**DMS Colloquium: Peter McGrath**
**Dec 01, 2023 04:00 PM**

**Refreshments 3:30 p.m., Parker 244**

Speaker: **Peter McGrath **(North Carolina State University)

Title: Calculus of Variations and the Bending Energy of Surfaces

Abstract: Beginning with the solution of the classical Plateau problem—the problem of finding an area-minimizing disk whose boundary is a prescribed simple closed curve in Euclidean 3-space—we will survey some applications of Calculus of Variations to solve geometric extremal problems. Particular emphasis will be placed on the problem of finding a smooth surface in 3-space with minimum total squared mean curvature and prescribed isoperimetric ratio and genus. Such minimizers of the total squared mean curvature (also called the bending energy) were proposed in 1970 by Biologist Peter Canham to model the shape of red blood cells and lipid bilayers.

**DMS Colloquium: Saba Gerami**
**Nov 16, 2023 04:00 PM**

**Refreshments will be served in Parker 244, 3:30-3:55pm**

Speaker: **Saba Gerami** University of Michigan.

Title: Beyond the Surface: Exploring the Content Underlying Instructional Tasks in Calculus

Abstract: Although lecture still dominates calculus instruction, incorporating more inquiry has been advocated to make the course more accessible. Despite this shift, existing research in undergraduate mathematics education has primarily focused on pedagogical variables, often overlooking the critical role of content. This oversight has not consistently yielded positive student outcomes. Alternatively, I shift my focus to the content instructors and students engage with in these classrooms. In this talk, I will share what I have learned about the teaching of calculus with inquiry through eight instructors’ instructional tasks for teaching derivatives. A pivotal finding emerges as the analysis reveals that, although instructional tasks may exhibit surface-level similarities and common patterns across instructors, there exists a high variability in work that is expected of students. This finding highlights the importance of understanding the nuanced ways in which instructors structure tasks to effectively implement inquiry-based teaching strategies in calculus education.

Faculty host: Melinda Lanius

**DMS Colloquium: Gregory G. Smith**
**Nov 10, 2023 04:00 PM**

**Refreshments 3:30 p.m., Parker 244**

Speaker: **Greg Smith** (Queen's University, Kingston, Ontario, CA)

Title: Sums of Squares and Projective Geometry

Abstract: A multivariate real polynomial is nonnegative if its value at any real point is greater than or equal to zero. These special polynomials play a central role in many branches of mathematics including algebraic geometry, optimization theory, and dynamical systems. However, it is very difficult, in general, to decide whether a given polynomial is nonnegative. In this talk, we will review some classical methods for certifying that a polynomial is nonnegative. We will then present novel certificates in some important cases.

This talk is based on joint work with Grigoriy Blekherman, Rainer Sinn, and Mauricio Velasco.

**DMS Colloquium: Yuesheng Xu**
**Nov 03, 2023 04:00 PM**

**Refreshments 3:30 p.m., Parker 244**

Speaker: **Yuesheng Xu**

Title: Multi-Grade Deep Learning and its Application in Solutions of Nonlinear Differential Equations

Abstract: The great success of deep learning has been widely recognized. From a mathematical perspective, such successes are mainly due to the powerful expressiveness of deep neural networks in representing a function. Deep learning requires solving a nonconvex optimization problem of a large size to learn a deep neural network. The current deep learning model is of a single-grade, that is, it learns a deep neural network by solving a single nonconvex optimization problem. When the layer number of the neural network is large, it is computationally challenging to carry out such a task efficiently. The complexity of the task comes from learning all weight matrices and bias vectors from one single nonconvex optimization problem of a large size. Inspired by the human education process which arranges learning in grades, we propose a multi-grade learning model: Instead of solving one single optimization problem of a large size, we successively solve a number of optimization problems of small sizes, which are organized in grades, to learn a shallow neural network for each grade. Specifically, the current grade is to learn the leftover from the previous grade. In each of the grades, we learn a shallow neural network stacked on top of the neural network, learned in the previous grades, which remains unchanged in training of the current and future grades. By dividing the task of learning a deep neural network into learning several shallow neural networks, one can alleviate the severity of the nonconvexity of the original optimization problem of a large size. When all grades of the learning are completed, the final neural network learned is a stair-shape neural network, which is the superposition of networks learned from all grades. Such a model enables us to learn a deep neural network much more effectively and efficiently. We provide several numerical examples of numerical solutions of the Burger equation (1D-3D), which demonstrate that the proposed multi-grade model significantly outperforms the traditional single-grade model.

**DMS Colloquium: Susmita Sadhu**
**Oct 27, 2023 04:00 PM**

**Refreshments 3:30 p.m., Parker 244**

Speaker: **Susmita Sadhu**, GCSU

Title: Complex oscillatory patterns and regime shifts in predator-prey models featuring multiple timescales

Abstract: Predator-prey models form basic building blocks for studying population cycles and under- standing complex interactions in ecological communities. The temporal patterns can be broadly viewed as oscillatory dynamics that alternate between small-amplitude and large-amplitude oscillations, and thus can be represented by mixed-mode oscillations or bursting dynamics that typically occur in higher-dimensional slow-fast dynamical systems. In the first part of my talk, I will consider predator-prey models in the framework of singularly perturbed system of equations and discuss mechanisms underlying different kinds of oscillatory patterns. I will show the existence of periodic traveling waves in a class of reaction-diffusion system of equations that have oscillatory local kinetics. In the second part of my talk, I will address ecological "regime shifts" or the phenomenon of "tipping" and the leading role of long transient dynamics in ecological timescales in explaining such regime shifts. I will discuss some analytical techniques to analyze long transient dynamics preceding a regime shift in a two-timescale three-dimensional predator-prey model with explicit interference competition between the predators. The methods are then used for identifying early warning signals of a large population transition leading to an outbreak or resulting in an extinction of one of the species. Finally, I will end with some challenges and future directions.

**DMS Colloquium: Gregory Berkolaiko**
**Oct 06, 2023 04:00 PM**

Speaker: **Gregory** **Berkolaiko**,** **Texas A&M University

Title: Spectral minimal partitions: Local vs global minimality

Abstract: In this overview talk we will explore a variational approach to the problem of Spectral Minimal Partitions (SMPs). The problem is to partition a domain or a manifold into k subdomains so that the first Dirichlet eigenvalue on each subdomain is as small as possible. We expand the problem to consider Spectral *Critical* Partitions (partitions where the max of the Dirichlet eigenvalues is experiencing a critical point) and show that a locally minimal bipartite partition is automatically globally minimal.

Extensions of this result to non-bipartite partitions, as well as its connections to counting nodal domains of the eigenfunctions and to a two-sided Dirichlet-to-Neumann map defined on the partition boundaries, will also be discussed.

The talk is based on several papers of Yaiza Canzani, Graham Cox, Bernard Helffer, Peter Kuchment, Jeremy Marzuola, Uzy Smilansky, and Mikael Sundqvist, with and without the speaker.

Faculty host: Selim Sukhtaiev