapid advancement of technologies such as high-speed processing hardware and software paradigms, advanced communication protocols, cloud-based storage, distributed processing, the Internet of Things (IoT), big data, machine-learning, data mining, and artificial intelligence, our world and systems in general are becoming ever more complicated and complex. Systems engineering provides the means for efficiently and robustly defining, designing, developing, implementing, supporting and eventually retiring/disposing of systems of all kinds, from small and simple, to Large, Dynamic, Complicated/Complex, and/or Adaptive systems. Systems Engineering can also be tailored and applied to medium or small systems, services, processes, and/or products and also has applicability in education and research. As such, and as an example, I present my research in Unconventional Sensing methods, with emphasis on Unconventional Imaging research as viewed through the lens of systems engineering. The particular area of emphasis for the research component of the talk is that for an atmopsheric turbulence compensation (ATC) algorithm that provides near diffraction – limited spatial resolution for optical imaging systems. The Well-Optimized Linear Finder (WOLF) algorithm, developed by William W. Arrasmith, at FIT applies to general linear, shift-invariant systems, has the potential for real-time (greater than 30 Hz) ATC for a wide class of imaging sensors, is software-dominant (highly protable), scales well with parallel processing technology, and can be implemented on a variety of computational platforms. The WOLF algorithm has potential for extension to colored imagery, hyperspectral, ultraspectral, and polarimetric imaging systems, and is applicable to coherent imaging systems (e.g., laser-based), and incoherent imaging systems (e.g., natural illumination like sun, moon, and stars).

An introduction to the mountain pass theorem

Kanishka Perera, Ph.D | Department of Mathematics & Systems Engineering, Florida Institute of Technology

Monday, November 6, 1:00-1:50 pm, Crawford Bldg, Room 210

Abstract:

This is an expository talk on the mountain pass theorem of Ambrosetti and Rabinowitz. The goal of the talk is to introduce this extremely useful tool from critical point theory to undergraduate students in the context of some very simple calculus problems. The talk will be accessible to students who have taken multivariable calculus and differential equations.

Spectral theory for electromagnetic scattering of slow light in 1D photonic crystals

Aaron Welters, Ph.D | Department of Mathematics & Systems Engineering, Florida Institute of Technology

Monday, October 23, 1:00-1:50 pm, Crawford Bldg, Room 210

Abstract:

For one-dimensional (1D) photonic crystals (PCs) with passive lossless nondispersive media, Maxwell’s equations for time-harmonic electromagnetic fields become, under separation of variables, linear periodic differential-algebraic equations (DAEs) in canonical form. In this talk, we discuss how the electrodynamics of 1D PCs is reduced to studying the spectral theory (with frequency as the spectral parameter) of the associated minimal and maximal differential-algebraic operators of these DAEs. Our recent results [22BW] (based on the 2022 PhD dissertation of my student, Bader Alshammari) will be presented on the self-adjointness of the maximal operator and its eigenvalues. We use this eigenvalue problem to highlight the differences in the spectral theory of canonical DAEs vs ODEs. To motivative the need for this spectral theory, we also discuss how it applies to our previous work [16SW] on scattering problems involving slow and frozen light when a defect is introduced into such photonic crystals.

[22BW] B. Alshammari and A. Welters, On the spectral theory of linear differential-algebraic equations with periodic coefficients, arXiv:2211.02134 [math-sp] (preprint, 2022). doi: 10.48550/arXiv.2211.02134 

[16SW] S. P. Shipman and A. Welters, Pathological scattering by a defect in a slow-light periodic layered medium, J. Math. Phys. 57, 022902 (2016). doi: 10.1063/1.4941137 

The Combinatorics of Finite Element Methods

Yakov Berchenko-Kogan, Ph.D. | Department of Mathematics & Systems Engineering, Florida Institute of Technology

Monday, August 28, 2023

Abstract: 

Finite element methods are a way of discretizing and numerically solving partial differential equations. Unfortunately, finite element methods can sometimes give wrong answers, even for simple problems. For example, they can give so-called "spurious modes", which are wrong solutions to an eigenvalue problem. Nonetheless, in the 1970s and 1980s, numerical analysts working in electromagnetism developed new finite element methods that avoided these issues. But, unbeknownst to them, they were actually reinventing spaces developed decades earlier by pure mathematicians working in geometry. Why were geometers working with finite element spaces?

In the 1500s, mathematicians noticed that, for any polyhedron, if one takes the number of vertices, subtracts the number of edges, and adds the number of faces, one always gets two. Meanwhile, in calculus, we learn that any divergence-free vector field B is equal to the curl of another vector field A. We also learn that this fact is false if our domain has a hole: the electric field of a point charge, defined everywhere except at the origin, is divergence-free but is not equal to the curl of any other vector field. By the 1930s, geometers understood that these seemingly unrelated facts are actually directly connected via what is now called de Rham's Theorem, a powerful link between the discrete world and the continuous world using the language of cohomology. Finite element methods are also, at their core, a link between the discrete world of computation and the continuous world of differential equations, and so, by the late 1980s, the fields of pure math and applied math ended up colliding.

