Calendar
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Shea Vela-Vick, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Adithyan Pandikkadan, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Tristan Reynoso, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Nilangshu Bhattacharyya, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Megan Fairchild, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Shea Vela-Vick, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Adithyan Pandikkadan, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Colton Sandvik, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Tristan Reynoso, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Nilangshu Bhattacharyya, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Adithyan Pandikkadan, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Colton Sandvik, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Shea Vela-Vick, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Megan Fairchild, Louisiana State University
Classical Knot Concordance
Posted August 30, 2024
Last modified September 16, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233TBD
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Nilangshu Bhattacharyya, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Tristan Reynoso, Louisiana State University
Classical Knot Concordance
Posted June 12, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Nilangshu Bhattacharyya, Louisiana State University
Classical Knot Concordance
Posted August 27, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233Organizational meeting
Posted August 30, 2024
Last modified September 3, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Huong Vo, Louisiana State University
Quasi-isometry and Milnor Schwarz theorem
In this talk, we will go over the proof of Milnor-Schwarz theorem, which states that a group G is quasi-isometric to a metric space X if it acts nicely on X. The definition of a quasi-isometry will be covered and so will other definitions relevant to the theorem.
Posted August 30, 2024
Last modified September 16, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Saumya Jain, Louisiana State University
Quasi-isometric invariance of hyperbolicity
In this talk, we will define $\delta$-hyperbolic spaces and show that geodesics and quasi-geodesics stay close in a hyperbolic space. We will then prove that hyperbolicity is a quasi-isometric invariant.
Posted August 30, 2024
Last modified September 23, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Rachel Meyers, Louisiana State University
Quasi-isometries and Boundaries of $\delta$-Hyperbolic Spaces
In this talk, we define the boundary of a $\delta$-hyperbolic space as a set of rays and describe the topology of X with this boundary. Further, we will prove if two spaces are quasi-isometric then the boundaries are homeomorphic.
Posted August 30, 2024
Last modified September 30, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Nilangshu Bhattacharyya, Louisiana State University
Some basic hyperbolic geometry
We will first discuss hyperboloid model for hyperbolic space and then discuss ball model and upper half space model. Furthermore, we will define the boundary at infinity and implicitly identify with spheres. We move on to talk about isometry group of hyperbolic spaces and classifying them (elliptic, parabolic and hyperbolic), with some examples. If time permits, I may state the Mostow's rigidity theorem.
Posted August 30, 2024
Last modified October 7, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Krishnendu Kar, Louisiana State University
Goodwillie-ingly Exploring Taylor Towers: Unravelling Functor Calculus
The study of the calculus of a function is the study of local behaviour of the function. One of the biggest features of calculus is approximation, i.e. given an unknown function; we may approximate to known functions to tell its property. A key example is Taylor series approximation, for a $n$ times differentiable function we can approximate to a polynomial of degree $n$. Goodwillie unlocked these deceptively simple yet so useful features of functions for functors, mostly to study K-theories. We may approximate a functor by suitable polynomial functors, and analogously, we get something called a Taylor tower. A key question here is how one might define the derivative of a functor in such a way it is commensurate with the original theory of calculus. Then, given a Taylor tower, we ask similar questions as we ask a Taylor series, how does Taylor tower converge in some analogous way? In this talk, we will explore some notions of Goodwillie’s calculus and answer some of the questions imposed.
Posted August 30, 2024
Last modified October 17, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Adithyan Pandikkadan, Louisiana State University
Construction of Hyperbolic Manifolds
In this talk, we will discuss two ways for constructing hyperbolic manifolds. We will begin by introducing hyperbolic surfaces, focusing on how to equip a hyperbolic structure on higher genus surfaces. Following this, we will discuss the construction of arithmetic hyperbolic manifolds which is a more general approach.
Posted August 30, 2024
Last modified October 17, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Megan Fairchild, Louisiana State University
Slicing Obstructions from 4-Manifold Theory
The orientable 4 genus of a knot is defined to be the minimum genus amongst all smoothly embedded surfaces in the 4-ball with boundary the knot. A knot is called slice if it bounds a smoothly embedded disk in the 4-ball. Invariants of knots, either classical or Heegaard Floer, are commonly used as lower bounds for the orientable 4 genus of knots. We will examine a different approach to showing knots are not smoothly slice, coming from 4-manifold theory.
Posted August 30, 2024
Last modified October 27, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Matthew Lemoine, Louisiana State University
A Brief Introduction to Khovanov Homology through an example
In this talk, we will discuss Khovanov Homology and how to compute this homology using an example with the trefoil knot. We will also discuss the relations between Khovanov Homology and the Jones Polynomial.
Posted August 30, 2024
Last modified November 4, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Nilangshu Bhattacharyya, Louisiana State University
Gromov norm of a compact manifold and straightening
We will define the Gromov norm of a compact manifold and straightening (every singular chain is naturally homotopic to a straight one).
Posted August 30, 2024
Last modified November 11, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Saumya Jain, Louisiana State University
Gromov norm is directly proportional to the volume
We will prove that the Gromov norm of a compact oriented hyperbolic manifold is directly proportional to its volume.
