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Curves Seminar

Thursday, October 19th, 2023

Time: 4:00 p.m.  Place: Jeffery Hall, Room 102

Speaker: Deepanshu Prasad

Title: Matrix Mutation and Cluster Algebra

Abstract: We'll extend the notion of mutation from quivers to certain class of matrices. In particular, we'll define the matrix mutation of extended skew-symmetrizable matrices. This will allow us to give an axiomatic setup of cluster algebras and put the motivating examples of Grassmannians of 2-planes and basic affine spaces in the framework of cluster algebras.

Dynamics, Geometry and Groups Seminar

Monday, October 16th, 2023

Time: 11:00 a.m.  Place: Jeffery Hall, Room 319

Speaker: Surena Hozoori (University of Rochester)

Title: Symplectic geometry of Anosov 3-flows

Abstract: Since their introduction in the early 1960s, Anosov flows have defined an important class of dynamics, thanks to their many interesting chaotic features and rigidity properties. Moreover, their topological aspects have been deeply explored, in particular in low dimensions, thanks to the use of foliation theory in their study. Although the connection of Anosov flows to contact and symplectic geometry was noted in the mid 1990s by Mitsumatsu and Eliashberg-Thurston, such interplay has been left mostly unexplored. I will present some recent results on the contact and symplectic geometric theory of Anosov flows in dimension 3. Time permitting, various related topics will be discussed, including the interplay of Anosov flows with Reeb dynamics, Liouville geometry, surgery theory, the presence of invariant volume forms, etc.

Curves Seminar

Thursday, October 5th, 2023

Time: 4:00 p.m.  Place: Jeffery Hall, Room 102

Speaker: Julia McClellan

Title: Introduction to Quivers and Quiver Mutations

Abstract: Now that we have seen some motivating examples, we introduce quivers and quiver mutations, which lie at the heart of the combinatorial framework for studying cluster algebras. Once we have these constructions, we will see some important properties of quiver mutations, and finally will look at the quivers associated to triangulations of polygons and wiring diagrams.

Math & Stats Department Colloquium

Friday, October 6th, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Ernesto Pérez-Chavela (ITAM)

Title: A walk with relative equilibria from the plane to the sphere $\mathbb{S}^2$

Abstract: The simplest solutions of the $N$–body problem are those where the mutual distances among the masses remain constant for all time, that is, the motions behave as a rigid body. For $N = 3$ on the Euclidean plane it is well known that there are exactly five relative equilibria: three collinear (Euler relative equilibria) and two planar forming an equilateral triangle (Lagrange relative equilibria).

In this talk, quickly I will describe the above relative equilibria, and I extend this concept to the sphere $\mathbb{S}^2$. The big difficulty to study relative equilibria on the sphere $\mathbb{S}^2$, that we call RE by short, is the absence of the center of mass as a first integral, since many of the standard methods used in the classical case don’t apply any more. Without the center of mass we do not know how to determine the rotation axis. I will show a geometrical method to study relative equilibria on the sphere (RE by short). We assume that the masses are moving under the influence of a general potential which only depends on the mutual distances among the masses. First we prove the existence of two new integrals of motion, which can be seen as an extension of the center of mass. These two new integrals allow us determine the rotation axis. For simplicity in the computations, we restrict our analysis to the case $N = 3$. Applying our method, we give some new families of Euler and Lagrange RE on the sphere for the cotangent potential (the natural extension of the Newtonian potential to the sphere).

Bio: Prof. Pérez-Chavela received his PhD from the Universidad Autónoma Metropolitana-Iztapalapa in 1991, and is now a professor and an emeritus national researcher at Instituto Tecnológico Autónomo de México (ITAM) in Mexico. His main research interests center around Hamiltonian systems, celestial mechanics, as well as more general dynamical systems and the qualitative theory of ODE.

Number Theory Seminar

Tuesday, October 3rd, 2023

Time: 4:00 p.m.  Place: Jeffery Hall, Room 319

Speaker: Sunil Naik (Queen's University)

Title: Brun's inequality for a geometric lattice.

Abstract: In his seminal paper of 1915, Viggo Brun introduced a sieve that is now known as Brun's sieve and it stems from Brun's inequality for the Mobius function. This sieve is a very powerful tool in modern number theory.

In an ongoing work with R. Murty, we prove Brun's inequality for a geometric lattice. Surprisingly, unlike in the classical case, it requires a recent work of K. Adiprasito, J. Huh and E. Katz.

Curves Seminar

Thursday, September 28th, 2023

Time: 4:00 p.m.  Place: Jeffery Hall, Room 102

Speaker: Calvin Fletcher

Title: Total positivity of flag minors

Abstract: Last week we saw our first piece of motivation for studying cluster algebras. In this talk we explore another motivational example, namely, total positivity of flag minors. These are an important subset of matrix minors and are related to the basic affine space by invariant theory. In much the same modus as last week we will construct a ring corresponding to these flag minors, study its generators and see how these generators form extended clusters with important combinatorial data. We will also compare and contrast the examples given to help motivate further the topic of cluster algebras.

Math & Stats Department Colloquium

Friday, September 29th, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Federico Salmoiraghi (ºÚÁϳԹÏ×ÊÔ´)

Title: Foliation, contact structures and Anosov flows in dimension 3.

Abstract: An example of the beautiful intertwine between hyperbolic dynamics, foliation theory, and contact geometry is given by an Anosov flow. Geometrically an Anosov flow is defined by two transverse invariant foliations with expanding and contracting behaviours. Much of our understanding of the structure of an Anosov flow relies on the study of the leaves space of the invariant foliations. Mitsumatsu first noticed that an Anosov vector field also belongs to the intersection of two transverse contact structures rotating towards each other. After giving the necessary background, I will show how to use this point of view to address questions in the theory of surgery on Anosov flows.

Bio: Federico Salmoiraghi joined ºÚÁϳԹÏ×ÊÔ´ as a Coleman postdoctoral fellow in 2022, before that he was postdoc at the Technion from 2019 to 2022 after receiving his PhD from Liousiana State University in 2019. Prof. Salmoiraghi’s work is at the intersection of contact geometry and dynamical systems.

Number Theory Seminar

Tuesday, September 26th, 2023

Time: 4:00 p.m.  Place: Jeffery Hall, Room 319

Speaker: Mike Roth (Queen's University)

Title: The Bugeaud-Corvaja-Zannier theorem and extensions.

Abstract: Two integers a and b are called multiplicatively independent if the only solution (m,n) to a^m = b^n is (m,n)=(0,0). In 2003 Bugeaud, Corvaja, and Zannier proved that, if a and b are multiplicatively independent, then for every epsilon > 0 the inequality log gcd(a^n -1, b^n -1) < (epsilon) n holds for all but finitely many n > 0. In this talk we will discuss the BCZ result and give a new proof of their theorem. This proof makes it easier to see the key ingredients which make the argument work. The method of proof applies equally well to prove an extension of the theorem due to Corvaja and Zannier, and a more general extension due to Aaron Levin.

This is joint work with David McKinnon at Waterloo.