Over the next few decades, the connection between finite element theory and differential geometry was more fully understood. In this talk, I will discuss the work of Arnold, Falk, and Winther in 2006-2010, who formalized this connection and explained why some finite element methods work well and others do not: Finite element spaces that respect de Rham's theorem will give correct answers; finite element spaces that do not respect de Rham's theorem might not.

An introduction to variational methods for nonlinear differential equations

Shibo Liu, Ph.D. | Department of Mathematical Sciences, Florida Institute of Technology
Friday, April 14, 2023

Abstract: Solving equations is a central problem in mathematics. Noting that even for polynomial equations one can not expect to get the expression for the solutions, by solving equation we mean proving the existence of the solutions. Many phenomena in nature obey variational principle. In this talk, I will briefly present how variational methods can be used to solve nonlinear differential equations arising from geometry, physics and other areas; outline an overview of the field; and mention some central components for working in this vigorous area of mathematical research (hoping to recruit some graduate students to work with me toward a MSc or Ph.D).

Minimax Technics in Optimal Control of Nonlinear Evolution Problems and Machine Learning

Ouayl Chadli, Ph.D. | Ibn Zohr University and University of Central Florida
Friday, March 31, 2023

Abstract: In this talk, we present some recent results from our contributions to the study of the existence and optimal control of Nonlinear Evolution Problems, as well as accelerated methods in Machine Learning Problems by a minimax inequality approach. Our study is motivated by the fact that many control problems arising in physical and engineering problems, whose state system is a variational inequality problem or a nonlinear evolution equation, or a hemivariational inequality problem, can be modeled as a control problem governed by a mixed minimax inequality. There are different techniques for studying the existence of solutions for optimal control problems governed by nonlinear evolution equations, variational inequalities, or hemivariational inequalities in the literature. However, the technique which I will describe in this presentation is completely different from the existing ones. It is based on a constructive Galerkin-type method using a minimax inequality associated with a maximal monotone bifunction and a pseudomonotone bifunction in the sense of Brezis, and a stability result associated with the Mosco convergence. In the second part, we discuss minimax technics in solving Stochastic Variational Inequalities (SVI). SVIs have recently found many applications in data analysis, especially in machine learning models helping to represent massive data compactly. We present a novel method for solving a monotone class of Stochastic Variational Inequalities (SVI). In our approach, we propose an accelerated algorithm combining the mirror-proximal method and a descent method for minimax inequalities through a compatible regularized gap function corresponding to the standard optimality criteria in the aforementioned problem. Our algorithm does not require a priori knowledge about smoothness or non smoothness of the objective function and the noise properties of the problem. Optimal error bounds are obtained and an application to an overlapped group lasso problem is provided.

On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth

Inbo Sim, Ph.D. | Department of Mathematics, University of Ulsan
Friday, February 17, 2023

Abstract: In this talk, we study the existence of multiple solutions to a generalized p(.)-Laplace equation with two parameters involving critical growth. More precisely, we give sufficient “local” conditions, which mean that growths between the main operator and nonlinear term are locally assumed for p(.)-sublinear, p(.)-superlinear, and sandwich-type cases. Compared to constant exponent problems (e.g., p-Laplacian and (p,q)-Laplacian), this characterizes the study of variable exponent problems. We show this by applying variants of the mountain pass theorem for p(.)-sublinear and p(.)-superlinear cases and constructing critical values defined by a minimax argument in the genus theory for sandwich-type case. Moreover, we also obtain a nontrivial nonnegative solution for sandwich-type case changing the role of parameters. Our work is a generalization of several existing works in the literature. This is based on joint work with K. Ho.

Symmetry breaking for a supercritical elliptic problem in an annulus

Francesca Colasuonno | Dipartimento di Matematica, Università di Bologna
Friday, February 3, 2023

Abstract:  In this talk, I will present an existence result for the Dirichlet problem associated with the elliptic equation -\Delta u + u = a(x)|u|^{p-2}u set in an annulus of R^N. Here p>2 is allowed to be supercritical in the sense of Sobolev embeddings, and a(x) is a positive weight with additional symmetry and monotonicity properties, which are shared by the solution that we construct. For this problem, we find a new type of positive, axially symmetric solutions. Moreover, in the case where the weight a(x) is constant, we detect a condition, depending only on the exponent p and on the inner radius of the annulus, that ensures that the solution is nonradial. In this setting, the major difficulty to overcome is the lack of compactness in a nonradial framework. The proofs rely on a combination of variational methods and dynamical system techniques. This is joint work with Alberto Boscaggin (Università di Torino), Benedetta Noris (Politecnico di Milano), and Tobias Weth (Goethe-Universitat Frankfurt).

