Mathematical Sciences > Colloquia

Mathematical Sciences Colloquium

The purpose of the colloquium series is to discuss research problems in pure and applied mathematics, operations research, STEM education, and related topics. For more information, or if you would like to give a talk, please contact Dr. Stanley Snelson at ssnelson@fit.edu.

2021/22

Date/Time	Speaker	Title/Abstract
Friday, May 6, 2022 3:00 - 4:00pm Room: 404 Crawford Building	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 3:00 - 4:00pm Virtual talk. Zoom ID: 967 1488 8896	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 3:00 - 4:00pm Room: 404 Crawford Building	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 3:00 - 4:00pm Room: 404 Crawford Building	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 3:00 - 4:00pm Room: 404 Crawford Building	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.

Date/Time

Speaker

Title/Abstract

Friday,
May 6, 2022

3:00 - 4:00pm

Room: 404 Crawford Building

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

3:00 - 4:00pm

Virtual talk. Zoom ID:

967 1488 8896

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

3:00 - 4:00pm

Room: 404 Crawford Building

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

3:00 - 4:00pm

Room: 404 Crawford Building

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

3:00 - 4:00pm

Room: 404 Crawford Building

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.

2020/2021

Date/Time	Speaker	Title/Abstract
Friday, April 23, 2021 3:00 - 4:00pm Zoom Meeting ID: 930 1022 9187	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 3:00 - 4:00pm Zoom Meeting ID: 930 1022 9187	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 3:00 - 4:00pm Zoom Meeting ID: 930 1022 9187	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 3:00 - 4:00pm Zoom Meeting ID: 930 1022 9187	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 3:00-4:00pm Zoom Meeting ID: 974 4116 5152	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 3:00-4:00pm Zoom Meeting ID: 947 0114 5457	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 activites, 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 3:00-4:00pm Zoom Meeting ID: 947 0114 5457	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 3:00-4:00pm Zoom Meeting ID: 947 0114 5457	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.

Date/Time

Speaker

Title/Abstract

Friday, April 23, 2021

3:00 - 4:00pm

Zoom Meeting ID: 930 1022 9187

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

3:00 - 4:00pm

Zoom Meeting ID: 930 1022 9187

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

3:00 - 4:00pm

Zoom Meeting ID: 930 1022 9187

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

3:00 - 4:00pm

Zoom Meeting ID: 930 1022 9187

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

3:00-4:00pm

Zoom Meeting ID: 974 4116 5152

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

3:00-4:00pm

Zoom Meeting ID: 947 0114 5457

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 activites, 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

3:00-4:00pm

Zoom Meeting ID: 947 0114 5457

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

3:00-4:00pm

Zoom Meeting ID: 947 0114 5457

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.

2018/2019

Date/Time	Speaker	Title/Abstract
Friday, April 26, 2019 3:00 - 4:00pm The MAC	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 3:00 - 4:00pm The MAC	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 3:00 - 4:00pm The MAC	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 3:00 - 4:00pm The MAC	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 3:00 - 4:00pm The MAC	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 3:00 - 4:00pm The MAC	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 3:00 - 4:00pm The MAC	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.

Date/Time

Speaker

Title/Abstract

Friday, April 26, 2019

3:00 - 4:00pm

The MAC

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

3:00 - 4:00pm

The MAC

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

3:00 - 4:00pm

The MAC

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

3:00 - 4:00pm

The MAC

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

3:00 - 4:00pm

The MAC

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

3:00 - 4:00pm

The MAC

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

3:00 - 4:00pm

The MAC

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.