Class summaries

Lecture Summaries

Lecture 39(12/1/23): In this lecture, we had a guest lecture on some introductory ideas in Topological Data Analysis, with some examples in hypothesis testing. 

Lecture 38(11/29/23): In this lecture, we had a guest lecture on ideas in planar folding and knot invariants which arise from these constructions.

Lecture 37(11/27/23): In this lecture, we finally finished the proof of the Fundamental Theorem of Lickorish and Wallace. Some important parts in this lecture were: 

• Defined Heegaard splittings and 3-dimensional handlebodies and discussed some examples.

• Proved that the homeomorphism type of 3-manifolds stablize by removing solid tori

• Proved the FTLW as a corollary.

Lecture 36(11/17/23): In this lecture, we continued our dicussion of TQFTs, with a guest lecture on gauge theory and Dijkgraaf-Witten theory. Some important parts in this lecture were: 

• Motivating TQFTs from a top-down view in terms of physical observables and relativistic QFTs

• Defining several notions in Gauge theory such as principle G-bundles, connections, etc.

• Defining Dijkgraaf-Witten theory and computing topological invariants. 

Lecture 35(11/15/23): In this lecture, we had a special topics lecture on topological quantum field theories. Some important parts in this lecture were:

• Motivating the study of TQFTs from a mathematical and physical viewpoint

• Giving the Atiyah-Segal axioms of a TQFT

• Discussed some preliminary results derived from the axioms

• Classified 1 and 2 dimensional TQFTs. 

Lecture 34(11/13/23): In this lecture, we had a guest lecture on the Uniformization theorem in complex analysis. Some important parts in this lecture were: 

• Defining a Riemann surface and giving some examples

• Stating the Uniformization theorem and giving a sketch of the proof

• Discussing some applications of the theorem

Lecture 32(11/6/23): In this lecture, we had the first of a two part guest lecture on Sard's theorem. Some important parts in this lecture were:

• Defining what it means to be measure 0

• Proving a lemma about when a set is measure 0

• Outlining the steps for proving Sard's theorem.

Lecture 31(11/3/23): In this lecture, we continued the proof of the fundamental theorem of Lickorish and Wallace. Some important parts in this lecture were

• Defining twist equivalence of curves 

• Determining when two curves are twist equivalent

• Relating twist equivalence to surgery and creation of manifolds

Lecture 30 (11/1/23): In this lecture, we discussed the idea of Spun-knots as a special case of 2-knots. Some important parts in this lecture were: 

• Defining higher knots and motivating their study

• Defining spun-knots as a special case of 2-knots

• Proved that the fundamental group of a spun-knot is isomorphic to the fundamental group of the corresponding (1-)knot

Lecture 29(10/30/23): In this lecture, we discussed handlebody theory and surveyed some ideas in low-dimensional topology. Some important parts in this lecture were:

• Defining a handlebody and determining when a handlebody decomposition exists

• Defined a cobordism and stated the h-cobordism theorem

• Discussed some other classification results of 3 and 4 manifolds

Lecture 28(10/27/23): In this lecture, we started the proof of the fundamental theorem of Lickorish and Wallace.

Lecture 27(10/25/2023): In this lecture, we introuced the idea of surgery theory, looking at Dehn Surgery as a special case. Some important parts in this lecture were:

• Defining surgery and Dehn surgery 

• Discussing what it measn to do Dehn surgery with (p,q) coefficients, meridans and longitudes

• Stated the fundamental theorem of Lickorish and Wallace

• Defined cosmetic surgery and discussed the cosmetic surgery conjecture.

Lecture 26(10/23/2023): In this lecture, we continued our discussions of the Jones polynomial.

Lecture 25(10/20/2023): In this lecture, we returned to Knot thoery and introduced the Jones polynomial of a knot as our second polynomial invariant.

Lecture 24(10/18/2023): In this lecture, we talked about ordinary homology theories and computed H_n(S^n) from an axiomatic viewpoint. Some important parts in this lecture were: 

• Defining chain complexes, exact sequences, and homology groups.

