Graduate Student Seminar
The graduate student seminar is a biweekly seminar, alternating with the colloquium, in which students give talks about their current research. For the Fall 2023 semester, the graduate student seminars will be held on Mondays at 1.00 p.m in Crawford room 210.
For more information, or if you would like to give a talk, please contact Dr. Yakov Berchenko-Kogan at firstname.lastname@example.org.
Existence of smooth solutions for the Landau equation with hard potentials
Shelly Ann Taylor is a graduate (PhD) student in Applied Mathematics at Florida Tech working with Dr. Stanley Snelson
The Landau equation was first documented in 1936, as an alternate version to the Boltzmann equation in the case of Coulomb interaction. It soon became (in combination with the Vlasov equation) the most important mathematical kinetic model in the theory of collisional plasma. Moreover, it is a nonlinear equation that combines transport in t and x, with non local diffusion in v. The aim of this presentation is to find a solution to the Landau equation that models the plasma particles. The solution f(t,x,v) will model the particle density of a plasma at time t > 0, location x in R3, and velocity v in R3.
Global existence with general initial data, poses a very challenging open problem. Therefore, the scope of this presentation primarily focuses on the nontrivial case of local existence. Particularly, the case of ``hard potentials,'' which is less understood than the other physical regimes, in terms of existence theory. Therefore, we prove the following: (i) precise upper bounds on the Landau coefficients in terms of weighted L∞ norms, (ii) pointwise lower bounds for solutions f for positive times, which is crucial for establishing ellipticity estimates for the diffusion matrix, and (iii) upper Gaussian bounds for solutions f of the Landau equation.
Time permitting, we will discuss plans for related future work.
Löewner, Legendre, and the Principal Pivot Transform - Oh My!
Kenneth Beard is currently attending Louisiana State University pursuing a Ph.D. in Applied Mathematics. He obtained his Masters Degree in Applied Mathematics from Florida Tech.
We present our recent results  on the (symmetric) principal pivot transform (PPT) improving on results in  which are motivated by the celebrated Löewner’s Theorem . In particular, we give the minimum hypotheses for which the PPT is matrix monotonic with respect to the Löewner partial order on Hermitian matrices. This result turns out to have natural connections in the theory of Herglotz-Nevanlinna functions. In addition, we present a new variational principle for the PPT which is intricately connected with the Legendre-Fenchel transform from convex analysis. We then use this minimization principle to establish concavity of the PPT for kernel equivalent positive semi-definite matrices. Through- out we utilize the fundamental properties of Schur complements and the Moore-Penrose pseudoinverse under the Löewner ordering. This is joint work with Aaron Welters (Florida Institute of Technology).
 K. Beard and A. Welters, “Matrix monotonicity and concavity of the principal pivot transform,” arXiv, 2023. doi: 10.48550/arXiv.2302.04293.
 J. Pascoe and R. Tully-Doyle, “Monotonicity of the principal pivot transform,” Linear Algebra and its Applications, vol. 643, pp. 161–165, 2022. doi: 10.1016/j.laa.2022.02.016.
 B. Simon, Loewner’s Theorem on Monotone Matrix Functions. Springer Cham, 2019. doi: 10.1007/978-3-030-22422-6.
Diffusion Model based Super-resolution for Medical Images
Mohammed Alsubaie is a graduate (Ph.D.) student in Operations Research at Florida Tech working under his advisor Dr. Xianqi Li
High-resolution medical imaging is critical for precise diagnosis and optimal patient care. Yet, obtaining such images often necessitates prolonged scanning durations, leading to discomfort for patients and increasing the risk of scan failures due to patient movements. To address these challenges, super-resolution techniques were proposed and offer a promising avenue for enhancing the quality of medical images. Traditional super-resolution methods often fall short in preserving intricate details inherent to medical images. This talk introduces deep learning approaches, particularly the diffusion models, to enhance the resolution of medical images.
We begin by discussing the fundamental challenges in medical image super-resolution and the limitations of conventional methods. Then, we talk about the deep learning approaches for image super-resolution. In particular, we delve into the principles of diffusion models and shed light on why they are promising in this application. Experimental results will be showcased, comparing our diffusion model-based approach with other SOTA super-resolution methods. The outcomes reveal notable improvements in preserving fine-grained details and critical structures, making it an invaluable tool for clinicians.
Attendees will gain an in-depth understanding of diffusion models and how they can revolutionize the super-resolution of medical images, paving the way for enhanced diagnostic precision and patient outcomes.
