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PDEs & Applications Seminar

Tuesday, November 21st, 2023

Time: 9:30 a.m.  Place: Jeffery Hall, Room 319

Speaker: Zachary Selk (���ϳԹ���Դ)

Title: Stochastic Representations of Solutions of the Wave Equation

Abstract: The Feynman-Kac formula represents solutions to parabolic PDE in a stochastic way through averages of diffusions. It has achieved widespread success in pure mathematics, physics, engineering and numerical methods through Monte-Carlo estimations, leading to thousands of papers since the 1950s. Similarly, solutions of elliptic PDE have stochastic representations through averages of exit times and have again found widespread success. It would be impossible to do justice to the literature of stochastic representations of solutions to either parabolic or elliptic PDE because they have been such vast and successful areas.

However, the hyperbolic case has been largely unstudied. In this talk, I will present a short (6-page!) preprint from Sourav Chaterjee on stochastic solutions to the wave equation https://arxiv.org/abs/1306.2382 It is one of three manuscripts I have found on stochastic solutions to the wave equation. It is insufficient in several ways and leads to several open questions which I will detail.

PDEs & Applications Seminar

Tuesday, November 28th, 2023

Time: 9:30 a.m.  Place: Jeffery Hall, Room 319

Speaker: Anirban Dutta (���ϳԹ���Դ)

Title: Stability properties for an abstract evolution equation

Abstract: In this talk, we will discuss the stability of the zero solution to a class of nonlinear evolution equations in Banach spaces. The approach is based on a linearization principle. The difficulty however is that zero is an eigenvalue of the relevant linear operator. So, the classical linearization principles do not hold. I will introduce a "generalized” linearization principle to study this problem, and obtain asymptotic stability results. Time permitting, we will discuss the instability result as well.

PDEs & Applications Seminar

Tuesday, December 5th, 2023

Time: 9:30 a.m.  Place: Jeffery Hall, Room 319

Speaker: Tyler Meadows (���ϳԹ���Դ)

Title: Numerical methods for 1-D biofilm models

Abstract: Microbial biofilms are communities of microorganisms organized into thin sheets adhered to a surface, often in aqueous environments. Bacteria and other microbes can form biofilms as a method to avoid antibiotics, or otherwise alter their environment to be more favourable. A standard model of biofilm growth is due to Wanner and Gujer (1986). They assume the film to be uniform in directions parallel to the adherence surface, which allows the biofilm to be modeled as one-dimensional. I will review their model, and discuss some of the difficulties that arise when trying to find numerical solutions. Most of these difficulties appear when trying to solve for the nutrient concentrations within the biofilm.

PDEs & Applications Seminar

Tuesday, January 23rd, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 319 (Via Zoom)

Speaker: Henry Shum (University of Waterloo)

Title: Numerical simulations of microswimmers

Abstract: Bacteria are relatively simple microorganisms that have evolved surprisingly effective modes of motility that have allowed them to survive and thrive in almost every environment on Earth, including inside the human body. These and other forms of natural locomotion have inspired designs for artificial motility from aircraft to nanorobots. Focusing on microscale motion in viscous fluids, we consider models for flagellated bacterial motility in which long, passive flagellar filaments are turned by a rotary motor to propel the cell forward like a propeller. We use a combination of boundary element methods and the method of regularized stokeslets to numerically solve the equations of Stokes flow around the swimmer and use a Kirchhoff rod model to describe the elastic dynamics of the flagellum. This model is applied to reveal the consequences of various morphological design parameters for bacteria swimming in free space and near solid boundaries. We also discuss a more general mathematical model for a microswimmer and apply it to describe a synthetic system of active droplets.

Number Theory Seminar

Monday, January 15th, 2024

Time: 3:00 p.m.  Place: Jeffery Hall, Room 319

Speaker: M. Ram Murty (Queen's University)

Title: LINEAR RELATIONS AMONG SPECIAL VALUES OF THE DIGAMMA FUNCTION

Abstract: The digamma function is the logarithmic derivative of the gamma function and its special values appear in the evaluation of periodic L-series. I will discuss some recent joint work with Abhishek Bharadwaj and Siddhi Pathak in this context.

