In 2025 Spring, the Math Department Colloquium Series will be held on Wednesdays 3:30 pm at CU 218. Tea and refreshments available at 3:00 p.m. in the Assmus Conference Room (CU 212).
If you have any questions, please reach out to the organizors Taeho Kim (tak422 AT lehigh DOT edu) and Ao Sun (aos223 AT lehigh DOT edu).
Date: 02/05/2025
Speaker: Hsuan-Wei Lee (Lehigh University)
Title: Partner-switching games with initial-network-dependent interaction patterns
Abstract: According to prior research on partner-switching games, altering partners could stabilize cooperation. Yet the role of per-edge (interactive diversity, ID) vs. per-node (interactive identity, II) strategies under a single co-evolutionary rule remains deficiently understood. In this talk, we propose an edge-based game as a natural extension of a well-studied node-based model in adaptive networks. By varying only the initial distribution of strategies, our framework can transition from interactive identity to interactive diversity without altering the update rules. We show that, in contrast to node-based scenarios, allowing each edge to carry its own strategy label grants defectors an exclusive "learning-by-rewiring" mechanism, thereby significantly enlarging the parameter region in which cooperation thrives. We find that cooperation in the edge-based (ID) setup can emerge even from low initial cooperation, because defectors may incrementally increase their cooperative ratio by severing disadvantageous edges. This effect is absent in node-based (II) games, which constrain each node to a single global strategy. Moreover, numerical evidence suggests that higher initial cooperation does not guarantee higher final cooperation in ID games, underscoring the non-monotonic and homophily-driven nature of cooperative cluster formation. Our results thus clarify how partner-switching, when combined with interactive diversity, yields complex threshold behaviors unseen in the conventional node-based setting. Finally, we use pair approximation to illustrate why accurate theoretical predictions require richer local information in edge-based models, where each directed edge may evolve independently.
Date: 02/19/2025
Speaker: Joe Kramer-Miller (Lehigh University)
Title: Transcendence theory and Einstein's final theorem
Abstract: A complex number is called transcendental if it is not the solution to any polynomial equation with rational coefficients. Similarly, a function is transcendental if it is not the solution to any polynomial equation whose coefficients are rational functions. The theory of such numbers/functions is a beautiful and mysterious field, with connections to number theory, algebraic geometry, and combinatorics. In this talk we survey the field and describe some important outstanding problems. In addition we will explain the last paper of Eisenstein before he died of tuberculosis at the untimely age of 29 and discuss some of our recent progress on the Eisenstein constant problem.
Date: 02/26/2025
Speaker: Daniel Ullman (George Washington University)
Title: Classic Problems
Abstract: The Problems Section of the American Mathematical Monthly has a newish subsection called "Classics". Each month we publish a problem whose charm, humor, or elegance elevates it to classic status as one of the greatest problems of all time. These are the sorts of problems that people hear once and then pass around eagerly to their mathematical friends. Many a restaurant napkin has been sacrificed in pursuit of solutions to these gems. And yet some of these are not as well known as one might expect. I'll offer up several of my favorites in this talk.
This presentation is suitable for a wide range of audiences, from undergraduates to research mathematicians.
Date: 03/05/2025
Speaker: Gigliola Staffilani (MIT)
Title: What do I see from my corner of wave turbulence theory?
Abstract: Wave turbulence theory is a vast subject and its goal is to formulate for us a multiscale picture of wave interactions. Phenomena involving interactions of waves happen at different scales and in different media: from gravitational waves to the waves on the surface of the ocean, from our milk and coffee in the morning to infinitesimal particles that behave like wave packets in quantum physics. These phenomena are difficult to study in a rigorous mathematical manner, but because of this challenge, mathematicians have developed interdisciplinary approaches that are powerful and beautiful. I will describe some of these approaches and I will outline along the way questions that remain open in spite of the great progress already made.
Date: 03/19/2025
Speaker: Dave Perkins (Colgate University)
Title: What do I see from my corner of wave turbulence theory?