Low-regularity solutions to nonlinear dispersive PDEs

Razvan Mosincat, Ph.D. | Department of Mathematical Sciences, Florida Institute of Technology
Friday, January 27, 2023

Abstract: Dispersive partial differential equations arise as models describing wave phenomena in various branches of physics and engineering such as plasma physics, fluid dynamics, quantum mechanics, and atmospheric sciences. Some of the canonical models are the Korteweg-de Vries, nonlinear Schroedinger, nonlinear wave, and wave maps equations. Fourier analytic methods play instrumental roles in the study of these equations with prescribed initial data, both for tackling the fundamental question of well-posedness (existence and uniqueness of solutions, and continuity of the flow map) as well as when trying to understand the long-time behavior of solutions. For low-regularity initial data, the techniques need to be adapted to each specific equation, exploiting as much as possible its structure. In the first part, we will talk about main ideas and tools employed in this field of PDEs and some of the current research directions and recent developments. In the second part, I will present my results on two such equations. The Dysthe equation was first derived as an asymptotic model for the water waves system in a certain regime, variants of it appearing in nonlinear optics as well. We establish global-in-time well-posedness and scattering for small initial data in the critical space of this equation. The Benjamin-Ono equation is a model PDE for the propagation of long one-dimensional waves at the interface of two layers of fluids with different densities. My recent work addresses the unconditional uniqueness of low-regularity solutions.

Morse theory and nonlinear Schrodinger equations

Shibo Liu, Ph.D. | Department of Mathematical Sciences, Florida Institute of Technology
Friday, January 20, 2023

Abstract: In the first part of this talk we quickly review some basic concepts in Morse theory, then in the setting of saddle point reduction, we investigate the relation between the critical groups of the original functional and the reduced functional. The abstract results are used to get multiple solutions for elliptic BVPs over bounded domain. In the second part, using Morse theory we discuss the existence of nonzero solutions for stationary nonlinear Schrodinger type equations with indefinite Schrodinger operator. Finally, by applying a variant of the Clark’s theorem we study a quasilinear Schrodinger equation whose nonlinear term can grow super critically. Infinitely many solutions with negative energy are obtained.

A Low-Complexity Algorithm in Phased-array Digital Receivers

Sirani M. Perera, Ph.D. | Department of Mathematics, Embry-Riddle Aeronautical University
Friday, December 2, 2022

Abstract: Phased-array multi-beam RF beamformers require calibration of receivers used in an array of elements before the signals can be applied to an analog beamforming network. The delay Vandemonde matrix (DVM) is an n × n matrix V(α) describing a network of true-time-delay linear combinations of the antenna outputs that are functions of α ∈ C, |α| = 1 that is suitable for analog realization of n simultaneous beams at low complexity. In this talk, we present a fast and exact algorithm to ef- ficiently reverse multi-beams at the DVM output such that calibration of the input low noise amplifiers can be achieved.

Following the structure of the DVM, we first present a sparse factorization to solve the delay Vandermonde systems. We use the proposed factorization to derive a fast algorithm with the arithmetic complexity of order O(n²) as opposed to O(n³). Next, we present numerical results for the forward accuracy of the proposed algo- rithm with different delays. Finally, the proposed algorithm is utilized to present a signal flow graph describing the architecture of an integrated circuit in connection to the phased-array digital receivers.

This work was partially funded by the National Science Foundation with the Award Number 1902283.

This is joint work with Arjuna Madanayake, Austin Ogle, Daniel Silverio, and Jacky Qi.

Half-Harmonic Gradient flow: Existence, Uniqueness and Bubbling

Jerome Wettstein, Ph.D. | Department of Mathematical Sciences, Florida Institute of Technology
Friday, November 18, 2022

Abstract:

The Dirichlet energy:

E(u) = \int | \nabla u |^2 dx,

belongs simultaneously to the simplest and most important energy functionals in the Calculus of Variations and, with suitable restrictions on the image space, Geometric Analysis. Its study may be used to answer one of the best known and classical problems regarding minimal surfaces, Plateau’s problem, and has lead to the discovery of a variety of techniques related to improved integrability by compensation that find application for a variety of critical, non-linear PDEs.

In 2011, Francesca Da Lio and Tristan Rivière introduced a fractional version of the Dirichlet energy, leading to a general formulation of s-Dirichlet energies:

E_{s}(u) = \int | (-\Delta)^{s/2} u |^2 dx,

which have connections to free-boundary minimal discs, self-repulsive knot energies and Dirichlet-to-Neumann operators for possibly degenerate elliptic problems. The resolution of various questions regarding the properties of half- and, in full generality, s-harmonic maps have lead to the introduction of a variety of tools from classical Geometric Analysis and integrability by compensation to the non-local realm, opening up new approaches and techniques leading to results that nowadays allow for the development of the non-local theory very closely to classical Dirichlet energy.

In this talk, we provide an overview and motivation for the study of fractional harmonic maps and go into some of the results found by the speaker pertaining to the well-posedness of the gradient flow associated with the 1/2-Dirichlet energy. The discussion will lead us to encounter many of the general features from local Geometric Analysis in a fractional setting and thus highlights the potential behind the introduction of non-local energies in the context of geometric problems.

