Events
Department of Mathematics and Statistics
Texas Tech University
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 Departmental Promotion Colloquium may be attended at 3 PM (UT-5) via this Zoom link.
Meeting ID: 957 2551 8877
Passcode: McCarthy
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 Departmental Promotion Colloquium may be attended at 3:30 PM (UT-5) via this Zoom link.
Meeting ID: 915 0082 7584
Passcode: Peace