Colloquia
Department of Mathematics and Statistics
Texas Tech University
Many problems in science and engineering involve multiple physical processes, where complex interactions between these components can result in dynamics over a wide range of time scales. This presents significant challenges for real-time simulation of these multiphysics problems. A prominent example is the existence of vastly differing time scales in atmospheric phenomena, ranging from a relatively slow advection to very fast gravity waves, which poses significant difficulties for real-time simulation of weather and long-term climate predictions. Developing fast and accurate time integration methods is thus crucial for a wide range of applications that rely on simulations of complex multiphysics systems.
In this talk, I will present my recent contributions to developing (construction, analysis, and implementation) new exponential and multirate time integration methods that significantly increase accuracy and efficiency in computational simulations of large-scale multiple time scale/multiphysics problems. I will also demonstrate their performance through numerical experiments across various applications, including visual computing, numerical weather prediction, and computational biology.
This colloquium may be viewed in the TTU Mediasite Catalog.
Finite field hypergeometric functions were first defined by Greene in the 1980s as analogues of classical hypergeometric series. These functions have nice character sum representations and, consequently, lend themselves naturally to counting problems. They also have strong links to modular forms. I will begin this talk with a description of these functions, outline their main applications and highlight their deficiencies. I will then discuss my work in defining new hypergeometric functions in both the finite field and p-adic settings, which resolves these deficiencies. These new functions are now widely used in my field, and I will describe their role in the development of new modularity results. Finally, I will outline the results of some of my most recent work involving these functions, including an application in graph theory leading to improved lower bounds for Ramsey numbers.
This colloquium may be viewed in the TTU Mediasite Catalog.
Biological systems are often shaped by stressors and how organisms respond to them. In ecological systems, populations are rarely influenced by just one factor. Instead, they are subjected to a combination of constraints. Broadly, my research program at Texas Tech focuses on population dynamics and community structures that are faced with multiple constraints or stressors. Ecological processes are naturally structured and depend on the flow and balance of essential elements such as carbon, nitrogen, and phosphorus. Alongside elemental constraints, populations may be subject to toxicants or pathogens. Understanding these interactions is essential for predicting how populations will respond to environmental changes and for developing strategies to mitigate the impacts of multiple stressors. Here I will present a variety of mathematical models that help us understand how heterogeneous populations and communities balance resources and live amongst multiple constraints. Using dynamical systems theory, I will highlight how techniques in stability and bifurcation analyses leads to biological insights. Moving forward from here, these modeling frameworks have the potential to investigate the influence of constraints on biodiversity and nutrient cycling and how they propagate throughout larger and more complex food webs.
This colloquium may be viewed in the TTU Mediasite Catalog.
In their seminal work, Freed and Hopkins studied the moduli space of topological, reflection positive, invertible, Euclidean field theories, providing a complete classification in terms of certain objects arising in stable homotopy theory. In this work, it was also conjectured that a similar classification holds in the case of nontopological field theories, and this conjecture is already being used in a variety of applications to condensed matter physics. In this talk, I will discuss a recent result which provides an affirmative answer to this conjecture. I will begin by reviewing motivation and background on reflection positive theories. Then I will state the conjecture and sketch of the proof.
This colloquium may be viewed in the TTU Mediasite Catalog.