Bessmertnyi Realizations, Representations, and Related Problems in Multiphase Composites

Aaron Welters, Ph.D. | Department of Mathematical Sciences, Florida Institute of Technology
Friday, October 28, 2022

Abstract: We discuss multivariate functions that can be represented as the Schur complement of a linear (matrix-, tensor-, or operator-valued) pencil, i.e., the class of Bessmertnyi realizable functions. We motivate this by showing that for multiphase composites, both the effective operator of the periodic conductivity problem in 2D and 3D (a quintessential example in the theory of composites) as well as the Dirichlet-to-Neumann map for a discrete electrical network are in this class in which the associated linear pencil is of positive semidefinite type. This naturally leads to multivariate (matrix-, tensor-, or operator-valued) Herglotz-Nevanlinna functions in this class and several open problems in realizability theory. Next, we present Bessmertnyi realizations as the ``universal" state-space model/realization [compared to others common in electrical engineering (e.g., Kalman-type, Fornasini-Marchesini, Givone-Roesser) that are just special cases of the Bessmertnyi realization]. Then we discuss our recent work [1], [2] on extensions of the Bessmertnyi realization theorem for multivariate rational functions, Schur complement algebra and operations, and their application in symmetric determinantal representations of polynomials. Finally, we will show how the latter relates to the open realization problems above. This is based on joint work with Anthony Stefan (Florida Institute of Technology).

[1] A. Stefan and A. Welters (2021). Complex Anal. Oper. Theory, 15, pp. 1--74. DOI: 10.1007/s11785-021-01150-2 [2] A. Stefan and A. Welters (2021). Linear Algebra Appl., 627, pp. 80--93. DOI: 10.1016/j.laa.2021.06.007

Numerical Methods in Differential Geometry

Yakov Berchenko-Kogan, Ph.D. | Department of Mathematical Sciences, Florida Institute of Technology
Friday, October 14, 2022

Abstract: In this talk, I will discuss some examples of numerical computation in differential geometry. The interaction between the fields of numerical analysis and differential geometry has been particularly fruitful in recent years: a notable example is the 2005 breakthrough in numerical relativity that was instrumental in LIGO's detection of gravitational waves in 2016. The bulk of my talk, however, will focus on a simpler geometric problem, namely, mean curvature flow. I will define and give intuition for mean curvature flow, and then I will discuss how computational methods can help us understand mean curvature flow singularities. Then, in the second part of the talk, I will shift focus to discuss some recent developments in numerical Riemannian geometry based on Regge calculus, a numerical relativity method from the 1960s. In numerical analysis terms, Regge calculus is a lowest-order method, but, in 2018, it was generalized to higher order. This development has the potential to lead to new numerical methods not only for general relativity but also for intrinsic Riemmanian geometry problems, such as Ricci flow, more generally.

Mathematical Description of DNA Configuration in Bacteriophage Capsids

Pei Liu, Ph.D. | Department of Mathematical Sciences, Florida Institute of Technology
Friday, October 7, 2022

Abstract: Bacteriophages densely pack their long dsDNA genome inside a protein capsid. The conformation of the viral genome inside the capsid is consistent with a hexagonal liquid crystalline structure. Experiments have confirmed that the details of the hexagonal packing depend on the electrochemistry of the capsid and its environment.

In this talk, I will introduce how we develop a mathematical model and analyze the model to study the equilibrium configurations of ds-DNA in a viral capsid. In our approach, the DNA is described by a unit helical vector field, tangent to an associated center curve, passing through properly selected locations. We postulate an expression for the energy of the encapsulated DNA based on that of columnar chromonic liquid crystals. A thorough analysis of the Euler-Lagrange equations yields multiple solutions to the corresponding boundary value problems. We demonstrate that there is a trivial, non-helical solution, together with two other solutions with nonzero helicity of the opposite sign. Using bifurcation analysis, we derive the conditions for local stability of such solutions and determine when the preferred coiling state is helical. The relevant bifurcation parameters are the ratio of the twist versus the bend moduli of DNA and the ratio between the sizes of the ordered and the disordered regions.

I will also introduce a computational approach that quantifies the relationship between DNA configurations inside bacteriophage capsids and the types and concentrations of ions present in a biological system. The numerical results show good agreement with existing experiments and molecular dynamics simulations.

Asymptotic behavior for a stochastic behavioral change SIR model

Son Luu Nguyen, Ph.D., Department of Mathematical Sciences, Florida Institute of Technology
Friday, September 16, 2022

Abstract: We introduce a new stochastic SIR (Susceptible, Infected, and Recovered) model in which the Susceptible population is divided into two subclasses S_0 and S_1. Individuals in S_1 change their behaviors to reduce their transmission rate and will contribute to lower the number of new infections. The long-term behavior of the system is investigated under the influence of various system parameters. We construct a threshold and show that if the threshold is negative then the SIR model converges to the disease-free case. This means that the infective class I(t) tends to 0 with the exponential rate meanwhile the susceptible class S_0(t) and the educated class S_1(t) come to the solutions of the boundary equations with an exponential rate.