Abstract: In 1843, Ada Lovelace published notes on a machine called The Analytical Engine that was never built, existing only in her friend Charles Babbage's mind (and on his thousands of pages of design notes). The machine was meant to run algorithms that were encoded on punch cards, making it a sort of universal computing device. Lovelace's notes contain a walkthrough of how the Analytical Engine could be used to recursively calculate Bernoulli numbers, an ambitious endeavor that has prompted many to grant her the title of world's first programmer. This talk is suitable for audiences ranging from undergraduate students to researchers in mathematics and computer science.
Date: 03/26/2025
Speaker: Yue Yu (Lehigh University)
Title: Nonlocal Attention Operator: Towards a Foundation Model for Physical Responses
Abstract: While foundation models have gained considerable attention in core AI fields such as natural language processing (NLP) and computer vision (CV), their application to learning complex responses of physical systems from experimental measurements remains underexplored. In physical systems, learning problems are often characterized as discovering operators that map between function spaces, using only a few samples of corresponding function pairs. For instance, in the automated discovery of heterogeneous material models, the foundation model must be capable of identifying the mapping between applied loading fields and the resulting displacement fields, while also inferring the underlying microstructure that governs this mapping. While the former task can be seen as a PDE forward problem, the later task frequently constitutes a severely ill-posed PDE inverse problem.
In this talk, we will explore the development of a foundation model for physical systems, by learning neural operators for both forward and inverse PDE problems. Specifically, we show that the attention mechanism is mathematically equivalent to a double integral operator, enabling nonlocal interactions among spatial tokens through a data-dependent kernel that characterizes the inverse mapping from data to the hidden PDE parameter field of the underlying operator. Consequently, the attention mechanism captures global prior information from training data generated by multiple systems and suggests an exploratory space in the form of a nonlinear kernel map. Based on this theoretical analysis, we introduce a novel neural operator architecture, the Nonlocal Attention Operator (NAO). By leveraging the attention mechanism, NAO can address ill-posedness and rank deficiency in inverse PDE problems by encoding regularization and enhancing generalizability. To demonstrate the applicability of NAO to material modeling problems, we apply it to the development of a foundation constitutive law across multiple materials, showcasing its generalizability to unseen data resolutions and system states. Our work not only suggests a novel neural operator architecture for learning an interpretable foundation model of physical systems, but also offers a new perspective towards understanding the attention mechanism.
Date: 04/02/2025
Speaker: Linghai Zhang (Lehigh University)
Title: The Influence of Physical Mechanisms on Global Solutions of Nonlinear Evolution Equations with Dissipations
Abstract: We study the influence of physical mechanisms (represented by dispersion coefficients, diffusion coefficients, incompressible conditions, spatial dimension, integrals of the initial functions, the integrals of the external forces, etc) on the global solutions of nonlinear evolution equations with dissipation.
Typical model equations include:
(1) the n-dimensional incompressible magnetohydrodynamics equations,
(2) the n-dimensional incompressible Navier-Stokes equations,
(3) the two-dimensional incompressible dissipative quasi-geostrophic equation,
(4) the Korteweg-de Vries-Burgers equation,
(5) the Benjamin-Bona-Mahony-Burgers equations.
In this talk, I will talk about my recent results on the Korteweg-de Vries-Burgers equation.
Date: 04/09/2025
Speaker: Feng Luo (Rutgers)
Title: Discrete uniformization problem for polyhedral surfaces
Abstract: The classical uniformization theorem for Riemann surfaces applies to all compact and non-compact connected surfaces. In the realm of discrete uniformization problems for polyhedral surfaces, progress has been made for compact surfaces. The major remaining issue is the discrete uniformization problem for non-compact surfaces. The challenges in this new setting include formulating the discrete uniformization problem for non-compact surfaces and determining the precise definition of non-compact polyhedral surfaces. This talk will discuss these problems and their relationships to the classical Schwarz lemma, the Cauchy rigidity theorem, and the Weyl problem. This is a joint work with Yanwen Luo.
Date: 04/30/2025
Speaker: Christine Breiner (Brown University)
Title: TBD
Abstract: TBD