Friday, Nov 18 — Holt 175, 4:00pm
Liam Watson — Univ. of British Columbia
Title: Counting Tori in Three-Manifolds
Abstract: I will describe some recent developments in Heegaard Floer homology in the context of manifolds with torus boundary. The talk will build from simple combinatorics, and not assume any knowledge of three-manifolds or their Floer-theoretical invariants.
Friday, Oct 28 — Holt 175, 4:00pm
Lauren Cappiello — CSU, Sacramento
Title: Challenges and Successes of Emergency Online Teaching
Abstract: As the COVID-19 pandemic took hold in the early months of 2020, education at all levels was pushed to emergency fully remote, online formats. This emergency shift affected all aspects of teaching and learning with very little notice and often with limited resources. Educators were required to convert entire courses online and shift to remote instructional approaches practically overnight. Students found themselves enrolled in online courses without choice and struggling to adjust to their new learning environments. This article highlights some of the challenges and successes of teaching emergency online undergraduate statistics courses, which serve students from a variety of backgrounds and abilities. In particular, we discuss challenges and successes related to (1) technology, (2) classroom community and feedback, (3), student engagement with course material, and (4) student workload and deadlines. We also reflect on the opportunity to continue to enhance and enrich the learning experiences of our students by utilizing some of the lessons learned from emergency online teaching as new permanent online statistics courses are developed and/or are moved back into the classroom.
Friday, Oct 21 — Holt 175, 4:00pm
Matthew Hedden — Michigan State University
Title: Knot theory and algebraic curves
Abstract: The modern study of knots and links has important roots in the theory of algebraic curves, where links encode subtle features of singularities. This thread was taken in interesting new directions in the 20th century, and the interaction between links in 3-dimensional manifolds and algebraic curves in complex surfaces continues to be a rich and beautiful area. In this talk I will survey the subject, from its inception in the early 1900’s to interesting advances which have occurred in the past decade.
Friday, Oct 14 — Holt 175, 4:00pm
Hiro Lee Tanaka — Texas State University
Title: Streams of water and the shapes of things
Abstract: If you pour water on a statue, the water will descend in predictable ways. Amazingly, if you know exactly how streams of water on a statue can break, you can reconstruct the statue's shape. This fact is a toy example of Morse theory, a field of topology that tries to understand shapes of things in terms of streams of least resistance (sometimes called gradient flows). I want to introduce this theory in some fun, concrete examples, and as time allows, talk about a mysterious, recent discovery that connects two disparate branches of math: The geometry of Morse theory encodes exactly the algebra of associativity.
Friday, Sept 22 -- Holt 175, 4:00pm
Diego Ricciotti -- CSU, Sacramento
Title: Mean Value Properties for harmonic and p-harmonic functions
Abstract: Harmonic functions are solutions of the Laplace equation, one of the most important linear partial differential equations that arises, for example, from the minimization of quadratic functionals. On the other hand, the minimization of functionals with super-quadratic (or sub-quadratic) growth leads to a nonlinear generalization of the Laplacian, called the p-Laplacian, whose solutions go under the name of p-harmonic functions. In this talk we will review some mean value properties that characterize harmonic functions, and we will explore recent results about asymptotic mean value properties that can be used to study and characterize p-harmonic functions.
Friday, Sept 16 -- Holt 175, 4:00pm
Jorge Garcia -- CSU, Channel Islands
Title: Constructible Sets
Abstract: Given an initial family of sets, we may take unions, intersections and complements of the sets contained in this family in order to form a new collection of sets; our construction process is done recursively until we obtain the last family. Problems encountered include the minimum number of steps required to arrive to the last family as well as a characterization of that last family; we solve all those problems. We define a class of simple families ( -minimal constructible) and analyze the relationship between partitions and separability (our new concept) that leads to interesting results such as finding families based on partitions that generate finite algebras. We prove a number of new results about -minimal constructible families such as every finite algebra of sets has a generating family which is -minimal constructible for all and we compute the minimum number of steps required to generate an algebra.
Friday, Sept 2 -- Holt 175, 4:00pm
Melissa Zhang -- Simons Laufer Mathematical Sciences Institute
Title: Using Algebra to Study the 4D Behavior of Knots
Abstract: A knot is a closed loop of string in 3-space. When subject to gentle wiggling and deforming, a knot does not lose its most inherent properties, making it a foundational object in many areas of topology. The simultaneous visual intuitiveness and rich diversity of knots allows them to also serve as bookkeeping tools for many other fields, including statistical mechanics, algebraic combinatorics, quantum computing, and many more.
Given two random pictures of complicated knots, it would be near impossible for a human to immediately determine whether the knots are different. For tasks like this, we use knot invariants, which assign diagrams of knots to some other mathematical object that we understand better. For instance, a famous knot invariant called the Jones polynomial assigns to each knot diagram a (Laurent) polynomial, and if given two different diagrams of the same knot, it will produce the same polynomial.
Knot invariants like the Jones polynomial are useful not only for telling knots apart but also for classifying them based on their similarities. A more recent enhancement of the Jones polynomial called Khovanov homology assigns a more complex algebraic object to a knot diagram. By studying knots through their Khovanov homology, one can now capture relationships between knots, or more precisely, the evolution of knots throughout 4D spacetime.
In this talk, we will get a taste of how Khovanov homology extends the Jones polynomial to the fourth dimension, and explore several ways the 4D behavior of knots makes them even more interesting.