New multiplicity results for critical p-Laplacian problems

Kanishka Perera, Ph.D. | Department of Mathematical Sciences, Florida Institute of Technology
Friday, September 2, 2022

Abstract: We prove new multiplicity results for the Brezis-Nirenberg problem for the p-Laplacian. Our proofs are based on a new abstract critical point theorem involving the Z_2-cohomological index that requires less compactness than the (PS) condition.

Friday,
May 6, 2022

Caterina Sportelli

Dipartimento di Matematica

University of Bari

Title: On critical growth elliptic problems with jumping nonlinearities

Abstract: In this talk we will present some new existence results for
critical growth elliptic problems involving jumping nonlinearities.
Firstly, we will recall the construction of the minimal and maximal
curves of the Dancer-Fučík spectrum in an abstract setting introduced by
Perera and Schechter. Then, we will present some new abstract linking
theorems and we will use them to obtain nontrivial solutions of our
problems.
Since our abstract results are of independent interest and can be used
to obtain nontrivial solutions of other types of problems with jumping
nonlinearities as well, in the last part of the talk we will examine a
particular critical growth problem in dimension N = 2 and we will derive
the existence of nontrivial solutions in this case, too.
These results are part of a joint work with Kanishka Perera (FIT).

Friday, January 21, 2022

Bruno Poggi, Ph.D.

Departament de Matematiques

Universidad Autonoma de Barcelona

Title: Two problems in the mathematical physics of the magnetic Schrodinger operator and their solutions via the Filoche-Mayboroda landscape function

Abstract: In two papers in the 90's, Z. Shen studied non-asymptotic bounds for the eigenvalue counting function of the magnetic Schrödinger operator in a few settings. But in dimensions 3 or above, his methods required a strong scale-invariant assumption on the gradient of the magnetic field; in particular, this excludes many singular or irregular magnetic fields, and the questions of treating these later cases had remained open.

In this talk, we present our solutions to these questions, and other new results on the exponential decay of solutions (eigenfunctions, integral kernels, resolvents) to Schrödinger operators. We will introduce the Filoche-Mayboroda landcape function for the (non-magnetic) Schrödinger operator, present its connection to the classical Fefferman-Phong-Shen maximal function, and then show how one may use directionality assumptions on the magnetic field to construct a new landscape function in the magnetic case. We solve Shen's problems (and recover other results in the irregular setting) by  putting these observations together.

Friday, October 29, 2021

Aaron Welters, Ph.D.

Department of Mathematical Sciences

Florida Institute of Technology

Title: Photonics, composites, and metamaterials: A view into the mathematical aspects of materials science

Abstract: In this talk, I will introduce the audience to some mathematical aspects of materials science from by highlighting my research in this area over the past few years. First, I will discuss my joint work with Steven Johnson (MIT) and Yehuda Avniel (MIT) on our proof in electromagnetism using Maxwell’s equation of the speed-of-light limitations in photonic crystals. Our proof is general enough to include a very broad range of material properties, including anisotropy, bianisotropy (chirality), dispersion, even delta functions or similar generalized functions. Along the way we will introduce an importance class of analytic functions known as Herglotz functions in which their fundamental properties have played an important role in my research. Next, I will discuss my research in the theory of composites and metamaterials, based on joint work with Graeme Milton (Univ. of Utah) and Maxence Cassier (Institut Fresnel) on two-component composites and continued-fraction expansions for effective tensors. Finally, I will conclude by discussing my recent work with Anthony Stefan (FIT) on progress toward resolving some open problems in realizability of multivariate rational functions as effective tensors in the theory of composites. In this regards, I will discuss our main result, an extension of the Bessmertnyi realization theorem and the important role that symmetries play in this theorem as motivated from effective tensor realization problems.

Friday, October 15, 2021

Mehdi Karimi, Ph.D.

Department of Mathematical Sciences

Florida Institute of Technology 

Title: Selected Modern Applications of Convex Optimization

Abstract: This talk starts with introducing three convex functions, namely the generalized power function, the vector relative entropy, and the quantum relative entropy. We give examples of practical optimization problems involving these functions including portfolio optimization, machine learning, and quantum key distribution. Then we explain how to put these seemingly different functions into the same framework we call "structural convex optimization". We introduce our software package for structural convex optimization and some of the challenges and open questions. In the last part of the talk, we see how convex optimization can be used for non-convex problems, specifically for those that arise in power network operations. We finish the talk with research opportunities in the field of optimization for power networks, which are demanding both theoretically and practically.

Friday, September 17, 2021

Xianqi Li, Ph.D.