• Giving the Eilenberg-Steenrod axioms of ordinary homology theories

• Computing H_n(S^n) from axioms

Lecture 23(10/16/2023): In this lecture, talked about (complex) projective space and gave a geometric interpretation to the Hopf map. Some important parts in this lecture were:

• Defining real and complex projective space

• Giving a CW decomposition of projective space (see homework 6 as well)

• Discussing homotopy groups of spheres

• Giving a visual/geometric interpretation of the Hopf map

Lecture 22(10/13/2023): In this lecture, we continued our discussion on Knot theory, specifically, about the Alexander Polynomial. Some important parts in this lecture were: 

• Discussing the various ways to compute the Alexander polynomial (Skein relation, seifert form, branched covering space, pi_1 of knot complement) 

• Defined the Skein relation for the Alexander polynomial and gave an example

• Defined the Seifert form of a Seifert surface of a knot, and the linking number of generators of H_1

• Gave various properties of Alexander polynomial from the Seifert perspective

Lecture 21(10/11/2023): In this lecture, we continued talking about Knot theory. Some important points in this lecture were:

• Gave the Seifert algorithm to find the Seifert surface of a knot

• Proved that the genus of a knot is additive under connect sums

• Defined prime knots and discussed the classification of knots

Lecture 20(10/9/2023): In this lecture, we started talking about Knot theory. Some important points in this lecture were:

• Defining a knot rigorously (and smooth isotopy)

• Defining Reidemiester moves and how they help prove knot invariants

• Defining the Seifert surface and genus of a knot

• Defining tricolorability, bridge number, and other elementary knot invariants

Lecture 19(10/6/2023): In this lecture, we finished talking about flows and discussion how the homotopy type of a manifold changes as we go across a critical point: Some essential points in this lecture were: 

• Finishing talking about behavior of manifolds away from critical points

• Stating theorem about homotopy type of a manifold through critical points

• Giving a proof sketch about theorem.

Lecture 18(10/4/2023): In this lecture, we finished the proof of the Morse lemma and started talking about flows. Some essential points in this lecture were:

• Proving the Morse lemma

• Defining one parameter group of diffeomorphisms

• Discussing some ODE theory and how to get flows from vector fields

• State theorem about behavior of manifolds away from critical points

Lecture 17(10/2/2023): In this lecture, we started talking about Morse theory and gave a geometric motivating example. Some essential points in this lecture were: 

• Looking at the torus as a motivating example for Morse theory

• Defining non-degenerate critical points and the index of a critical point

• Defining Morse functions

• Stating and proving an important lemma. Stating the Morse lemma

Lecture 16(09/29/2023): In this lecture, we proved that the fundamental group of S^1 was isomorphic to the integers. Some essential points in the lecture were: 

• Computing \pi_1(S^1) = Z and proving the isomorphism

• Describing the wirtinger presentation of a knot

Lecture 14 (09/25/2023): In this lecture, we started our discussion of algbraic topology. Some essential points in this lecture were:

• Defining what a homotopy is and what a homotopy equivalence is

• Defining deformation retract

Lecture 13 (09/22/2023): In this lecture, we finished up our discussion of smooth manifolds. Some essential points in the lecture were:

• Defining what the differential of a smooth function is, and realizing it as a map between tangent spaces

• Defining the tangent space of an arbitrary smooth manifold via the differential of coordinate charts

• Defining the tangent bundle of a smooth manifold

Lecture 12 (09/20/2023): In this lecture, we continued our discussion about smooth manifolds. Some essential points in the lecture were: 

• Defining what it means for a map between manifolds to be smooth (squeezing with coordinate charts)

• Checking that smooth maps are well defined (i.e. independent of choice of coordinate chart)

• Defining geometric tangent vectors in R^n and extracting some essential properties

• Defining a derivation and proving the (vector space) isomorphism between the space of geometric tangent vectors and derivations at a point in R^n.

Lecture 11 (09/18/2023): In this lecture, we introduced the definition of a smooth manifold and manifolds with boundary. Some essential points in the lecture were:

• Revewing calculus on R^n, defining what it means for a function f: R^n -> R^m to be differentable (and smooth)

• Defining a transition map (of coordinate charts) and what it means to be smoothly compatible

• Defining a smooth (maximal) atlas and showing the existence and uniquenss of smooth maximal atlases

• Defining a smooth structure, smooth manifold, and manifold with boundary. 

