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 2022-2023, 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!
Subscribe to the 1-2-3 mailing list to get notifications!
Time: Monday 4 - 5pm
Location: Padelford C401 and on Zoom
Zoom Link: https://washington.zoom.us/j/92849568892
Note: the Zoom recordings will expire 3 months after the date of the each talk
Speaker: Dan Guyer
Abstract: In 1935, Erdős and Szekeres posed (and answered) the following question. For each n>2, is it possible to put enough points in general position in the plane so that it is guaranteed that some n-subset of these points forms a convex n-gon? This question is indeed true, and the exact number of points needed was conjectured by Erdős and Szekeres in 1935. Later, natural generalizations of this question were formed in higher dimensions. We will begin with the base cases of this problem in two and three dimensions. Then, we will discuss a purely graph theoretic Ramsey argument that immediately proves the existence of such point sets. Finally, we will explore geometric cup-cap arguments that have been used to provide tighter bounds on the size of these point sets.
Zoom recording (UW sign-in required).
Speaker: Emily Casey
Abstract: What is a rectifiable set? We can think of a rectifiable set as the image of countably many "nice'' maps—in particular, maps that have nice differentiability properties. So, it would make sense that at almost every point of these rectifiable sets there should be a tangent. But, is there? And if so, in what sense? We seek to understand these questions through three key examples. This talk is purely expository and is based on a survey: Rectifiability; A Survey by Pertti Mattila.
Zoom recoding (UW sign-in required).
Speaker: Nelson Niu
Title: Sheaves and Toposes in Logic, or “What’s Bigger Than 2? A Topological Space”
Abstract: In math, every statement is either true or false; an element is either in a set or out of it. But real life is not so simple: the statement “it is raining” is only ever true on a subset of the 2-sphere on which we live, and I am in Padelford only at certain times of the day on certain days of the week. Can all this be formalized mathematically? It turns out that by leaving the world of sets behind, we’ll enter an alternate world at once strange and familiar, in which there are many more truth values possible than simply “true” and “false”—as many, in fact, as there are open sets in a given topology. Each such world is a special category called a topos, in which sets are replaced by sheaves. No prior knowledge of category theory, sheaves, or toposes necessary: we’ll build everything up from sets and functions.
Zoom recoding (UW sign-in required).
Speaker: Brian Nugent
Abstract: Morphisms between varieties can seem pretty mysterious at first. Luckily, linear systems give a useful and wonderfully geometric way of looking at morphisms into projective space. In this talk, we will discuss linear systems on P^n and see how they lead us to other interesting geometric phenomena such as blowing up and a curve that intersects itself -1 times.
Zoom recoding (UW sign-in required).
Speaker: Tony Zeng
Abstract: Desmos is a popular online graphing calculator commonly used in educational settings. It has many features that your household TI-84 would have, including most common calculator functions and the ability to plot graphs of functions. Moreover, it is built to be friendly to direct user interaction and animations. In this talk, we will explore some of Desmos's lesser-known features, including colors, lists, and actions, which give us a great deal more power and allow us to create far more interesting visualizations.
Zoom recoding (UW sign-in required).
Speaker: Haocheng Cai
Title: An Introduction to D-Modules Without Algebraic Geometry
Abstract: The theory of D-modules, developed in the 1970s, is renowned for its success in proving many significant theorems, including the Riemann Hilbert correspondence. In this presentation, I will introduce the foundational object in the theory of D-modules: the nth Weyl algebra. By treating it as a ring of differential operators on the polynomial ring, we will explore its properties and establish the connection between modules over the Weyl algebra (D-modules) and systems of partial differential equations. While a familiarity with partial derivatives is necessary, knowledge of partial differential equations is not required for understanding.
Zoom recoding (UW sign-in required).
NO Seminar -- Room Conflict.
Speaker: Natasha Crepeau
Abstract: A divisor on a graph is an assignment of integers to the vertices of the graph. We can go from one divisor to another by performing a sequence of "chip-firing moves". One special type of divisor are break divisors, which are induced by a spanning tree and orientation of edges not in the tree. In this talk, we will examine properties of break divisors and define generalized break divisors. We will also discuss connections to Jacobians and results in the genus 1 case.
Zoom recoding (UW sign-in required).
