The MSU student geometry/topology seminar for fall 2019 is every Monday at 3-4pm in C304 Wells Hall. It is co-organized by Zhe Zhang and myself (Keshav Sutrave).
| Date | Speaker | Title | Description |
|---|---|---|---|
| Dec 2 | Sanjay Kumar | An Introduction to Link Invariants from Tangle Operators | In the early 90’s, Reshetikhin and Turaev constructed topological invariants of 3-manifoolds and of framed links in 3-manifolds using quantum groups. In this talk, I will introduce their approach with specific examples and show how they relate to known link invariants such as the Jones polynomial. |
| Nov 18 | Sarah Klanderman | Computations in Topological CoHochschild Homology | Hochschild homology (HH) is a classical algebraic invariant of rings that can be extended topologically to be an invariant of ring spectra, called topological Hochschild homology (THH). There exists a dual theory for coalgebras called coHochschild homology (coHH), and in recent work Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology (coTHH). In this talk we will discuss coTHH calculations and the tools needed to do them. |
| Nov 11 | Wenchuan Tian | Properties of Busemann function on manifolds with nonnegative sectional curvature outside of a compact set | Busemann functions are useful. Cheeger and Gromoll used them to prove the splitting theorem for manifolds with nonnegative ricci curvature that contains a line. Yau used them to prove that complete noncompact manifolds with nonnegative Ricci curvature have at least linear volume growth. In a paper called "Positive Harmonic Functions on Complete Manifolds with Non-Negative Curvature Outside a Compact Set" Peter Li and Luen-Fai Tam also used Busemann function to show the existence of positive harmonic functions. I will talk about Li and Tam's proof of properties of Busemann function. The proof only uses Toponogov theorem and cosine law. The results of the proof is useful for the subsequent analysis part of the paper. |
| Nov 4 | Gorapada Bera | Introduction to Riemannian Holonomy | The holonomy group of a Riemannian manifold exhibits various geometric structures compatible with the metric. In 1955, M.Berger classified all possible Riemannian holonomy groups. Studying all these are more than one semester subject. So, in this talk after a brief introduction we overview very basics of these holonomy groups. |
| Oct 28 | Arman Tavakoli | Introduction to the Yang-Mills equation | The Yang-Mills equation is a celebrated topic that is studied in differential geometry and particle physics. We will motivate the equation as a generalization of Maxwell's equations, define the relevant geometrical objects and discuss their properties |
| Oct 21 | Brandon Bavier | The Ends of Hyperbolic Manifolds | When studying knots, we can often get a lot of information by removing the knot from space, and looking at the knot complement. It's pretty natural to ask, then, what happens to the area close to the removed knot? We call these areas cusps, and, in the case of hyperbolic knots, the cusp alone can tell us quite a lot. In this talk, we will give an introduction to these cusps, including their uses in topology, as well as how to find invariants from them. |
| Oct 14 | Zhe Zhang | An introduction to intersection forms: Taking K3 surface as an example | I’ll define intersection product both on 4 manifolds and in the algebraic geometry setting, then introduce the blow up technique and give some easy examples. After that I will jump to K3 surface, give definition and constructions, and talk a little bit about the elliptic fibrations of K3. If I still have time, I will talk about the relation between intersection form and characteristic classes. |
| Oct 7 | Joe Melby | Complexity, 3-Manifolds, and Zombies | An important invariant of a path-connected topological space X is the number of homomorphisms from the fundamental group of X to a finite, non-abelian, simple group G. Kuperberg and Samperton proved that, although these invariants can be powerful, they are often computationally intractable, particularly when X is an integral homology 3-sphere. More specifically, they prove that the problem of counting such homomorphisms is #P-complete via a reduction from a known #P-complete circuit satisfiability problem to a model which constructs X from a well-chosen Heegaard surface and a mapping class in its Torelli group. We will introduce the basics of complexity for counting problems, summarize the reduction used by K-S to bound the problem of counting homomorphisms, and discuss some of the topological and quantum computing implications of their results |
| Sep 30 | Keshav Sutrave | Some kind of introduction to special relativity | (Soft) The beginning of Einstein's theory of special relativity, which gives us a way of doing physics in different reference frames (observers in motion). Specifically: "What happens when you turn on a flashlight while already moving at half the speed of light?" I will introduce time dilation and length contraction, event simultaneity, and touch on the problem in electromagnetism, using many examples. |
The MSU student geometry/topology seminar for spring 2019 is every Wednesday at 4-5pm in C304 WH. It is co-organized by Woongbae Park, Zhe Zhang, and myself (Keshav Sutrave).