Lecture 10 (09/15/2023): In this lecture, we had a fun lecture introducing some ideas in knot theory, building up to defining the mapping class group of a torus. Some essential points in the lecture were: 

• Defining smooth and ambient isotopy.

• Defining the Dehn twist map and seeing examples of Dehn twists on a torus.

• Defining the mapping class group Mod(T^2) of a torus. 

Lecture 9 (09/13/2023): In this lecture, we started our formal introduction to topological manifolds and started building up several key ideas used later on. Some essential points during this lecture were:

• Defining what a manifolds is rigorously, with pathological counterexamples to see why we need each part of the definition

• Defining coordinate charts, transition functions, and what an atlas is for a topological manifold

• Defining what it means to be locally compact and proving that manifolds are locally compact. 

Lecture 8 (09/11/2023): In this lecture, we discussed the idea of connectedness (and path connectedness) in topological spaces, as well as motivated the study of manifolds. Some essential points during the lecture were:

• Defining connectedness and path connectedness, and giving examples of when they are not equivalent (broken comb)

• Proving several properties of (path)connectedness under continuous maps, leading to the proof that no open subset of R is homeomorphic to an open subset of R^2

• Roughly defining a manifold and giving some motivating examples

Lecture 7 (09/08/2023): In this lecture, we continued our discussion of compactness as a crucial property of topological spaces. Some essential points during the lecture were:

• Defining the idea of sequential compactness and in which cases this characterization is equivalent to compactness

• Proving that a compact subspace of a Hausdorff space is closed

• Proving Heine-Borel 

Lecture 6 (09/06/2023): In this lecture, we finished the definition of the quotient topology and gave a few examples including S^1 and T^2 as quotients of R and R^2, respectively. We also started discussing the idea of compactness and proved several theorems about compact spaces. Some essential points during this lecture were: 

• Getting intuition about what the quotient topology actually is 

• Defining what it means for a space to be compact and motivating why we want to work with compact spaces (local properties --> global properties)

• Showing how we can use "shortcuts" to prove a space is compact, rather than find finite subcovers for each open cover

• Introducing the definition of a Hausdorff space and how this ties in with the discussion of compactness. 

Lecture 5 (09/01/2023): In this lecture, we defined what a basis of a topology is and talked about how to get new topologies from existing topologies via some cannoncial constructions. Some essential points during this lecture were: 

• Defining a basis of a topology, with R and open intervals as the key example

• Defining how and why we pick the subspace and product topology (both finite and infinite)

• Defining the quotient topology 

Lecture 4 (08/30/2023): In this lecture, we defined what it means to be a topological space and gave various examples of topologies on finite and infinite spaces. Some essential points during this lecture were:

• Using open balls in R^2 to motivate the axioms of a topology

• Defining what the requirements of a topological space are

• Defining what it means for a topology to be metrizable

• Defining what it means for two topologies to be coaser or finer than one another

Lecture 3 (08/28/2023): In this lecture, we reviewed the concept of a metric space in order to motivate the axioms of what a topology on a set is. Some essential points during this lecture were:

• Defining a metric and what it means for two metric spaces to be "equal" (isometry)

• Examples of non-isometric spaces (Max metric on R vs. Euclidean metric)

• Defining what it means to be an open set in a metric space

• Defining delta-epsilon continuity in metric spaces

Lecture 2 (08/25/2023): In this lecture we continue the "proof" for why the Trefoil is fibered. Some essential points during this lecture were:

• Defining the trefoil and give intuition about what smooth isotopy means

• Defining a torus knot and showing that a trefoil is a (2,3)-torus knot

• Define what a Seifert surface for a knot is, give some examples for the unknot

• Define what it means to be fibered and show that the unknot and trefoil are fibered

Lecture 1 (08/23/2023) : In this lecture we introduce the course and go over course logistics, as well as discuss the motivation for studying low dimensional topology. Afterwards we introduce some necessary ideas in order to show that the Trefoil knot is fibered. Some of these ideas include: 

• Defining what it means to be a knot

• What does the 3-sphere S^3 look like?

• Defining Stereographic Projection and how to use this idea to view the n-sphere

• Why do we want to work in S^3 instead of R^3?