Department of Mathematical Sciences

Florida Institute of Technology 

Title: Variational Methods VS Deep Learning-based Super-resolution for MR Spectroscopic Imaging

Abstract: MR Spectroscopic Imaging (MRSI) is a highly versatile metabolic imaging technique that can perform non-invasive measurements for approximately twenty metabolites in the human brain. Metabolite levels in disease vary and have different time course compared to anatomical changes, providing molecular information about disease mechanisms not available from structural imaging. However, the spatial resolution obtained in MRSI is limited by low metabolite concentrations. Low spatial resolution MRSI may miss small lesions or blur the boundaries and heterogeneity of large lesions. While the acquisition of MRSI with the same resolution as anatomical imaging is not possible, upsampling MRSI can bridge the resolution gap. In this talk, I will present our recently developed variational methods and deep learning methods for super-resolution MRSI. Particularly, I will talk about the possibility of combining these two types of methods under certain circumstances. I will also compare and show their performance using both simulated and in vivo data.

 Friday, April 23, 2021

Ryan White, Ph.D.

Department of Mathematical Sciences

Florida Institute of Technology 

Title: In-Orbit Object Detection with Computer Vision

Abstract: The proliferation of dysfunctional satellites and space junk in orbit, and the risks they entail, have generated much interest in the use of autonomous chaser satellites for in-orbit missions, such as servicing satellites and capturing space junk. Devising guidance and navigation systems for chaser satellites requires the ability to identify and localize in-orbit objects in real-time with limited local computational resources. This talk focuses on some methods and recent progress on an active project to use computer vision to detect components of satellites, such as solar panels and antennas, in real-time from camera feeds using convolutional neural networks, object detectors, and a new approach to improve their performance under heavy computational constraints.

Friday, April 16, 2021

Thomas Marcinkowski, Ph.D.

Department of Mathematical Sciences

Florida Institute of Technology

Title: The Theory and Practice of Assessing Environmental Literacy (EL) in the Context of Environmental Education (EE): A Review of Selected Projects and Papers

Abstract: This presentation will introduce and briefly summarize assessment work in the area of Environmental Literacy (EL) in which Dr. Marcinkowski has been involved over the past 20 years. The first part of this presentation, entitled ‘Background,’ describes conceptual features of EE and EL. The second part, entitled ‘Assessments.’ presents selected findings from the NELA Project in the U.S. and from assessments in other countries, and introduces recent dissertation work undertaken here.

Friday, April 9, 2021

William Feldman, Ph.D.

Department of Mathematics

University of Utah

Title: Interfaces in inhomogeneous media: pinning, hysteresis, and facets

Abstract: I will discuss some models for the shape of liquid droplets on rough solid surfaces. The framework of homogenization theory allows to study the large scale effects of small scale surface roughness, including interesting physical phenomena such as contact line pinning, hysteresis, and formation of facets.

Friday, March 12, 2021

Christopher Henderson, Ph.D.

Department of Mathematics

University of Arizona

Title: Front propogation and nonlocal interations

Abstract:  Reaction-diffusion equations arise as models of systems in which spreading and growing forces interact in nontrivial ways, often creating a front (i.e., a moving interface).  In many applications it is natural to consider nonlocal interactions, for example, bacteria feel a "pull" towards their neighbors.  Mathematically this leads to a number of new features and technical difficulties; in particular, the comparison principle, which states that two initially ordered solutions remain ordered, no longer applies.

After giving an overview of classical results, I will discuss a few examples of nonlocal reaction-diffusion equations, including some where the nonlocality is subtle and nonobvious.  The main goal of the talk is to determine how and when the long-range interactions of the nonlocal terms can influence the behavior of the fronts.

Friday, March 5, 2021

Sorin Alexe, Ph.D.

QML Alpha, LLC

Title: Quantitative Machine Learning for Stock Markets

Abstract: Recently, there is an increased interest in applying Machine Learning and Artificial Intelligence to the construction of investment strategies. Here we introduce a Quantitative Machine Learning Platform for alpha discovery and strategy construction. There are three research layers that can help researchers finding signals that are hidden in large amounts of data:

• Interactive Desing
• Machine Learning Tools
• Computer Simulations

The platform offers researchers tools for applying supervised and unsupervised Machine Learning techniques to large collection of data describing time-series systems, where uncertainty plays an important role. Modeling with a programming-free GUI allows researchers to focus on discovery of models based on discriminative and generative Machine Learning algorithms and integrate them in complex decision-making tools. A high level of automation is achieved via the definition of a structured search space and investigation of large samples of strategy candidates.

 A brief introduction on portfolio construction will be followed by a demo session on alpha design. For those of you who become interested in learning more about quantitative investment and model construction we can provide access to use this platform to investigate new ideas, discover new models and apply hedging techniques to balance the reward to risk ratio.

Friday, November 20, 2020

Antonnette Gibbs, Ph.D.

Department of Mathematical Sciences

Florida Institute of Technology

  Title: Socio-Critical Mathematics Modeling and the Role of Mathematics in Society

Abstract: Socio-critical modeling is one of the six modeling perspectives in mathematics education. The socio-critial perspective on mathematical modeling is distinguished by its focus on:

• Critical view of mathematical models and the consequences of these models in society;
• Applying modling to real-life problems (social, political, economic, etc.);
• Empowerment of students and development of reflexive discourse among students.