| Date | Speaker | Title | Description |
|---|---|---|---|
| Apr 17 | Gorapada Bera | Introduction to Seiberg Witten invariants on three manifolds | Although Seiberg Witten invariant originally introduced for four manifolds, but its three dimensional version is also interesting. After a brief discussion on the definition of the Seiberg Witten invariant on three manifolds we will see some results from literature equating this invariant to some known invariants of three manifolds. |
| Apr 10 | Joseph Melby | Knot Genus and Complexity | A classical problem in knot theory is determining whether or not a given 2-dimensional diagram represents the unknot. The UNKNOTTING PROBLEM was proven to be in NP by Hass, Lagarias, and Pippenger. A generalization of this decision problem is the GENUS PROBLEM. We will discuss the basics of computational complexity, knot genus, and normal surface theory in order to present an algorithm (from HLP) to explicitly compute the genus of a knot. We will then show that this algorithm is in PSPACE and discuss more recent results and implications in the field. |
| Mar 27 | Dongsoo Lee | Knot Floer homology obstructs ribbon concordance. | Zemke shows that the map on knot Floer homology induced by a ribbon concordance is injective in his paper. I will be talking about its applications and proof. |
| Mar 13 | Hitesh Gakhar | Sliding window embeddings | Classically, Sliding Window Embeddings were used in the study of dynamical systems to reconstruct topology of underlying attractors from generic observation functions. In 2015, Perea and Harer studied persistent homology of sliding window embeddings from L^2 periodic functions. We define a quasiperiodic function as a superposition of periodic functions with incommensurate frequencies. As it turns out, sliding window embeddings of quasiperiodic functions are dense in high dimensional tori. In this talk, I will present some results for the quasiperiodic case. |
| Feb 27 | Woongbae Park | Analysis of measures converging to a Dirac-delta measure in Riemann surfaces | This is a 2-dim case of general energy concentration phenomenon. If we have sequence of harmonic maps with bounded energy defined on a Riemann surface, Uhlenbeck compactness theorem says its subsequence converges away from at most finite points, called bubble points. At the bubble point energy concentrates, so we may blow up the point to capture energy distribution on the bubble. But energy may concentrate again on the bubble, so careful touch is needed to finish this blow up process. In this talk I will introduce a way to choose two marked points with desired properties. The typical example is of harmonic map case, but it may be applied to other energy concentrating cases. |
| Feb 20 | Zhe Zhang | A construction of the Deligne-Mumford orbifold | Isomorphism classes Mg,n of stable nodal Riemann surfaces of arithmetic genus g with n marked points. A marked nodal Riemann surface is stable if and only if its isomorphism group is finite. A natural construction based on the existence of universal unfoldings endows the Deligne-Mumford moduli space with an orbifold structure. Here we use the methods of differential geometry rather than algebraic geometry. |
| Feb 13 | Keshav Sutrave | Harmonic map heat flow | The harmonic map flow equation and Eells-Sampson theorem |
| Feb 6 | Abhishek Mallick | Computations in equivariant Floer homology | Following Hendricks-Lipshitz-Sarkar we discuss the construction of equivariant Floer homology and a few cases where it can be computed. |
The MSU student geometry/topology seminar for fall 2018 is every Wednesday at 4-5pm in A202 WH. It is co-organized by Woongbae Park, Zhe Zhang, and myself (Keshav Sutrave).
