1-2-3 Seminar 2024
1-2-3 Seminar is a student-ran seminar at the University of Washington that takes place every winter and spring quarter. This seminar is a place where we talk about topics near and dear to ourselves, geared towards engaging audiences that are graduate students across different fields. The format of each talk will be three examples in increasing complexity (1-2-3), presented with an emphasis on quality and engagement.
For the year 2023-2024, the 1-2-3 Seminar is organized by Leo Mayer (leomayer@uw.edu) and Haoming Ning (hning99@uw.edu). If you would like the opportunity to present, please contact us!
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For speakers: we encourage you to choose any topic of math that you like for your talk! There're only two requirements for your 50-minute talk:
Your talk should be formatted around three examples of increasing complexity (1-2-3). Theorems (e.g. Hensel's Lemma) are not considered examples!
Your talk should be accessible to graduate students in other fields. People with just a background from first-year courses should understand at least the first half of your talk.
Check out the previous years' schedule if you need ideas!
Spring 2024 Schedule
Time: Friday 2:30 - 3:30 pm
Location: Padelford C401 and on Zoom
Zoom Link: https://washington.zoom.us/j/92849568892
Week 1 - Mar 29
Speaker: Justin Bloom
Abstract: Lie algebras play a fundamental role in the representation theory of groups. The connection between real and complex representations for Lie groups and their corresponding Lie algebras is understood classically, but we will be interested in representations over positive characteristic. In this setting it is important to introduce the concept of a restricted Lie algebra. A particular connection I’ve been interested in is how a given group and restricted Lie algebra can have the same abelian category of representations, but with different tensor products. A group or a restricted Lie algebra may appear innocent enough while having wild representation type, making it very difficult to say anything reasonable about these sorts of things. Nevertheless, we will do what we can, even conjecture if we must!
Zoom Recording (UW sign-in required)
Week 2 - Apr 5
Speaker: Garrett Mulcahy
Abstract: Originally posed in the 18th century by Gaspard Monge, the problem central to optimal transport is to transport a unit mass in some shape (i.e. a probability measure) into another shape while doing the least amount of work. This 1-2-3 talk will walk through the computation of optimal transport maps in explicit settings and conclude with a brief discussion of the geometry of Wasserstein space (the space of probability measures). This expository talk should be accessible to all graduate students; all necessary probability and analysis notions will be introduced in the talk.
Week 3 - Apr 12
Speaker: Tony Zeng
Abstract: Desmos is an online graphing calculator and a versatile tool for mathematical visualization. It has the standard graphing calculator features of defining variables and functions, plotting points, curves, and polygons, and much more. It even includes features that allow for more sophisticated graphs such as physics simulators, 3d rendering, and cellular automata. In this talk, we will explore how these features allow for graphs which more heavily rely on direct user interaction in several fun applications.
Zoom Recording (UW sign-in required)
Week 4 - Apr 19
Speaker: Tyson Klingner
Title: Eigenvalues and Much More
Abstract: Eigenvalues are one of the most foundational concepts in Mathematics and are applicable across the board. In this talk we will build up from eigenvalues associated to endomorphisms of vector spaces all the way to the generalised notion of an eigenvalue to a twisted endomorphism of vector bundles, which is called a spectral curve.
Zoom Recording (UW sign-in required)
Week 5 - Apr 26
Speaker: Tracy Chin
Title: Real Algebraic Geometry
Abstract: This talk will be a survey of a hierarchy of classes of real polynomials. Along the way, we'll investigate the underlying geometry and see some surprising connections to combinatorial objects. We'll start from real-rooted univariate polynomials, then make our way through real stable polynomials to Lorentzian and log-concave polynomials, building more evidence along the way that all math is secretly linear algebra.
Zoom Recording (UW sign-in required)
Week 6 - May 3
Speaker: Cameron Wright
Abstract: In this talk we examine some aspects of the modern theory of hyperplane arrangements, a theory which stars an interesting interplay between combinatorics and topology. Over the course of the talk, we will familiarize ourselves with two main characters associated with any arrangement: the intersection poset and the complex complement. The former is a combinatorial object associated to the arrangement and the latter is an interesting topological space given by taking the complement of the arrangement in a complex vector space. Over the reals, the complement of an arrangement consists of finitely many contractible polyhedra; over the complex numbers, the complement of an arrangement is connected in general and has nontrivial topology. We shall see that several interesting topological invariants of the complex complement are determined by combinatorial invariants associated to the arrangement.