Posted August 30, 2024
Last modified November 18, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Samuel Weiner, Louisiana State University
A Survey of Topological Graph Theory
Although graphs are often thought of as strictly combinatorial objects, many substantial developments in graph theory have come by considering their topological properties. For instance, we may define the genus of a graph G to be the minimum integer n such that G embeds into an orientable surface of genus n. Kuratowski's Theorem is a seminal result that precisely determines the class of all genus-2 graphs; a recent result due to Robertson and Seymour characterizes the much broader class of all graphs of bounded genus. We will explore these and other results that lie at the intersection of topology and graph theory. The speaker will assume no prior graph theory knowledge.
Posted August 30, 2024
Last modified December 2, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Locket 233
Huong Vo, Louisiana State University
Mostow's Rigidity Theorem
Mostow's Rigidity Theorem states that two connected, compact, oriented hyperbolic manifolds of dimension at least 3 that are homotopy equivalent are isometric. In this talk, we will review key steps and finish the proof of this theorem.
Posted February 3, 2025
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm
Megan Fairchild, Louisiana State University
Rachel Meyers, Louisiana State University
Introduction to the h-Cobordism Theorem
We state the h-Cobordism theorem and go over the motivation and background to understand the statement of the theorem. Additionally, we discuss its relevance and impact to the generalized Poincarè conjecture in dimensions 5 and higher.
Posted February 3, 2025
Last modified February 10, 2025
Informal Geometry and Topology Seminar Questions or comments?
12:30 pm
Porter Morgan, University of Massachusetts Amherst
Obtaining exotic 4-manifolds through torus surgery
Let M be a closed, smooth, oriented 4-manifold. In this talk, we’ll explore how to construct an irreducible copy of M using torus surgery; this means that we construct a 4-manifold X that’s homeomorphic to M, but not diffeomorphic to it, and also that X is irreducible in the sense that it can’t be expressed as a non-trivial connect sum. We’ll first describe a general strategy for finding irreducible copies. Then we’ll define torus surgery, and go through an example of building an irreducible copy with this tool. If time permits, we’ll talk about some other surgery techniques that can be used to build irreducible copies.
Posted February 3, 2025
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm
Krishnendu Kar, Louisiana State University
TBD
Posted February 3, 2025
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm
Huong Vo, Louisiana State University
TBD
Posted February 3, 2025
Last modified March 10, 2025
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm
Emmanuel Astante, Louisiana State University
Rota's conjecture and Geometric Lattices
Rota defined homology groups for certain subsets, called cross-cuts, of a lattice. He showed that the value of the Euler characteristic associated with this homology theory depends only on the lattice, not on the choice of the cross-cut. It was conjectured that the homology groups themselves depend only on the lattice. First, we will prove Rota's conjecture. Using this result, we determine the structure of the homology groups of an important class of lattices called geometric lattices.
Posted February 3, 2025
Last modified March 17, 2025
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm
Saumya Jain, Louisiana State University
Handle Trading
Equipped with the tools developed in the previous talks, we will begin by outlining the idea of the proof of the h-cobordism theorem. We will see that if the algebraic "d-pairing" can be realized geometrically, then the proof follows. To this end, we will explore a way to handle low and high handles, introducing handle-trading.
Posted February 3, 2025
Last modified March 24, 2025
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm
Peter Ramsey, Louisiana State University
The Orlik-Solomon Algebra and the Cohomology Ring of Hyperplane Arrangements
A hyperplane is a subspace of codimension one in a given vector space. A finite collection of hyperplanes is called a hyperplane arrangement. The compliment of such an arrangement in complex space defines a connected manifold whose topology can be studied via its cohomology ring. A fundamental result by Brieskorn, Orlik, and Solomon shows that this cohomology ring can be computed in a purely combinatorial way using the Orlik-Solomon Algebra. In this talk, we will explore this construction and, if time permits, discuss its implications for the Poincaré polynomial.
Posted February 3, 2025
Last modified April 7, 2025
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm
Adithyan Pandikkadan, Louisiana State University
Whitney Trick
In the previous talk, we outlined the proof of the h-cobordism theorem. The key step is realizing the algebraic intersection number +1 between the attaching sphere of the k-handle and the belt sphere of the (k+1)-handle as an actual geometric intersection. Achieving this requires eliminating pairs of intersection points with opposite signs by the "Whitney Trick". In this talk, we will focus on understanding the "Whitney Trick" in detail and how it enables these critical geometric manipulations.
Posted February 3, 2025
Last modified April 22, 2025
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm
Matthew Lemoine, Louisiana State University
Topological Data Analysis and The Persistent Laplacian
In this talk, we will go through some basic information about Topological Data Analysis (TDA) such as Persistent Homology with the goal of getting to the Persistent Laplacian and how these tools are used to analyze data.
Posted February 3, 2025
Last modified April 22, 2025
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm
Nilangshu Bhattacharyya, Louisiana State University
Proof of h-cobordism, Whitney trick and issues in the 4 dimension.
In this presentation, I will begin by recapping the complete proof of the h-cobordism theorem, which states that in dimensions greater than four, a homotopically trivial, simply connected cobordism between two simply connected compact manifolds is smoothly trivial. As a corollary, this implies the higher-dimensional Poincaré conjecture. A central tool in the proof is the Whitney trick, which is effective in higher dimensions. However, in dimension four, a framing obstruction naturally arises, presenting significant challenges. In the latter part of the presentation, I will discuss some of the technical aspects and difficulties associated with applying the Whitney trick.