Fall 2021
Fall 2021
Fall 2021
- September 2 -- Grad Student Panel
- September 9 -- Kewen Wang
Title: Tate's Thesis
Title: Tate's Thesis
Abstract: In 1950, Tate gave an elegant way to reformulate the proof of the functional equation of Hecke L-function (i.e the zeta function twisted by a Hecke character) in his PhD Thesis. In his thesis, instead of considering the Hecke L-function, which seems like merely the information on the archimedean place, he lifted the zeta function on the ideles of the number field. In this talk, I will give a brief recall of the proof of the functional equation of the Riemann Zeta function, and talk about the basic idea of how Tate came up with the local functional equations and how he used it to derive the global functional equation for L-functions.
Also, I will cover some basic properties for Adeles and Ideles for number fields and thus there will be no prerequisite for this talk.
Abstract: In 1950, Tate gave an elegant way to reformulate the proof of the functional equation of Hecke L-function (i.e the zeta function twisted by a Hecke character) in his PhD Thesis. In his thesis, instead of considering the Hecke L-function, which seems like merely the information on the archimedean place, he lifted the zeta function on the ideles of the number field. In this talk, I will give a brief recall of the proof of the functional equation of the Riemann Zeta function, and talk about the basic idea of how Tate came up with the local functional equations and how he used it to derive the global functional equation for L-functions.
Also, I will cover some basic properties for Adeles and Ideles for number fields and thus there will be no prerequisite for this talk.
- September 30 -- Zihao Liu
Title: Injective hulls of certain discrete metric spaces and groups
Title: Injective hulls of certain discrete metric spaces and groups
Abstract: Injective metric spaces share many properties with CAT(0) spaces. It has been proved that every metric space X has an injective hull E(X). In this talk, I will present the known result that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E(X) is a locally finite polyhedral complex with finitely many isometry types of n-cells, isometric to polytopes in $l_{\infty}^n$. This applies to a result for injective metric spaces analogous to a result for CAT(0) spaces.
Abstract: Injective metric spaces share many properties with CAT(0) spaces. It has been proved that every metric space X has an injective hull E(X). In this talk, I will present the known result that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E(X) is a locally finite polyhedral complex with finitely many isometry types of n-cells, isometric to polytopes in $l_{\infty}^n$. This applies to a result for injective metric spaces analogous to a result for CAT(0) spaces.
- October 14 -- Shizhe Liang
Title: Generalized Transversal Structures and Applications to Drawing
Title: Generalized Transversal Structures and Applications to Drawing
Abstract: In 1989, Walter Schnyder showed that every maximal plane graph admits a special decomposition into 3 trees, which are now known as the Schnyder woods. Using this structure, Schnyder presented a straight-line grid drawing algorithm for all planar graphs. Since then, graph theorists have discovered many other structures with a flavor similar to Schnyder woods. Two of them are Bernardi and Fusy's Schnyder decompositions for d-angulations, and Kant and He's regular edge labelings. In this talk, I will show that they are just two special cases of a more general structure, and discuss some possible ways to derive new algorithms based on this bigger, more versatile structure.
Abstract: In 1989, Walter Schnyder showed that every maximal plane graph admits a special decomposition into 3 trees, which are now known as the Schnyder woods. Using this structure, Schnyder presented a straight-line grid drawing algorithm for all planar graphs. Since then, graph theorists have discovered many other structures with a flavor similar to Schnyder woods. Two of them are Bernardi and Fusy's Schnyder decompositions for d-angulations, and Kant and He's regular edge labelings. In this talk, I will show that they are just two special cases of a more general structure, and discuss some possible ways to derive new algorithms based on this bigger, more versatile structure.
- October 21 -- Ray Maresca
Title: Equivalence of Serre Duality and Auslander-Reiten Triangles in 'nice' Categories
Title: Equivalence of Serre Duality and Auslander-Reiten Triangles in 'nice' Categories
Abstract: Classically, algebraic geometry and Serre duality was done over commutative rings; however, more recently people have been interested in noncommutative algebraic geometry. As a result, people have been interested in finding connections between noncommutative algebraic geometry and path algebras of quivers. It turns out that the two fields are extremely closely related as their bounded derived categories under some assumptions are equivalent. But in this talk we will discuss work done by Idun Reiten and Michel Van den Bergh in 2001. In particular we will stay far away from algebraic geometry and quickly run through some category theory, define Serre functors and A-R triangles, and finally state a couple results from Reiten and Van den Bergh's paper "Noetherian hereditary abelian categories satisfying Serre duality".
