First Year Seminar
Meeting Times: Biweekly on Friday 4pm-5pm
Organizers: Emma Dooley, Joshua Lehman, Leon Menger, Feibai Ren, Fuxiang Yang
September 15th
Speaker: Leon Menger
Title: Axiomatic TQFTs
Abstract: The talk will entail a brief introduction to symmetric monoidal categories and then construct the cobordism category as an example. Finally the requirements of a TQFT in physics are discussed and a functorial description is given. (Warning: Audience participation outside of consuming pizza may be involved!)
September 29th
Speaker: Joshua Lehman
Title: Thurston's Proof of the Poincaré-Hopf Index Theorem
Abstract: The Poincaré-Hopf Index Theorem, whose origins trace back to complex function theory, is a crown jewel of topology - it tells us that the Euler characteristic (something global) of a smooth manifold can be computed by means of only local observations (using vector fields). In this talk, I aim to share with you an amazing proof of this result, due to William Thurston. This proof strikes at the heart of why the result is really true. The talk will be highly non-technical with lots of pictures.
October 13th
Speaker: Fuxiang Yang
Title: Introduction to Algebraic Geometry: Conics
Abstract: In this talk, we will look at the classification problem and enumerative problem in Algebraic Geometry via conics. A conic is a plane curve defined by a degree 2 polynomial. We will classify all irreducible conics up to isomorphisms and discuss why five points (in general position) determine a conic.
October 27th
Speaker: Pengkun Huang
Title: Trivial Math 101: Subtraction, Division and Inversion (Localization of categories)
Abstract: In this talk, I will start with introducing localizations of algebra structures, which is how we get integers and rational numbers. We will see that they are really determined by some universal properties. By mimicking the universal properties, we will observe this idea is easily generalized to define localization of categories, which means inverting a class of morphisms in a category. Using this construction, we will see how to define derived categories, homotopy category of spaces. If time permitted, I will also introduce the Sullivan’s arithmetic square for p-localizations, which is one of main tools of homotopy theories to solve problems.
November 11th
Speaker: Sulin Hu
Title: How logic does magic
Abstract: In this talk I will introduce some basic motivations and concepts in model theory. As an application I will present a striking proof of the Ax-Grothendieck theorem, a result on complex numbers.
November 17th
Speaker: Tan Özalp
Title: Fodor’s Train
Abstract: A crash course on ordinals (some of them) and an introduction to basic set theory concepts like club and stationary sets of ordinals to formulate and solve a riddle.
December 1st
Speaker: Chen-Kuan Lee
Title: Hodge theory of harmonic forms on a compact oriented Riemannian manifold
Abstract: The main purpose of this talk is to give the audiences a glimpse of how PDE works on geometry. We will briefly introduce hodge theory on manifold. If time permits, we will also discuss the concept of elliptic operator.
February 9th
Speaker: Tan Özalp
Title: Arrow's impossibility theorem
Abstract: We prove Arrow's impossibility theorem.
February 23rd
Speaker: Emma Dooley
Title: An Introduction to Intersection Theory
Abstract: In this talk I will give a very basic introduction to intersection theory, a subject which is at the heart of algebraic geometry. I will introduce the Chow ring (modulo a lot of detail) and state (or compute, time-permitting) the Chow ring of some spaces of interest. The goal is to present some concrete problems in enumerative geometry that can be solved using the ideas and techniques from my talk.
March 8th
Speaker: Fuxiang Yang
Title: Introduction to Algebraic Geometry: Riemann-Roch Theorem
Abstract: The Riemann-Roch Theorem is an important theorem in Complex Analysis and Algebraic Geometry. It gives a formula to compute the dimension of the space of rational functions (or meromorphic functions) with certain conditions on a smooth projective curve (or compact Riemann surface). In this talk, I will go step by step in understanding the statement of the Riemann-Roch Theorem.
April 5th
Speaker: Feibai Ren
Title: Analysis on Homogeneous Spaces
Abstract: This will be an introduction to analysis on homogeneous spaces. I will briefly describe fundamental problems in this field stated in Nice, 1970. I aim to provide readable, self-contained introduction to the theory of differential operators on Lie groups, based on the original papers by Harish-Chandra 53’, Schwartz 56’ and Gelfand 50’.
April 19th
Speaker: Joshua Lehman
Title: Infinite Groups as Geometric Objects
Abstract: A general principle of life is that you understand a group by the way that it acts on things - but what does this look like in the realm of infinite groups? My goal is to convince you that there is a meaningful way to think of infinite groups as geometric objects (a slogan I've cribbed from Gromov's ICM article).