The soci-critical modeling is constituted in practice through student reglexive discussions during mathematical modeling activities; however, little guidance can be found in the literature on how to design and plan classroom activities that would stimulate such reflexive discussions.

This talk will report findings from a project that examined student reflexive discussions about the role of mathematics in society based on a designed pedagogical tool. The project involves 27 community students' collaborative mathematics modeling activities, and student interation data was analyzed by the Constuctivist Grounded Theory (CGT) approach. The talk will highlight four concepts and an emerging theory from data along with key examples from students discourses. I will close the talk by contextualizing this project in a broader body of work focused on mathematics classroom projects that foster intellectually productive and critial citizens. This talk is based on findings from my dissertation research (Ph.D. advisor: Dr. Joo Young Park, Florida Institute of Technology).

Friday, November 6, 2020

Jian Du, Ph.D.

Department of Mathematical Sciences

Florida Institute of Technology

Title: Permeability, Elasticity, and Platelet Binding Kinetics in Arterial Thrombosis

Abstract: The stability of a platelet thrombus under flow depends strongly on the local hemodynamics and on the thrombus’ structural properties such as porosity, permeability, and elasticity. We develop a two-phase continuum model to investigate the biomechanics of thrombus stability in fluid channels. It is among the few existing models that are capable of explicitly tracking the formation and breaking of interplatelet molecular bonds, which directly determine the viscoelastic property of the thrombus and govern its ability to resist fluid drag. We characterize the stability/fragility of thrombi for various flow speeds, porosities, bond concentrations, and bond types.

Friday, October 16, 2020

Stanley Snelson, Ph.D.

Department of Mathematical Sciences

 Florida Institute of Technology

Title: Mathematical approaches to kinetic theory: history, recent progress, and open problems

Abstract: Kinetic theory seeks to understand the physical properties of matter by studying statistical averages of its constituent particles. Mathematically, this leads to a differential equation for the particle density function (i.e. the density of particles at time t, location x, and velocity v). This talk will focus on two classical kinetic differential equations: the Boltzmann equation (1872) which models diffuse gases, and the Landau equation (1936) which models plasmas. Despite their long history, these equations have proven difficult to understand mathematically, and the question of global existence vs. breakdown remains open (in the general case) for both equations. I will give an overview of recent progress on these equations, focusing especially on the program of conditional regularity, which gives physically meaningful conditions under which the solution can be extended past a given time. The talk will end with a discussion of some promising directions for future research. 

Friday, April 26, 2019

Roby Poteau

Department of Mathematical Sciences

Florida Institute of Technology

Title: Identification of Parameters in Systems Biology

Abstract: Systems Biology is an actively emerging interdisciplinary area between biology and applied mathematics, based on the idea of treating biological systems as a whole entity which is more than the sum of its interrelated components.  One of the major goals of systems biology is to reveal, understand, and predict such properties through the development of mathematical models based on experimental data. In many cases, predictive models of systems biology are described by large systems of nonlinear differential equations. Quantitative identification of such systems requires the solution of inverse problems on the identification of parameters of the system. This dissertation explores the inverse problem for the identification of the finite dimensional set of parameters for systems of nonlinear ordinary differential equations (ODEs) arising in systems biology. Two numerical methods are implemented. The first method combines the ideas of Pontryagin optimization or Bellman's quasilinearization with sensitivity analysis and Tikhonov's regularization. The method is applied to various biological models such as the classical Lotka-Volterra system, bistable switch model in genetic regulatory networks, gene regulation and repressilator models from synthetic biology. The numerical results and application to real data demonstrate the quadratic convergence. The method proved to be extremely effective in moderate scale models of systems biology. The results are published in a recent paper in Mathematical Biosciences, 305(2018), 133-145. To address adaptation and scalability of the method for large-scale models of systems biology the modification of the method is pursued by embedding a method of staggered corrector for sensitivity analysis and by enhancing multi-objective optimization which enables application of the method to large-scale models with practically non-identifiable parameters based on multiple data sets, possibly with partial and noisy measurements. The modified method is applied to benchmark model of three-step pathway modeled by 8 nonlinear ODEs with 36 unknown parameters and two control input parameters. The numerical results demonstrate the geometric convergence with minimum five data sets and with minimum measurements per data set. The method is extremely robust with respect to partial and noisy measurements, and in terms of required number of measurements for each components of the system. Optimal choice of the Tikhonov regularization parameter significantly improves convergence rate, precision and convergence range of the algorithm. Software package qlopt is developed for both methods and posted in GitHub. MATLAB package AMIGO2 is used to demonstrate advantage of qlopt over most popular methods/software such as lsqnonlin, fmincon and nl2sol.

Friday, April 19, 2019

 Robert Talbert, Ph.D.

Department of Mathematics

Grand Valley State University

Title: Flipped learning in theory and practice for mathematics

Abstract: Flipped learning is a pedagogical model in which learners get first contact with new ideas through guided and structured self-learning activities prior to group meetings, allowing significantly more attention to be paid during class meetings to more advanced topics through challenging active learning tasks done collaboratively. Flipped learning combines the best traditions of tutorial and case study methods with modern technologies and evidence-based teaching practices, to help create highly effective learning environments for all learners. In this talk, we will trace the origins of flipped learning, present a framework for flipped learning based in self-determination theory, and illustrate real-life examples from applications to mathematics courses. 

