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 2024-2025, 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!
Time: Friday 2:30 - 3:30 pm
Location: Padelford C401 and on Zoom
Zoom Link: https://washington.zoom.us/j/92849568892
Note: the Zoom is open to all, but the recording requires an UW sign-in and expire 3 months after the date of the each talk. If you are outside of UW and would like to access to the recording please email us.
Speaker: Jack Kendrick
Abstract: Symmetry arises often when learning from high dimensional data. For example, data sets consisting of point clouds, graphs, and unordered sets appear routinely in contemporary applications and exhibit rich underlying symmetries. Moreover, many functions which we hope to learn from these data sets are well-defined regardless of the ambient dimension of the data. In this talk, we will explore how the phenomenon of representation stability can be exploited to learn invariant functions that generalize well as we vary the underlying dimension of data. In particular, we will discuss three examples of machine learning models: equivariant neural networks, invariant polynomial regression, and invariant kernels.
Zoom Recording (UW sign-in required)
Speaker: Charlie Magland
Abstract: When studying representations of a group or group scheme, we can find new representations by "adding" (taking a direct sum) or "multiplying" (taking a tensor product) two representations. In this way, we have a structure on the category of representations similar to a ring. With some additional conditions to make this nice, we call such a category a (symmetric) tensor category. In this talk we will look at examples of other tensor categories and see how representation theory leads to a conjectured classification of all symmetric tensor categories of moderate growth.
Zoom Recording (UW sign-in required)
Speaker: Ethan MacBrough
Title: Algebraic surfaces
Abstract: An algebraic surface is a smooth projective variety of dimension 2 (in this talk, over C). An extensive study of these objects by the classical Italian school culminated in the 1910s with Enriques' coarse classification into 8 types, but many basic questions about surfaces (especially the surfaces of "general" type) remain unsolved. In this talk I will explain the Enriques classification and present a few explicit examples of surfaces with beautiful hidden geometry.
Zoom Recording (UW sign-in required)
Speaker: Yirong Yang
Abstract: Ternary graphs, or trinity graphs, or however people from different areas call them, are graphs that avoid induced cycles of length divisible by 3. The talk mainly consists of many pictures, a healthy dose of graph theory, and some buzzwords from the ITM, through which we explore the topology of the independence complex of these graphs.
Zoom Recording (UW sign-in required)
Speaker: Jackson Morris
Abstract: Localization techniques are essential to modern mathematics. In commutative algebra and algebraic geometry, localization of rings lets one "zoom-in" on a problem to a particular place, and in nice situations one can assemble all these simpler zoomed-in problems to answer the original one. This idea of "zooming-in" or "inverting irrelevant information is extremely powerful, and its categorical generalizations lead to deep understandings of global structures. In this talk, I will introduce the notion of localization from ring theory and then move on to its categorical analogue, giving examples along the way. I will end with a case-study of brief overview of the telescope conjecture, a long and fabled story about the difference between two particular families of localizations.
Zoom Recording (UW sign-in required)
Speaker: Ting Gong
Abstract: In this talk, I am going to introduce and give intuitions about many geometric objects via examples. We are going to start with ideas in manifolds, step towards schemes and algebraic spaces, and finally talk about stacks and "gerbes" via the many variations of the example of a bug eyed line.
Zoom Recording (UW sign-in required)
Speaker: Jay Reiter
Title: Descent, derived descent, and the Adams spectral sequence
Abstract: Classical faithfully flat descent is a powerful technique from algebraic geometry that allows us to prove things about “harder objects” by extending them to “easier objects,” and then coming back down. In this talk, we’ll see how an even richer theory of descent can be obtained by passing to derived categories. We will present the resulting theory of “derived descent” as an entry point to “derived algebraic geometry,” the theory obtained by replacing ordinary commutative rings with homotopical objects called E-infinity rings. Along the way, we’ll give an algebra-forward introduction to spectra as “the derived category of the sphere” and see how the mechanism for doing derived descent is exactly the Adams spectral sequence.
Zoom Recording (UW sign-in required)
Speaker: Cameron Wright
Abstract: Complex analytic geometry is among the most successful fusions of analysis and algebra in mathematics, and much of this is reliant on the fact that the usual absolute value on $$\mathbb{C}$$ is highly unique. But what happens when we try to do analytic geometry with other absolute values or other fields? In the (more-typical) non-archimedean situation, the metric spaces defined on rational points are particularly badly-behaved, and we are in need of a "fix" to have a coherent notion of analytic geometry. We survey some of these pathological properties, describe how they can be remedied using inspiration from functional analysis, and see how this remedy affords us powerful analytic techniques familiar from the complex setting.
Zoom Recording (UW sign-in required)
Speaker: Andrew Tawfeek
Cancelled
Speaker: Dan Guyer
Abstract: Kneser’s conjecture was open for 20 years before Lovász used the Borsuk-Ulam Theorem to (tightly) bound the chromatic number of all Kneser Graphs. Since then, there has been much work expanding this theory by Babson-Kozlov and others. Our goal will be to get a taste of the general ``test map” strategy and illustrate a few combinatorial results that quickly fall to the power of these techniques. Such results will consist of bounds for the chromatic number of (hyper)graphs and embeddability results for simplicial complexes.