Abstract: Classically, algebraic geometry and Serre duality was done over commutative rings; however, more recently people have been interested in noncommutative algebraic geometry. As a result, people have been interested in finding connections between noncommutative algebraic geometry and path algebras of quivers. It turns out that the two fields are extremely closely related as their bounded derived categories under some assumptions are equivalent. But in this talk we will discuss work done by Idun Reiten and Michel Van den Bergh in 2001. In particular we will stay far away from algebraic geometry and quickly run through some category theory, define Serre functors and A-R triangles, and finally state a couple results from Reiten and Van den Bergh's paper "Noetherian hereditary abelian categories satisfying Serre duality".
- October 28 -- Jiajie Zheng
Title: Metric Theory of Numbers and Recurrence in Ergodic Theory
Title: Metric Theory of Numbers and Recurrence in Ergodic Theory
Abstract: One of the most fundamental results in ergodic theory is the Poincare Recurrence Theorem, which asserts that almost all points in measurable dynamical systems return close to themselves under a measure-preserving map. In Michael Boshernitzan's 1993 paper, Quantitative Recurrence Results, he proved one of the first results concerning the speed of recurrence. Now, generalizations of Boshernitzan's quantitative recurrence are still of interest in ergodic theory. In this talk, I will talk about the motivation behind recurrence in ergodic theory and some of the recent progress concerning this topic.
Abstract: One of the most fundamental results in ergodic theory is the Poincare Recurrence Theorem, which asserts that almost all points in measurable dynamical systems return close to themselves under a measure-preserving map. In Michael Boshernitzan's 1993 paper, Quantitative Recurrence Results, he proved one of the first results concerning the speed of recurrence. Now, generalizations of Boshernitzan's quantitative recurrence are still of interest in ergodic theory. In this talk, I will talk about the motivation behind recurrence in ergodic theory and some of the recent progress concerning this topic.
- November 4 -- Rebecca Rohrlich
Title: Intro to Toric Varieties
Title: Intro to Toric Varieties
Abstract: I will give the basic definition and construction of toric varieties.
Abstract: I will give the basic definition and construction of toric varieties.
- November 11 -- Alex Semendinger
Title: The Basics of Bass-Serre Theory
Title: The Basics of Bass-Serre Theory
Abstract: Bass-Serre theory is a way to study the action of groups on trees and is an important tool in geometric group theory. I will present the basic ideas, including graphs of groups, their fundamental groups, and what they're good for.
Abstract: Bass-Serre theory is a way to study the action of groups on trees and is an important tool in geometric group theory. I will present the basic ideas, including graphs of groups, their fundamental groups, and what they're good for.
- November 18 -- Yu Xin
Title: The Langlands Correspondence
Title: The Langlands Correspondence
Abstract: The Langlands correspondence is mostly the connection between Galois representations and automorphic representation. This lecture aims to introduce the main statements of the Langlands conjectures in a conceptual sense on the local and global cases.
Abstract: The Langlands correspondence is mostly the connection between Galois representations and automorphic representation. This lecture aims to introduce the main statements of the Langlands conjectures in a conceptual sense on the local and global cases.
- December 2 -- Simon Huynh
Title: Introduction to Delay Differential Equations with a Stability and Bifurcation Analysis of a Nonlinear Metal Drilling Model
Title: Introduction to Delay Differential Equations with a Stability and Bifurcation Analysis of a Nonlinear Metal Drilling Model
Abstract: Delay Differential Equations (DDEs) are a type of differential equations in which the dynamics of the systems depend not only on the present but also on the past state. They provide a more realistic way to study real-world phenomena, some of which have previously been modeled by ODEs. In a specific area of research, DDEs can naturally be used to model regenerative chatter, a phenomenon that happens in metal cutting and drilling. We will explore the properties of time-delay dynamical systems including stability of solutions and bifurcation. We will also look at an analysis of a nonlinear DDE model for chatter in metal drilling by Stone and Campbell. [https://link.springer.com/content/pdf/10.1007/s00332-003-0553-1.pdf].
Abstract: Delay Differential Equations (DDEs) are a type of differential equations in which the dynamics of the systems depend not only on the present but also on the past state. They provide a more realistic way to study real-world phenomena, some of which have previously been modeled by ODEs. In a specific area of research, DDEs can naturally be used to model regenerative chatter, a phenomenon that happens in metal cutting and drilling. We will explore the properties of time-delay dynamical systems including stability of solutions and bifurcation. We will also look at an analysis of a nonlinear DDE model for chatter in metal drilling by Stone and Campbell. [https://link.springer.com/content/pdf/10.1007/s00332-003-0553-1.pdf].
Acknowledgment: I want to thank Prof. Touboul for introducing me to the topic and for his support this semester.
Acknowledgment: I want to thank Prof. Touboul for introducing me to the topic and for his support this semester.