Intro to Research at UCR seminar 2025

Tuesday 3:30-4:30 in 268 Skye Hall

Schedule:

• January 7: Peter Samuelson

Title: Knot invariants from quantum groups

Abstract: A knot is an embedding of the circle into R^3, considered up to isotopy. In the early 80’s, Jones constructed a polynomial invariant of knots using operators algebras. This construction was generalized by Reshetikhin and Turaev using quantum groups, and our goal for this talk is to give an introductory overview of their construction.

• January 14: Maziar Raissi

Title:Physics-Informed Machine Learning: Bridging Data and Scientific Principles

Abstract: Physics-Informed Machine Learning (PIML) is an emerging paradigm that integrates fundamental scientific laws with data-driven models to achieve enhanced accuracy, interpretability, and generalization in diverse applications. This talk explores the synergy between traditional physics-based modeling and modern machine learning techniques, highlighting how physical principles such as conservation laws, symmetry, and governing equations can guide and constrainlearning algorithms. We will discuss foundational concepts, showcasereal-world applications in fields such as fluid dynamics, climatemodeling, and material science, and examine the advantages of PIML inscenarios with sparse or noisy data. Attendees will gain insights into key methodologies, including embedding physics into neural network architectures, enforcing constraints through regularization, and leveraging differentiable programming. The talk will conclude with a discussion on current challenges and future directions, emphasizing the transformative potential of PIML in advancing both science and technology.

• January 21: Mykhailo Potomkin

Title: Introduction to theory of confined active matter

Abstract: Active matter refers to systems of motile agents capable of spontaneous, persistent motion. Examples of active matter range from the nanoscale to human-scale systems, exhibiting fascinating universal properties across all the scales. In this talk, I will focus on the active rod model, which is used to describe the dynamics of motile microorganisms and bio-mimetic micro-particles. Unlike passive systems, active rods subjected to an imposed flow in confined environments exhibit intriguing phenomena, such as boundary accumulation and upstream swimming. These behaviors play a significant role in determining how active rods are transported through microchannels. I will present a multiscale PDE approach that enables the direct computation of the probability distribution function for the location and orientation of active rods. The talk will conclude with some open questions.

• January 28: Morgan Weiler

Title: J-holomorphic curves in low dimensions

Abstract: Symplectic geometry is the mathematics of classical physics. We will introduce J-holomorphic curves, the geodesics of symplectic geometry, and explain two of their applications to symplectic embedding and dynamics problems. The two recent (2010s) results I will discuss in detail are not my own, but instead provide the foundations for my work and motivate much of the field today. 

• February 4: Carl Mautner

Ttile: Representations, singular spaces and matroids

Abstract:  This talk will be an introductory glimpse into the representation theory of finite groups and a connection with the geometry of singular spaces.  Time permitting I will explain how this connection leads, by analogy, to new structure in the combinatorial world of matroids.

• February 11: Jia Gou

Title:  From Signaling Pathways to Cellular Behavior: Applications of Mathematical Biology Models

Abstract: In this talk, I will present my research on mathematical biology modeling across various scales. I will begin by discussing signaling pathways in the Drosophila wing disc, focusing on their roles in growth control and cell polarity determination. I will then explore the integration of growth signaling pathways with mechanical responses. In addition, I will address modeling cell morphology and motility, emphasizing the interaction between cells and their substrate. Different modeling approaches will be introduced, and recent developments in my work will also be discussed. 

• February 18: Amir Moradifam

Title: Inverse Wave Problems and Mean Field Equations

Abstract: In the first part of the talk, I will talk about the inverse problem of determining both the source of a wave and its speed inside a medium from boundary measurements of the wave equation. This problem arises in photoacoustic and thermoacoustic tomography and has important applications in medical imaging. 

In the second part,  I will talk about the Sphere Covering Inequality, which is a geometric inequality for surfaces embedded in three dimensions.  This inequality states that if two distinct surfaces, with Gaussian curvature less than 1,  are conformal to the Euclidean unit disk and share the same conformal factor on the boundary, then their total area must be at least 4π. In other words, these surfaces, when properly rearranged, must cover the entire unit sphere. I will also discuss how this inequality leads to solutions of some long standing conjectures. 

• February 25: No seminar Today Go To the Burton Jones Lecture Wednesday 4-5.

• March 4: Filippo Mazzoli

Title: Properly convex domains in RP^2 and projective convex hulls in the symmetric space of SL(3,R)

Abstract: The projective plane, together with the group of projective transformations, provides a unifying framework in which Euclidean, elliptic, and hyperbolic geometry naturally emerge. In this talk, we will see how projective geometry relates to the symmetric space of the Lie group SL(3,R) and how several interesting geometric objects naturally arise from the data of a properly convex domain in the projective plane. Part of this talk is based on joint (ongoing) work with Martin Bobb and Max Riestenberg.

• March 11: Yiwei Wang

Title: Machine Learning-Enhanced Variational Modeling and Computation

Abstract: Many problems in physics, materials science, biology, and data science can be studied using variational models, which describe a system in terms of its energy and how that energy changes over time. In this talk, we will explore how machine learning can help both in building these models and in developing effective numerical methods to solve them. The talk will have two parts. In the first part, I will introduce structure-preserving numerical methods that use neural networks for spatial discretization, ensuring that key physical properties are maintained in computations. In the second part, I will discuss how combining physics-based models with data-driven techniques can lead to better variational models to tackle complex problems in science and engineering.