Zoom Recording (UW sign-in required)
Time: Friday 2:30 - 3:30 pm
Location: Padelford C401 and on Zoom
Zoom Link: https://washington.zoom.us/j/92849568892
Note: the Zoom is open to all, but the recording requires an UW sign-in and expire 3 months after the date of the each talk. If you are outside of UW and would like to access to the recording please email us.
Speaker: Andrew Aguilar
Title: When do Lie algebras lie?
Abstract: Much of what can be said about Lie groups comes from studying their Lie algebras. But when does the Lie algebra not give us everything? We will look at the relationship between affine group schemes and their Lie algebras over a field of positive characteristic and highlight the differences from the characteristic 0 case. As motivation, we will look at examples related to the special linear group and do some concrete computations.
Zoom Recording (UW sign-in required)
Speaker: Michael Zeng
Title: Chow groups, quotients, and equivariant intersection theory
Abstract: Chow groups are algebraists' version of homology. They enable us to talk about homologous cycles and intersection of ‘submanifolds’ purely within the confines of schemes, providing generalizations to these notions to varieties over more general fields. However, we often need to consider varieties with interesting symmetries, and this machinery breaks for quotient spaces coming from those group actions.
In this talk, we are going to learn about a ‘fix’ to the above problem via equivariant Chow groups. We will compute concrete examples of chow groups of singular quotients and provide hints towards the category of motivic spaces and intersection theory on quotient stacks. Notes
Zoom Recording (UW sign-in required)
Speaker: Zawad Chowdhury
Abstract: What do random walks, Gale duality and toric ideals have in common? They can all be used to prove results about graphical designs! Graphical designs are defined using graph theory, but they touch fields as far-flung as probability theory, polytopes and algebraic geometry! In this talk we'll see three different graphical design examples and use them to visit connections to three different fields of math.
Zoom Recording (UW sign-in required)
Speaker: Alex Waugh
Abstract: One can attempt to study a (pointed) space X by probing it with (pointed) maps from other (pointed) spaces Y. It turns out that probing with maps from Y = S^n provides a lot of information about X. In particular, if we relax "maps" to "maps up to homotopy", the collection of such maps form the nth homotopy group of X. These are strong invariants in the sense that a map between connected CW-complexes is an equivalence if it induces isomorphisms on nth homotopy groups for each n. However, when we pass to "maps up to homotopy" we are destroying interesting information about maps between Y and X. In this talk we will study the space of maps from spheres S^n to "nice" spaces X, how this relates to the study of homotopy groups and homology groups of X, and, time permitting, how the study of such spaces naturally lead to the first theorem in stable homotopy theory, the Freudenthal suspension theorem.
Zoom Recording (UW sign-in required)
Speaker: Jonathan Niño-Cortes
Abstract: Higher-rank numerical ranges are a generalization of the classical numerical range, which has been extensively studied in linear algebra and operator theory. This talk will introduce the concept of higher-rank numerical ranges and explore their properties from the perspective of real algebraic geometry. To illustrate the key concepts, I will present three concrete examples that showcase the diverse behavior and rich structure of higher-rank numerical ranges.
Zoom Recording (UW sign-in required)
Speaker: Arkamouli Debnath
Abstract: Fundamental group is one of the first algebraic invariants one learns in algebraic topology. Beyond the definition, one learns how to associate the fundamental group with covering spaces. The issue with this approach in algebraic geometry is the lack of a universal cover (due to our requirement for finite type schemes). We use an analogue of covering spaces, namely finite etale maps, to define the Etale fundamental group. Yet another issue with this, is the requirement that the base point be in a separable extension rather than the field itself (i.e the difference between a rational point and a point in a field extension). We resolve that by introducing the Nori fundamental group. In this talk I'll give examples of all 3 kinds of "fun"damental groups.
Zoom Recording (UW sign-in required)
Speaker: Mallory Dolorfino
Abstract: Last week, we saw some strange and, from some points of view, undesirable qualities of the etale fundamental group. Namely, the etale fundamental group of a point is usually not trivial, and the etale fundamental group of SpecZ is 0. From a number theoretic perspective, these qualities of the etale fundamental group make it a great tool for studying the arithmetic properties of schemes. In this talk, we will elucidate this fact by computing the etale fundamental groups of certain fields, generalizing the fact about SpecZ to rings of integers in arbitrary number fields, and exploring the arithmetic implications of Grothendieck's section conjecture.
Zoom Recording (UW sign-in required)
Speaker: Cameron Wright
Abstract: As any matroid theorist will enthusiastically tell you, matroids exhibit many strong and subtle relationships with other areas of mathematics. The past two decades have brought a resurgence of interest in matroid theory, with a crowning jewel being the positive solution of Mason's conjecture in 2018 (by two independent groups). To the surprise of many, one proof of this result uses deep tools from algebraic geometry. We provide an introduction to the theory of matroids through graph theory and polyhedral geometry before building toward some deep connections with algebraic geometry which were used in the proof. In particular, we will see that modern matroid theory benefits from relations with the geometry of toric varieties, intersection theory, and Hodge theory.
Zoom Recording (UW sign-in required)
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