Dartmouth Geometry Seminar
Winter-Spring 2024
Thursday 1:20-2:20 PM
307 Kemeny Hall
This term, the Geometry seminar will use the same weekly slot as the Functional Analysis seminar.
The schedule below includes both Geometry and Functional Analysis talks. Functional analysis talks are marked in blue.
Special meeting times are marked in red.
Schedule
Date Speaker Title
January 4 Jane Wang
(The University of Maine)
Slope gap distributions: from Farey fractions to translation surfaces
January 18 Surena Hozoori
(The University of Rochester)
Anosov contact metrics, Dirichlet optimization and entropy
January 25 Mohammad Latifi-Jebelli
(Dartmouth College)
An elementary prelude to quantum fields, I
February 8 Tamara Kucherenko
(City College of New York and CUNY Graduate Center)
Realization of Anosov Diffeomorphisms on the Torus
February 15 Mohammad Latifi-Jebelli
(Dartmouth College)
An elementary prelude to quantum fields, II
February 22 Ben Call
(The University of Illinois Chicago)
Mixing Properties of Equilibrium States for Systems Beyond Negative Curvature
February 29 Ian Adelstein
(Yale University)
Geometric Manifold Learning
March 28 Thomas Barthelmé
(Queen's University)
Group actions on bifoliated planes and classification of (pseudo)-Anosov flows in dimension 3
April 25 Kathryn Lindsey
(Boston College)
TBD
May 9 Joel Rosenfeld
(University of South Florida)
TBD
May 16 Diana Davis
(Philips Exeter Academy)
TBD
May 23 Daniel Mitsutani
(The University of Chicago)
TBD
May 30 Dave Constantine
(Wesleyan University)
TBD

Abstracts

January 4, 2024: Jane Wang "Slope gap distributions: from Farey fractions to translation surfaces"

Abstract: The Farey fractions of order n are an ordered list of reduced fractions between 0 and 1 with denominators at most n. For example, F_4 = {0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1}. A natural question that one could ask is: how random are these sets of fractions? One way to measure their randomness is to ask for a function called their limiting gap distribution. While the gap distribution of the Farey fractions has been known for some time, recent work of Athreya and Cheung rederived the gap distribution using dynamical techniques relating to dynamics on moduli spaces of translation surfaces.
Translation surfaces are geometric surfaces that are locally Euclidean except at finitely many cone points. In this talk, we will explain what the slope gap distribution of a translation surface is and survey some known results about slope gap distributions, including how one can use dynamical properties of the horocycle flow to compute the slope gap distributions of special translation surfaces called Veech surfaces (the Farey fractions are related to the square torus, an example of a Veech surface). We'll then discuss results showing that the slope gap distributions of Veech surfaces have to satisfy some nice properties. This project is joint work with Luis Kumanduri and Anthony Sanchez.

January 18, 2024: Surena Hozoori "Anosov contact metrics, Dirichlet optimization and entropy"

Abstract: In 1984, Chern and Hamilton initiated the study of various energy functionals on contact 3-manifolds. In particular, given an almost complex structure, they define its Dirichlet energy as the L^2-norm of its variations along the associated Reeb flow and ask when the minimum of such energy functional is achieved. We classify all the minimizers of the Dirichlet energy functional, giving a complete answer to the Chern-Hamilton question. Furthermore, in the case of Anosov contact manifolds, we show that the optimization of this functional is closely related to Reeb dynamics and can be computed in terms of its entropy.

January 25, 2024: Mohammad Latifi-Jebelli "An elementary prelude to quantum fields, I"

Abstract: In my two talks on quantum field theory, I aim to explore the mathematical aspects of the theory while keeping it relatable for physicists. During the first talk, I'll discuss the Lagrangian formulation of Euclidean field theories. We'll also dive into Feynman diagrams, exploring their importance from a probabilistic standpoint. At times, we'll consider the idea that quantum field theory might have been discovered through a mathematical lens (much like calculus) with potential applications beyond the realm of physics. In my second talk, I plan to visit measure theory in infinite dimensions and review rigorous constructions of quantum field theories.