Zoom Recording (UW sign-in required)
Week 7 - May 10
Speaker: Alex Waugh
Abstract: In a first course in algebraic topology, one is often introduced to the cup product as a means of strengthening cohomology as an invariant. That is, two spaces may have the same cohomology groups, but fail to have the same cohomology rings (and are therefore not homotopy equivalent). One can ask if there is a further refinement of cohomology which can be used to distinguish spaces which have the same cohomology rings, but are not homotopy equivalent. We will motivate such a refinement via examples leading to "stable cohomology operations". Finally, we will use these operations to show that pi_4(S^3) is nontrivial.
Zoom Recording (UW sign-in required)
Week 8 - May 17
Speaker: Ting Gong
Title: Stacking bug eyes
Abstract: In this talk, I am going to introduce and give intuitions about many geometric objects via examples. We are going to start with schemes, step towards algebraic spaces, and finally talk about stacks and "gerbes" via the many variations of the example of a bug eyed line.
Week 9 - May 24
Speaker: Junaid Hasan
Title: Discrete Curvatures
Abstract: In this talk we will begin by introducing the classical notion of curvature on Riemannian manifolds and then transition to surveying several recent analogs of curvatures on simplicial complexes and graphs some of which have found applications in network analysis and community detection.
Zoom Recording (UW sign-in required)
Week 10 - May 31
Speaker: Herman Chau
Abstract: In this talk, we'll explore 3 elementary and different ways to look at the same object: rhombic tilings of a regular polygon, sorting networks, and 0-1 matrices called weaving patterns. These different perspectives will help us answer some enumerative questions about these objects.
Zoom Recording (UW sign-in required)
Winter 2024 Schedule
Time: Friday 2:30 - 3:30 pm
Location: Padelford C401 and on Zoom
Zoom Link: https://washington.zoom.us/j/92849568892
Week 2 - Jan 12
Speaker: Linhang Huang
Title: Statistical physics: what and how to study for random stuff
Abstract: The central limit theorem is used frequently in our daily lives, but its statement may still seem surprising. Why can we approximate so many distributions using the bell curve, even for non-continuous or asymmetric ones like students' grades? In (mathematical) statistical physics, we often come across similar phenomena - when random objects converge, the limits (which in some sense are universal) observe new patterns or symmetries. This talk will highlight some examples of such phenomena, along with a brief introduction to the measure-theoretic philosophy of probability and the necessary topological infrastructure.
Zoom Recording (UW sign-in required)
Week 3 - Jan 19
Speaker: Alex Wang
Abstract: One of the most important aspects of the real numbers is that it is complete: every Cauchy sequence of real numbers converges to a real number. Now that we have the attention of the analysts, we can begin an exploration of other ways to complete the rational numbers! This leads us to the p-adic numbers, a collection of complete fields which are closely related to prime numbers, and allow us to capture arithmetic information in an algebraic way. We'll discuss the structure and properties of the p-adics, some techniques to solve problems over these fields, and how they generalize to a broader class of objects known as local fields.
Zoom Recording (UW sign-in required)
Week 4 - Jan 26
Speaker: Andrew Tawfeek
Abstract: Given a tropical cycle X, one can talk about a notion of ''tropical" vector bundles on X having tropical fibers. By restricting our attention to bounded rational sections of these bundles, one can develop a good notion of characteristic classes that behave as expected classically. We present further results on these characteristic classes and use these properties to prove a Porteous' formula for these bundles, which gives a determinantal expression of the fundamental class of degeneracy loci of a (tropical) bundle morphism in terms of their Chern classes.
Zoom Recording (UW sign-in required)
Week 5 - Feb 2
Speaker: Arkamouli Debnath
Title: What is a moduli space?