Math & Stats Department Colloquium

Friday, January 12th, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Weijing Tang (CMU)

Title: Survival Analysis via Ordinary Differential Equations

Abstract: Survival analysis is an extensively studied branch of statistics with wide applications in various fields. Despite rich literature on survival analysis, the growing scale and complexity of modern data create new challenges that existing statistical models and estimation methods cannot meet. In the first part of this talk, I will introduce a novel and unified ordinary differential equation (ODE) framework for survival analysis. I will show that this ODE framework allows flexible modeling and enables a computationally and statistically efficient procedure for estimation and inference. In particular, the proposed estimation procedure is scalable, easy-to-implement, and applicable to a wide range of survival models. In the second part, I will present how the proposed ODE framework can be used to address the intrinsic optimization challenge in deep learning survival analysis, so as to accommodate data in diverse formats.

Bio: Weijing Tang is an Assistant Professor in the Department of Statistics and Data Science at Carnegie Mellon University. Her research interests include statistical network analysis, machine learning, and survival analysis with applications to health and social sciences. She has received multiple awards for her research work, including the ASA Nonparametric Statistics, Statistical Learning and Data Science, and ENAR Distinguished Student Paper Awards. Prior to CMU, she was a Postdoctoral Researcher at Harvard University. She received her Ph.D. from the University of Michigan and a B.Sc. from Tsinghua University.

Curves Seminar

Thursday, November 30th, 2023

Time: 4:00 p.m.  Place: Jeffery Hall, Room 102

Speaker: Alexandre (Sasha) Zotine

Title: Classifying Cluster Algebras of Finite Type

Abstract: We'll shift gears by beginning to discuss the classification of cluster algebras of finite type. In this talk, we'll particularly consider the rank one and two cases, which will involve more explicit calculations.

Math & Stats Department Colloquium

Friday, December 1st, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Zihang Lu (���ϳԹ���Դ)

Title: Integrating Multidimensional Longitudinal Features to Discover Disease Phenotypes: A Bayesian Consensus Clustering Approach

Abstract: Clustering longitudinal features is a common research goal in health studies to identify distinct developmental patterns that reflect disease phenotypes and to facilitate targeted intervention. Compared to clustering a single longitudinal feature, integrating multiple longitudinal features allows additional information to be incorporated into the clustering process, which reveals co-existing developmental patterns and generates deeper biological insight. In this talk, I will discuss a newly developed Bayesian clustering approach for clustering multidimensional and high-dimensional longitudinal features with complex data structures. Results from analyzing birth cohort data to discover respiratory phenotypes will be presented and discussed.

Bio: Dr. Zihang Lu is an Assistant Professor in the Department of Public Health Sciences at ���ϳԹ���Դ, with a cross-appointment to the Department of Mathematics and Statistics. He completed an MSc and PhD in Biostatistics from the University of Toronto. Dr. Lu’s research focuses on statistical and machine learning methods motivated by high-dimensional and large health data with complex structures. His research is supported by funding from the Natural Sciences and Engineering Research Council of Canada and the Canadian Institutes of Health Research.

Dynamics, Geometry and Groups Seminar

Monday, November 27th, 2023

Time: 11:00 a.m.  Place: Jeffery Hall, Room 319

Speaker: Chi Cheuk Tsang (UQAM)

Title: Birkhoff sections for Anosov flows

Abstract: A global section to a flow on a 3-manifold is a closed cooriented embedded surface that is positively transverse to the flow lines. A Birkhoff section is a generalization where one allows the surface to admit boundary components tangent to the flow. Using Birkhoff sections, one can convert between dynamical information of 3-dimensional flows and 2-dimensional maps. A classical result of Fried states that every transitive Anosov flow admits a Birkhoff section. The natural next question is how simple of a Birkhoff section can we find. In this talk, we discuss some recent progress on this question. If time permits, we will also explain some tools and ideas used in the proofs of the results that we mention.

Curves Seminar

Thursday, November 23rd, 2023

Time: 4:00 p.m.  Place: Jeffery Hall, Room 102

Speaker: Luke Steverango

Title: Tropical Semifields and Seed Patterns

Abstract: In this talk, we will examine another way of encoding the data we have used previously, specifically focusing on how to encode the data from the frozen variables in our extended exchange matrix. In this case, we introduce the tropical semifield and will express the exchange relations in terms of a new operation called tropical addition. The advantage of this new framework is that it allows us to perform calculations for arbitrary extensions of a given exchange matrix.