Schedule

Dirichlet Energy Problem 

Abstract. The Dirichlet problem generally asks whether it's possible to find a function that satisfies some PDE & boundary condition on some domain. We will look at solving the Laplace equation with Dirichlet boundary conditions (on smooth compact RM)- equivalent to minimizing normalized Dirichlet energy. In this talk, I'll overview a method using some functional analysis. The solution is related to "hearing the shape of a drum". 

Emma Bollinger, Oklahoma State University

Monomial Ideals with Characteristic-Dependent Projective Dimension 

Abstract. Let R = k[x_1,...,x_p] and I be a squarefree monomial ideal over R with g minimal generators. The projective dimension is the length of the minimal resolution of I, which sometimes depends on the characteristic of k. Given integers n,i such that g ≥ 3n-1 and 3 ≤ n ≤ i ≤ g-n+1, we provide an algebraic construction of an ideal whose projective dimension is equal to i if and only if the characteristic divides n.  If time permits, we will explore the combinatorics behind the construction. 

Reid Buchanan, Oklahoma State University

What is a Scheme? 

Abstract. We begin by introducing the notion of a scheme. We then discuss projective morphisms and line bundles, focusing on how they give rise to embeddings into projective space. If time permits, we will explore how syzygies reflect the geometry of a smooth projective variety. 

Nawal Hazarika, Oklahoma State University

Proper maps of annuli  

Abstract. We study proper holomorphic maps of annuli in complex Euclidean spaces, domains with U(n) as the automorphism group. Such maps are always rational and extend to proper maps of balls. We first prove that a proper map of annuli from n dimensions to N dimensions where N < (n + 1 choose 2) is always unitarily equivalent to an affine embedding. This inequality is sharp as the homogeneous map of degree 2 satisfies the boundary point. This yields the first gap interval for proper holomorphic maps between annuli, which surprisingly is much longer than the well-known first gap interval (n, 2n - 1) for ball maps. We give a complete classification of rational proper maps of annuli of degrees 1 and 2.

Abdullah Al Helal, Oklahoma State University

Visualizing Elementary Moves on 3-manifold Triangulations and Spines 

Abstract. While triangulations of 3-manifolds may be more commonly used in research and computations, their dual, called (special) spines, are often more convenient in proofs. Elementary moves, such as the bistellar flips, are relevant both in theoretical and practical settings. These moves are local modifications to the structure that preserve the topology of the manifold. Upon exploring the duality of triangulations and spines, this talk will illustrate several elementary moves from both perspectives. 

Rylee Dennis, Oklahoma State University

Manifolds with No Conformal Killing Fields   

Abstract. We will begin this talk with Riemannian manifolds and demonstrate that there exist manifolds admitting no conformal Killing vector fields. We then explain how analogous nonexistence results extend to the Lorentzian setting. In particular, by a theorem of Richard Schoen, together with Piotr T. Chruściel and Robert Beig, one knows that the absence of conformal Killing vector fields is in fact a generic property of Lorentzian metrics. This provides a natural geometric setting in which no nontrivial infinitesimal conformal symmetries occur. 

Siddiqur (Milon) Rahman, Oklahoma State University

Stability and Bifurcation Analysis of a Behavioral-Entomological Model 

Abstract. Mosquito-borne diseases remain a major public health threat due to the lack of a universal vaccine. As a result, eliminating breeding sites is a key strategy for controlling mosquito populations, and household decision-making plays an important role in this process.

We develop a series of behavioral-entomological models that integrate mosquito population dynamics with household control behavior. In this talk, we focus on stability and bifurcation analysis of these models. We show that the system admits equilibria that can lose stability through Hopf bifurcation, which leads to sustained oscillations driven solely by behavioral-ecological feedback.

Rubayet Rahman, Oklahoma State University

Phase retrievability of Super Operators 

Abstract. Super operators are linear operators acting on vector spaces of linear operators. In quantum information theory, quantum channels form a particularly important class of such operators, as they describe how quantum systems evolve over time. A central problem in this setting is quantum state recovery: given only the final state of a quantum system, can we determine the initial state that produced it? Recent work has addressed this question for the simplest class of states (called pure states) using tools from frame theory. In practice, however, quantum states are mixed (not pure), and extending these techniques to mixed states has proven substantially more difficult within that framework. This talk introduces a generalized notion of phase retrievability for super operators and applies it to construct a quantum channel that recovers mixed states. 

Jeremiah Isu, University of Oklahoma

On the rigidity of Landau solutions to the steady-state Navier-Stokes equations 

Abstract. The goal of this talk will be to give an expository presentation of a 2012 result due to Vladimír Šverák, which demonstrated the rigidity of Landau solutions to the three-dimensional steady-state Navier-Stokes equations. The result is a particularly nice demonstration of the usefulness of differential geometric language in PDEs, and this talk will also cover some preliminary materials in differential geometry and fluid PDEs. 

Tyler Labus, Oklahoma State University

Forecasting Challenges Across Time Horizons: Challenges and Solutions for Short- and Long-Term Predictions 

Abstract. Accurate disease forecasting is essential for decision-making across both short-term response and long-term public health planning. However, predictive performance and modeling strategies vary substantially across forecast horizons. This talk examines the key differences between short-term and long-term time series forecasting, highlighting their respective advantages, limitations, and methodological challenges. Short-term forecasts tend to be more accurate and well-suited for operational decision-making, but they are sensitive to noise, reporting delays, and sudden behavioral changes. In contrast, long-term forecasts support strategic planning and scenario analysis, yet face challenges related to structural changes, nonstationary, and increasing uncertainty over time. Drawing on data-driven examples motivated by mobility-informed and epidemiological modeling, this presentation discusses appropriate modeling approaches, evaluation metrics, and practical considerations for selecting forecasting strategies across different time horizons. The goal is to provide a clear understanding of how forecast horizon influences model design, interpretation, and real-world applicability. 

Dr. Haridas Kumar Das, Oklahoma State University

Movie

The Man Who Knew Infinity