Initial-Periodic Problems for Three-Dimensional Linear Hyperbolic Systems
Najma Alarbi is a graduate (PhD) student in Applied Mathematics at Florida Tech working with Dr. Tariel Kiguradze.
Bessmertnyĭ Realizations of Effective Tensors for Metamaterial Synthesis: Conjectures and Counterexamples
Anthony is a graduate (Ph.D.) student in applied mathematics at Florida Tech working under his advisor Dr. Aaron Welters.
We discuss the current progress towards my dissertation on the representability and analytic characterization of effective tensors for isotropic n-phase composites using a Hilbert space approach as described, for example, in  and . The key properties of such tensors is that they are homogeneous degree-1 rational functions of n complex variables that belong to a special class known as Herglotz-Nevanlinna functions. My research is on proving the converse of functions with these properties, that is, the realizability problem for effective tensors (cf. [Ch. 7, 3]) which is important in metamaterial synthesis. Recently, a theorem was claimed by M. Bessmertnyĭ (in a preprint posted to arXiv ) to have been proven which would have essentially allowed us to do this. Unfortunately, we discovered that a certain lemma of his (attempting to rehabilitate a similar claim by T. Koga’s ), that he uses in the proof of this theorem, is false. We show this by constructing a counterexample (which also provides a counterexample, as we will show, to the result of T. Koga) that utilizes the stability properties of the basis generating polynomial to the Vámos matroid. These properties have been shown, over a decade, by several authors investigating matroids with the half-plane property and generalizations of the Lax Conjecture for hyperbolic polynomials. In this talk, we provide an overview of how these necessary properties for the counterexample were proved, such as being a homogeneous multi-affine stable polynomial and the existence of a certain real nonnegative polynomial that cannot be written as a polynomial sum-of-squares (SOS)  (which is interesting in its own right in light of Hilbert’s 17th problem  and the history associated with it). This talk is joint work with my advisor Dr. Aaron Welters.
-  G. W. Milton, The Theory of Composites. SIAM, Philadelphia, PA, 2023.
-  Y. Grabovsky, Composite Materials. Mathematical Theory and Exact Relations. IOP Publishing, 2016.
-  A. Stefan, Schur complement algebra and operations with applications in multivariate functions, realizability, and representations, M.S. Thesis, FIT, 2021.
-  M. F. Bessmertnyĭ, (2021). Representation of rational positive real functions of several variables by means of positive long resolvent. arXiv:2103.02541
-  T. Koga, (1968). Synthesis of finite passive n-ports with prescribed positive real matrices of several variables, in IEEE Transactions on Circuit Theory, vol. 15, no. 1, pp. 2-23. DOI:10.1109/TCT.1968.1082780
-  Wikipedia. 2008. “Polynomial SOS,” Wikimedia Foundation. Modified May 4, 2023. https://en.wikipedia.org/wiki/Polynomial_SOS
-  Wikipedia. 2005. “Hilbert’s seventeenth problem,” Wikimedia Foundation. Modified Sep. 2, 2023. https://en.wikipedia.org/wiki/Hilbert%27s_seventeenth_problem
Variational Discretization Methods for Curvature Flows on Riemannian Manifolds
Zofia Adamska, Pedro Estrada Gallegos, and Ricardo Garcia
The speakers are students working on a research project in computational differential geometry with Dr. Yakov Berchenko-Kogan. Zofia Adamska and Ricardo Garcia are students from Caltech visiting under the Summer Undergraduate Research Fellowship (SURF) program and Pedro Estrada Gallegos is a student from Universidad Nacional Autόnoma de México.
The study of geometric flows has several applications that span a wide variety of scientific disciplines. For example, these geometric evolution equations were used to model processes such as the annealing of metal sheets, evolution of reaction diffusion systems, or the behavior of cellular automata. Furthermore, geometric flows are applied to problems in computer vision and image processing. We present a variational method to compute curve shortening and curve straightening flows on arbitrary Riemannian manifolds. We apply our method to manifolds with different metrics and verify convergence to expected theoretical results. Furthermore, we introduce an algorithm that uses partitions of unity to evolve curves in manifolds covered by multiple charts, an example being noncontractible loops in tori. The methods developed in this project could have the potential to deepen our understanding of the geometry of non-trivial Riemannian manifolds, such as their geodesics and minimal submanifolds. These insights can open new avenues of investigation and inspire further developments in the field of discrete and numerical differential geometry.