Abstract: One usually encounters moduli spaces in Algebraic or Complex Geometry, but the concept itself is categorical and can hence be used in almost every branch of mathematics. We will first introduce the idea of a "family" which vaguely speaking is a map from X ---> Y such that each fibre has some special property P. It turns out that we would like to have "families" of various objects of interest, because instead of studying each of them individually, studying them collectively is more systematic and gives stronger understanding. A moduli space, will then be defined as a very special "family" such that each point on it parameterizes a certain object of interest. In fact one is usually introduced to the projective space as something that parameterizes lines in the plane passing through the origin. No background knowledge in any of the above will be required to understand examples 1 and 2 of my talk. Example 3, time permitting, will be related to complex geometry, with very little background assumed.
Zoom Recording (UW sign-in required)
Week 6 - Feb 9
Speaker: Justin Bloom
Title: Modular Representations or: How I Learned to Stop Worrying and Love the Klein 4 group
Abstract: Exploring phenomena in representation theory including tensor structures and support varieties. We will make extensive use of the Klein 4 group for explicit computations, and briefly discuss why this group is so special!
Zoom Recording (UW sign-in required)
Week 7 - Feb 16
Speaker: Michael Zeng
Title: 27 Lines on a Cubic Surface —— via Classifying Spaces!?
Abstract: A famous result in classical enumerative geometry says that there are exactly 27 lines on a cubic surface. Meanwhile, a classifying space classifies principal G-bundles. This talk aims to explore a connection between these seemingly unrelated concepts. We will see how classifying spaces give rise to a theory of equivariant integrals, which in turn provides a novel way of addressing the question: “How many lines are there on a cubic surface?”
Zoom Recording (UW sign-in required)
Week 8 - Feb 23
Speaker: Jackson Morris
Title: What the Hopf?
Abstract: There are many things in math named after Heinz Hopf. What are they? Where do they come from? How do they fit together? In this talk, we will talk about and connect 3 main examples (with some sub-Hopf examples along the way): the Hopf fibration, H-spaces, and Hopf algebras. I Hopf to see you there!
Zoom Recording (UW sign-in required)
Week 9 - Mar 1
Speaker: Joe Rogge
Title: Euclidean Steiner Trees
Abstract: Given a finite set of points S in a metric space, an interconnection network is a collection of paths in which it is possible to start at any point of S and travel to any other point in S. A Steiner tree is an interconnection network of minimum total length, where the minimum is taken over all possible interconnection networks. Steiner trees in Euclidean space (equipped with the l_2 metric) are structurally completely characterized and simultaneously very difficult to compute in practice---NP-hard, in fact. The same is true when Euclidean space is equipped with the l_1 metric. In this talk, we will explore the elegant geometry of Steiner trees in the l_1 and l_2 metrics, see a linear time reconstruction algorithm, and discuss open problems.
Zoom Recording (UW sign-in required)
Week 10 - Mar 8
Speaker: Cody Tipton
Abstract: One of the first examples of Koszul duality in the world of quadratic algebras is the duality between the symmetric algebras S(V) and the exterior algebra \Lambda(V^*). This arises from taking relations in S(V) and finding the orthogonal complement with respect to some non-degenerate bilinear form. Since these algebras are Koszul (having some minimal graded resolution) then there is some equivalence between some subcategories of their derived categories. We will first introduce the conept of Koszul duality in the realm of algebras and describe some important examples.
Next, we will introduce the extension of Koszul duality in the world of operads and give a few implications of the relationships between their respective algebras, specifically to the deformation theories of them. The most important examples of Koszul duality will come from the operads Ass, Com, and Lie, the operads encoding associative, commutative associative, and Lie algebras respectively. In particular, the algebra Ass is Koszul dual to itself and Com is Koszul dual to Lie.
Finally, we will introduce the two operads I have been working with, which are proved to be Koszul dual. Specifically, we will talk about the operad n-Lie_d, the operad encoding n-Lie algebras of degree d, and its Koszul dual n-Com_{-d+n-2}, the operad encoding algebras with n-arity operations of degree -d+n-2 that satisfy relations coming from a Young tableaux of shape (n,n-1).Zoom Recording (UW sign-in required)