February 8, 2024: Tamara Kucherenko "Realization of Anosov Diffeomorphisms on the Torus"

Abstract: We consider area preserving Anosov maps on the two-dimensional torus within a fixed homotopy class. We show that the set of pressure functions for Anosov diffeomorphisms with respect to the geometric potential is equal to the set of pressure functions for the linear Anosov automorphism with respect to H\"{o}lder potentials. We use this result to provide a negative answer to the $C^{1+\alpha}$ version of the question posted in the paper by A. Erchenko: ``if two homotopic smooth area preserving Anosov difeomorphisms have identical pressure functions for their geometric potentials, must they be $C^\infty$ conjugate?" (This is based on joint work with Anthony Quas.)

February 15, 2024: Mohammad Latifi-Jebelli "An elementary prelude to quantum fields, II"

Abstract: In my second talk, I am planning to give an introduction to measure theory in infinite dimensions and review the rigorous constructions of quantum field theories. The infinite-dimensional Gaussian measures play a pivotal role, and they are defined based on finite-dimensional Gaussian projections. It turns out that the representation of infinite-dimensional measures using a density function is problematic, but one can incorporate Fourier domain to establish a rigorous foundation.

February 22, 2024: Ben Call "Mixing Properties of Equilibrium States for Systems Beyond Negative Curvature"

Abstract: I will first introduce some known results about equilibrium states for the geodesic flow in negative curvature. I will then discuss some tools and techniques that can be used to show similar results in settings beyond negative curvature, including rank one nonpositively curved manifolds and translation surfaces. Then, time permitting, I will discuss upcoming work showing the Bernoulli property for a class of equilibrium states, which shows that they are ​"the most random" they can be in a qualitative sense. This is joint work with Dave Constantine, Alena Erchenko, Noelle Sawyer, and Grace Work.

February 29, 2024: Ian Adelstein "Geometric Manifold Learning"

Abstract: With the advent of high dimensional high throughput data in many fields including biomedicine, social science, physics, finance, etc, there is an increasing need to understand the topology (shape, structure) and geometry (intrinsic distance, curvature) of the data. This process starts with the manifold hypothesis: that high dimensional data actually live on or near a much lower dimensional manifold. One can then study the geometric and topological properties of this manifold to reveal important features of the data. The primary goal of this talk will be to introduce the main bridge between the data manifold and its geometry: the spectral theory for the graph Laplacian. I will also discuss some recent projects that explore the geometry of the data manifold.

March 28, 2024: Thomas Barthelmé "Group actions on bifoliated planes and classification of (pseudo)-Anosov flows in dimension 3"

Abstract: An old problem in dynamical systems is to try to classify Anosov flows up to orbit-equivalence. This question is particularly interesting in dimension 3 where we both have lots of examples and a rich, but still poorly understood, relationships between the dynamics of the flow and the topology of the manifold. By a result of T. Barbot, classifying Anosov flows (or more general pseudo-Anosov flows) in dimension 3 up to orbit equivalence restricts to classifying, up to conjugacy, certain actions of $\pi_1(M)$ on the orbit space, a topological plane with two transverse foliations. In this talk, I will recall the above and discuss a new complete invariant for transitive (pseudo)-Anosov flows which often reduces to just knowing which conjugacy classes in $\pi_1(M)$ are represented by periodic orbits of the flow. If time permits, I’ll talk about some applications with link to contact geometry. This is all joint work with Kathryn Mann, Steven Frankel and Sergio Fenley.

April 25, 2024: Kathryn Lindsey "TBD"

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May 9, 2024: Joel Rosenfeld "TBD"

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May 16, 2024: Diana Davis "TBD"

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May 23, 2024: Daniel Mitsutani "TBD"

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May 30, 2024: Dave Constantine "TBD"

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2016-2017

2018-2019

Fall 2023