GIST will be held in Maloney 560. Contact Matt or Siddharth if you want to speak or attend!
Title: The Thick-Thin Decomposition in Dimensions 2,3 and 4
Abstract: Hyperbolic manifolds can be decomposed into two parts based on a natural notion of local "thickness," called injectivity radius. We'll look at injectivity radius in some surfaces to motivate the thick-thin decomposition for hyperbolic n-manifolds. As an application of this, we'll discuss a way to visualize the geometry of hyperbolic link complements. Finally, we'll discuss some results on cusp shapes of hyperbolic four manifolds.
Title: We are but our tools: Euclidean geometry and the art of folding paper
Abstract: There are fewer things more satisfying to me than Euclidean geometry. One thing that is, is paper-folding! Join me as I talk about them both, and maybe a little mathematical historiography along the way. Pictures and audience engagement will abound!
Title: Very good orbifolds and Selberg’s lemma
Abstract: An orbifold is a space locally modeled on some quotient of R^n by a finite group. An orbifold is called very good if it has a finite cover that’s a manifold. We’ll discuss good and bad orbifolds, and in particular, that hyperbolic orbifolds are very good by Selberg’s lemma, which states the existence of a finite index torsion-free subgroup in a finitely generated linear group.
Title: Progressions and sumsets in combinatorics/dynamics
Abstract:
Q: What's something that can answer combinatorial questions?
A: Combinatorics.
Follow-up Q: What's something that SHOULD answer combinatorial questions?
A: Dynamics!
Value judgements aside, we will look at some questions about fitting arithmetic configurations into certain subsets of the natural numbers. Methods from dynamical systems have yielded new approaches to them throughout the 20th century. I hope this talk is a nice tour of the history of partition regularity and its interfaces with other math. Only one piece of prior knowledge is required, which is how to count all natural numbers in finite time (so we can work through examples efficiently).
Title: LLC for GLn
Abstract: At the turn of the millenia, Harris and Taylor announced the proof of Local Langlands Conjecture for General Linear Group GLn, the proof was then simplified by Henniart and a purely local proof was given by Scholze. In this talk, we will introduce the main geometric objects that are involved in the construction of this grand bridge between the p-adic analytic and algebraic world. Partaking participants should present with their preferred prime p.
Title: An intro to instanton Floer theory and property p conjecture
Abstract: I will introduce a certain TQFT coming from the instanton equation and how it relates to the fundamental group of a homology sphere. Then I will show how one can use this theory to prove the property p conjecture.
Title: The Queen of Mathematics
Abstract: Gauss called number theory “the queen of mathematics.” I don’t know why he would say this, but it’s provocative. My talk will be a historical overview of number theory starting from antiquity via major results and major shifts in tools. We will see number theory go from its fundamentals to the applied discipline that it is today. From counting, to arithmetic, to algebra, to analysis, to geometry, and beyond. My hope is that it can help us understand mathematical trends as a whole as we investigate one of the oldest fields of mathematics.
Title: Dressing manifolds with hyperbolicity
Abstract: Every GT student may have heard (and repeated!) phrases like "most three-manifolds are hyperbolic" or "a random three-manifold is hyperbolic," but I don't believe that anyone gained a single bit of understanding about hyperbolic structures after hearing those statements.
The goals for the talk are (in increasing order of ambition):
0) Understand why most surfaces admit a hyperbolic structure.
1) Why does your favorite hyperbolic and finite volume 3-manifold admit a hyperbolic structure? (only acceptable answer: the figure 8-knot complement)
2) Are there different ways to construct this hyperbolic structure? And if not, what fails?
Do expect some severe wave-handing and incomplete statements.
Title: Theorems without proofs, and proofs without theorems
Abstract: What is a proof? This question is innocuous most of the time, but becomes dangerous when stakes are high. Many famous proofs in the past few decades have stirred controversy as a result of differing standards, and many famous mathematicians have made sweeping statements about their own philosophies on proofs. In this talk, we won't be concerned with metaphysical answers to our initial question. Instead, we'll use it as a starting point to probe aspects of our mathematical culture, like: what does a proof contribute to our knowledge, or who has access to the tools of proofs? We'll let famous controversial proofs guide our inquiry, from the abc conjecture to the 3- and 4-dimensional Poincare conjectures, and we'll discuss the views of some visionary mathematicians on the role of proof in math.
Title: Infinite Chess
Abstract: Chess is usually played on an 8×8 board, which is boring. In this talk we still play chess on an infinite board. We will study some chess positions. And then, we will go through the proof that every countable ordinal can be realized by a position in 3-dimensional infinite chess.
For this talk, it’ll be useful if you know how pawns, bishops, rooks, queens, and kings move in chess. Knights will not appear in this talk so don’t worry about them.
Title: Measoamerican Mathematics
Abstract: I'll talk about how most Mesoamerican cultures used math and how important it was with respect to Astronomy and Divinity