| Date | Speaker | Title | Description |
|---|---|---|---|
| Dec 5 | Zhe Zhang | Bott's paper about geometric quantization | Raoul Bott - "On some recent interactions between mathematics and physics" (1985). It is a mathematical point of view about how quantum phenomena naturally arises when we use Feynman’s idea of path integral and try to give a rigorous definition of the electromagnetic potential. This in turn gives us a new interpretation of a symplectic manifold as space of flat connections over a Riemann surface. |
| Nov 28 | Brandon Bavier | Generalizing Alternating Knots | One of the nicer properties a knot can have is to be alternating. These knots tend to be easy to work with, and can give us several nice results about the whole class. Unfortunately, many knots are not alternating, causing general proofs about them to be difficult at best. In this talk, we will take a look at a couple of different ways we can broaden the class of alternating knots, and see what we can get from these different definitions of alternating. |
| Nov 21 | Gorapada Bera | Introduction to Seiberg-Witten Invariants | After a brief introduction of Seiberg-Witten equations on closed smooth four manifolds, we will see how moduli space of solutions leads to an oriented compact manifold and a topological invariant (Seiberg-Witten Invariant) for the four manifold. Then for the purpose of computation of this invariant on Kähler manifolds, we will rewrite the equation in terms of complex geometry and see for most of the Kähler Surfaces the answer will be in terms of algebraic geometric criterion of the surface. Most of the technical details will be omitted but some brief sketches will be there. I will follow John Morgan's Book on Seiberg Witten equations. |
| Nov 14 | Hitesh Gakhar | Künneth formulae in persistent homology | The classical Künneth formula provides a relationship between the homology of a product space and that of its factors. In this talk, I will briefly review persistent homology and show Künneth-type theorems for it. That is, for two different notions of products, we show how the persistent homology of a filtered product space relates to that of the factor filtered spaces. |
| Nov 7 | Abhishek Mallick | Equivariant Floer Homology | We will discuss constructions given by Seidel-Smith and Hendricks-Lipshitz-Sarkar. |
| Oct 31 | Wenchuan Tian | A Compactness Theorem for Rotationally Symmetric Riemannian Manifolds with Positive Scalar Curvature | Gromov conjectured that sequences of compact Riemannian manifolds with positive scalar curvature should have subsequences which converge in the intrinsic flat sense to limit spaces with some generalized notion of scalar curvature. In light of three dimensional examples discovered jointly with Basilio and Dodziuk, Sormani suggested that one add an hypothesis assuming a uniform lower bound on the area of a closed minimal surface. We have proven this revised conjecture in the setting where the sequence of manifolds are 3 dimensional rotationally symmetric warped product manifolds. This is a project given by professor Christina Sormani, and is joint work with Jiewon Park and Changliang Wang. |
| Oct 24 | Woongbae Park | Bubbling of harmonic maps | Harmonic map is a generalization of harmonic function but has different behavior. I briefly introduce harmonic map and explain one of the differences, called bubbling. This is special kind of singularity only occurs under certain conditions. I explain how we deal with bubbling in different favors, and prove some details. |
| Oct 17 | Michael Shultz | The homology polynomial and pseudo-Anosov braids | Every orientation preserving homeomorphism of a compact, connected, orientable surface S is isotopic to a representative that is periodic, reducible, or pseudo-Anosov (pA). In the last case, the representative is neither periodic nor reducible and the surface admits two (singular) transverse measured foliations. The pA representative "stretches" with respect to one of these measures by a number called the stretch factor. The homology polynomial, introduced by Birman, Brinkmann, and Kawamuro, is an invariant of the isotopy class and contains the stretch factor as it's largest real root. It can also distinguish some distinct pA maps with the same stretch factor. In this talk I will discuss the ideas behind the homology polynomial and how it is obtained. As time permits I will discuss some examples involving pA braids and touch on a connection with the Burau representation. |
| Oct 10 | Sanjay Kumar | Turaev-Viro invariants via quantum representations of the mapping class group | The Turaev-Viro invariants are an infinite family of real valued 3-manifold invariants originally defined by state sums of a triangulation. Using SO(3)-TQFT, I will demonstrate an equivalent formulation in terms of traces of quantum representations and discuss its possible advantages in studying mapping tori of surfaces. |
| Oct 3 | Keshav Sutrave | Take a walk on a Riemann surface | This talk will be an introduction to Riemann surfaces, including branched covering and monodromy in this setting. I will prove Riemann's existence theorem of branched covers, illustrate this using algebraic curves, and finish with Riemann-Hurwitz. |
| Sep 26 | Nick Ovenhouse | Noncommutative Geometry and Character Varieties | Roughly speaking, noncommutative geometry studies noncommutative rings and algebras from a "geometric" perspective. I will discuss some philosophies and approaches to the subject, which leads to the study of character varieties, which I will define and discuss. |
| Sep 19 | Zhe Zhang | Elliptic Regularity of J-Holomorphic Curves | One of the fundamental estimates for the L^p theory of elliptic operators is Calderon-Zygmund inequality. I’ll follow Mcduff & Salamon’s book for the proof of regularity theorem, raising the order of nonlinear Cauchy Riemann equation and making use of mean value property. |