Friday,  April 12, 2019

Adam Prinkey

Department of Mathematical Sciences

Florida Institute of Technology

Title:   Qualitative Analysis of the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption

Abstract:  This talk presents full classification of the evolution of the interfaces and asymptotics of the local solution near the interfaces and at infinity for the nonlinear double degenerate parabolic equation of turbulent filtration with absorption u_t=(|(u^m)_x|^p-1(u^m)_x)_x-bu^β. The nonlinear partial differential equation above is a key model example expressing competition between nonlinear diffusion with gradient dependent diffusivity in with slow (mp>1) or fast (0<mp<1) regime and nonlinear state dependent reaction (b>0) forces. If interface is finite, it may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters m, p, b, and β, and asymptotics of the initial function near its support. In the fast diffusion regime strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The proof of the resulrs is based on rescaling laws for the nonlinear PDEs and blow-up techniques for the identification of the asymptotics of the solution near the interfaces, construction of barriers using special comparison theorems in irregular domains with characteristic boundary curves.

Friday, February 22, 2019

Benjamin F. Akers, Ph.D.

Department of Mathematics & Statistics

Air Force Institute of Technology

Title:  Asymptotics and Numerics for Modulational Instabilities of Traveling Waves.

Abstract:  The spectral stability problem for periodic traveling waves for water wave models is considered. The structure of the spectrum is discussed from the perspective of resonant interaction theory. Modulational asymptotic expansions are used to predict the location of instabilities in frequency-amplitude space. These predictions explain numerical results in [1]. Asymptotics results are presented in the potential ow equations [2] as well as weakly nonlinear models [3]. The asymptotic predictions are compared to the results of a direct numerical simulation of the modulational spectrum.

References

[1]  Nicholls, David P., Spectral data for travelling water waves: singularities and stability, Journal of Fluid Mechanics, 624 (2009), 339-360.

[2] Akers, Benjamin F., Modulational Instabilities of periodic traveling waves in deep water, Physica D: Nonlinear Phenomena, 300 (2015), 26-33.

[3] Akers, Benjamin F. and Milewski, Paul A., A Model Equation for Wavepacket Solitary Waves Arising from Capillary-Gravity Flows, Studies in Applied Mathematics, 122 (2009), 249-274.

Friday, October 12, 2018

Alexandru Tamasan, Ph.D.

Department of Mathematics

University of Central Florida

 Title: Current Density based Impedance Imaging (CDII)

Abstract: In this talk I will present the inverse hybrid problem of CDII, where the electrical conductivity of body is to be recovered from of the magnitude of one current density field. Physically we use a connection between the current density field generated from the boundary, with interior measurements of the magnetization in a MRI machine. The basic idea is to first recover the voltage potential, which solves a generalized 1-Laplacian. Mathematical methods involved in solving this problem combines ideas from Riemannian geometry with geometric measure theory, and touch on some algebraic topology. 

Friday, October 5, 2018

Yalchin Efendiev, Ph.D.

Department of Mathematics

Texas A&M University 

Title: Data Integration in Multiscale Simulations

Abstract: In this talk, I will discuss several data integration techniques in multiscale simulations. I will give a brief overview of multiscale simulation concepts that will be used. These multiscale techniques are designed for problems when the coarse grid does not resolve scales and contrast. I will describe the relation between multiscale and upscaling methods. I will describe three data integration techniques. The first one, Bayesian multiscale modeling, will sample basis functions and incorporate available data. In the second approach, we will use deep learning techniques to design and modify existing multiscale methods in the presence of data and nonlinearities.

Friday, September 14, 2018

Ugur G. Abdulla, Ph.D.

Department of Mathematical Sciences

Florida Institute of Technology 

Title: Breast Cancer Detection through Electrical Impedance Tomography and Optimal Control of Elliptic PDEs -- Invitation to Research

Abstract: In this talk I am going to discuss the inverse Electrical Impedance Tomography (EIT) problem or Calderon problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the electrodes for a given electrode current. The inverse EIT problem presents an effective mathematical model of breast cancer detection based on the experimental fact that the electrical conductivity of malignant tumors of the breast is significantly different from conductivity of the normal tissue. I am going to introduce a mathematical model of the inverse EIT problem as a PDE constrained optimal control problem in a Sobolev-Besov spaces framework, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The state vector is a solution of the second order elliptic PDE in divergence form with bounded measurable coefficients under mixed Neumann/Robin type boundary condition. Some recent results on the existence of the optimal control, Frechet differentiability in the Besov space setting, derivation of the he formula for the Frechet gradient, optimality condition, and extensive numerical analysis in the 2D case through implementation of the gradient method in Banach spaces will be presented. Talk will end with the formulation of some major open problems and the perspectives of future advance.