Speaker: Vasily Ilin
Abstract: The usual approach to solving the heat equation and its various Fokker-Planck friends is SDE Monte Carlo simulation. In this talk we will see a (mostly) deterministic algorithm that relies on approximating the gradient-log pdf. The talk will feature a little bit of everything except AG: PDEs, optimal transport, probability, neural networks, and algorithms.
Zoom recoding (UW sign-in required).
NO Seminar -- University Holiday.
Time: Friday 2:30 - 3:20pm
Location: Padelford C401 and on Zoom
Zoom Link: https://washington.zoom.us/j/92849568892
Speaker: Justin Bloom
Title: Support and Structure
Abstract: We will look at a geometric tool called support, and explore, through examples of modules over PIDs, its connection to classification (the structure theorem). We will compute cohomology groups; this talk is dual to the presentation on local cohomology given in Algebraic Structures last quarter. We will only compute using elementary abelian groups by hand, but some of us may see shadows of the theorems discussed in that presentation!
Zoom recording (UW sign-in required).
Speaker: Leo Mayer
Abstract: I will give a brief overview of the theory of varieties and motivate why the theory of schemes is more flexible. I will then give an overview of the theory of Gröbner bases, and as an application I will construct a variety/scheme whose points correspond to configurations of points in the plane. No background in algebraic geometry is assumed.
Zoom recording (UW sign-in required).
Speaker: Alex Wang
Title: Number Fields
Abstract: The integers Z satisfy many interesting properties: we have the rational root theorem, all integers admit a unique factorization, and Z is integrally closed in Q, its field of fractions. However, as we consider larger fields that contain Q, not all of these facts remain true. In this talk, we'll explore number fields, which are finite extensions of Q, and how these number-theoretic properties generalize.
Zoom recording (UW sign-in required).
Speaker: Cameron Wright
Abstract: We survey the geometric-combinatorial theory of triangulations and subdivisions of polytopes. This topic is of classical interest in the context of combinatorics, and indeed of classical childhood interest for many of us (as witnessed by the success of toys like Legos). However, over the past century it was discovered that this subject possesses several connections to other areas of mathematics. We study subdivisions from three perspectives of increasing abstraction, and will see at the end that these perspectives provide interesting applications. Along the way we will encounter characters from enumerative and geometric combinatorics, as well as from algebraic geometry if time permits.
Zoom recording (UW sign-in required).
Speaker: Jackson Morris
Title: Spectral Sequences
Abstract: We will investigate spectral sequences, a powerful tool that allows us to approximate some unknown homology. After developing some general theory, we will compute some singular cohomology and group cohomology using specific spectral sequences.
Zoom recording (UW sign-in required).
Speaker: Alex Galarraga
Abstract: Beginning with a series of examples, we will explore the arithmetic of postcritically finite polynomials, a dynamically defined class of polynomials. We will end with some work in progress and a small result.
Zoom recording (UW sign-in required).
Speaker: Yirong Yang
Abstract: How many combinatorially distinct n-vertex triangulations of a d-dimensional sphere are there? In this talk, we will have a crash course in simplicial complex, going through basic definitions and tools, and use them to build A LOT OF spheres (a construction due to Nevo, Santos, and Wilson) to achieve the currently known best lower bound for the case when d is odd!
Zoom recording (UW sign-in required).
Speaker: Josh Hinman
Abstract: Serge Lang said, in praise of Grothendieck’s Éléments de géométrie algébrique: “A theorem is not true any more because one can draw a picture, it is true because it is functorial.” With all due respect to Lang and Grothendieck, I disagree – in my opinion, a theorem often is true because one can draw a picture. Nowhere is this clearer than in the study of polytopes. In my talk, I’ll demonstrate how to visualize higher-dimensional polytopes, show you some of my favorite ones, and discuss a few results about them. By the end, I hope to convince you that solving a math problem often just means drawing the right picture.
Zoom recording (UW sign-in required).
Speaker: Andrew Tawfeek
Title: An Honest-to-Goodness Explanation of Tropical Geometry
Abstract: You may have heard those odd combination of words: "tropical geometry", but what in the world is it? Where did it come from and why are various people from combinatorics, number theory, and geometry, all suddenly talking about it? Is it secretly a big ploy for mathematicians across fields to get money for hosting conferences in tropical locations? In this talk, we'll finally get our answer to these pressing questions: and see an example for why each of these subjects cares for this subject.
Zoom recording (UW sign-in required).