Demetre Kazaras : If Ricci is bounded below, then mass is in control!
- Geometry and Topology ( 519 Views )The ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields. In a previous work, we showed how to compute this invariant (in 3 spatial dimensions) by studying harmonic functions. Now I'll use this formula to consider the question: How flat is an asymptotically flat manifold with very little total mass? In the presence of a lower bound on Ricci curvature, we make progress on this question and confirm special cases of conjectures made by Ilmanen and Sormani.
Calvin McPhail-Snyder : Making the Jones polynomial more geometric
- Geometry and Topology ( 352 Views )The colored Jones polynomials are conjectured to detect geometric information about knot complements, such as hyperbolic volume. These relationships ("volume conjectures") are known in a number of special cases but are in general quite mysterious. In this talk I will discuss a program to better understand them by constructing holonomy invariants, which depend on both a knot K and a representation of its knot group into SL_2(C). By defining a version of the Jones polynomial that knows about geometric data, we hope to better understand why the ordinary Jones polynomial does too. Along the way we can obtain more powerful quantum invariants of knots and other topological objects.
Subhankar Dey : Cable knots are not thin
- Geometry and Topology ( 239 Views )Thurston's geometrization conjecture and its subsequent proof for Haken manifolds distinguish knots in S^3 by the geometries in the complement of the knots. While the definition of alternating knots make use of nice knot diagrams, Knot Floer homology, a knot invariant toolbox, defined by Ozsvath-Szabo and Rasumussen, generalizes the definition of alternating knots in the context of knot Floer homology and defines family of quasi-alternating knots which contains all alternating knots. Using Lipshitz-Ozsvath-Thurston's bordered Floer homology, we prove a partial affirmation of a folklore conjecture in knot Floer theory, which bridges these two viewpoints of looking at knots.
Tye Lidman : Homology cobordisms with no 3-handles
- Geometry and Topology ( 229 Views )Homology cobordisms are a special type of manifold which are relevant to a variety of areas in geometric topology, including knot theory and triangulability. We study the behavior of a variety of invariants under a particular family of four-dimensional homology cobordisms which naturally arise from Stein manifolds. This is joint work with Ali Daemi, Jen Hom, Shea Vela-Vick, and Mike Wong.
Shubham Dwivedi : Geometric flows of $G_2$ structures
- Geometry and Topology ( 202 Views )We will start by discussing a flow of isometric $G_2$ structures. We consider the negative gradient flow of the energy functional restricted to the class of $G_2$ structures inducing a given Riemannian metric. We will discuss various analytic aspects of the flow including global and local derivative estimates, a compactness theorem and a monotonicity formula for the solutions. After defining an entropy functional we will prove that low entropy initial data lead to solutions that exist for all time and converge smoothly to a $G_2$ structure with divergence free torsion. We will also discuss finite time singularities and the singular set of the solutions. Finally, we will discuss the isometric flow "coupled” with the Ricci flow of the underlying metric, which again is a flow of $G_2$ structures, and discuss some of its properties. This is a based on two separate joint works with Panagiotis Gianniotis (University of Athens) and Spiro Karigiannis (University of Waterloo).
Daniel Stern : Scalar curvature and circle-valued harmonic maps
- Geometry and Topology ( 218 Views )We introduce a new tool for relating the scalar curvature of a Riemannian manifold to its global geometry and topology, based on the study of level sets of harmonic functions and harmonic maps to the circle. We will explain how these ideas lead to simple new proofs and improvements upon some well-known results in three-manifold geometry and general relativity, previously studied primarily via minimal surface and Dirac operator methods.
Colleen Robles : A refinement of the Lefschetz decomposition for hyperkahler manifolds
- Geometry and Topology ( 179 Views )The cohomology (with complex coefficients) of a compact kahler manifold M admits an action of the algebra sl(2,C), and this action plays an essential role in the analysis of the cohomology. In the case that M is a hyperkahler manifold Verbitsky and Looijenga—Lunts showed there is a family of such sl(2,C)’s generating an algebra isomorphic to so(4,b_2-2), and this algebra similarly can tell us quite a bit about the cohomology of the hyperkahler. I will describe some results of this nature for both the Hodge numbers and Nagai’s conjecture on the nilpotent logarithm of monodromy arising from a degeneration. This is joint work with Mark Green, Radu Laza and Yoonjoo Kim.
Diana Davis : Periodic paths on the pentagon
- Geometry and Topology ( 171 Views )Mathematicians have long understood periodic billiard trajectories on the square table, which occur when the slope of the trajectory is rational. In this talk, I'll explain my work on periodic trajectories on the regular pentagon, describing their geometry, symbolic dynamics, and group structure. The periodic trajectories are very beautiful, and some of them exhibit a surprising "dense but not equidistributed" behavior. There will be lots of pictures. This is joint work with Samuel Lelièvre.
Natalia Kolokolnikova : Thom polynomial and its K-theoretic generalization
- Geometry and Topology ( 166 Views )Global singularity theory originates from problems in obstruction theory. Consider the following question: is there an immersion in a given homotopy class of maps between two smooth compact manifolds M and N? We can reformulate this question as "is the set of points, where a generic smooth map between M and N is not an immersion, empty"? This set is the simplest example of a singularity. Alternatively, we can ask a question whether the cohomology class of this set is 0 or not. Turns out, there is a universal polynomial depending only on the dimensions of M and N and on the type of singularity, that, when evaluated in the corresponding characteristic classes of M and N, computes the cohomology class of a singularity. This polynomial is called the Thom polynomial, and it is the central notion of singularity theory. In my talk I will give an introduction to singularity theory, define the classic Thom polynomial and talk about different approaches to its K-theoretic generalization.
Julian Chaidez : Essential tori In spaces of symplectic embeddings
- Geometry and Topology ( 148 Views )The problem of when and how one symplectic manifold can be symplectically embedded into another is notoriously subtle, even when the spaces in question are relatively simple. Gromov's non-squeezing theorem and McDuff's Fibonacci staircase are examples of this phenomenon. One can interpret these results as realizing the principle that "variations of quantitative symplectic parameters alter the topology of symplectic embedding spaces." In this talk, we explain recent work (joint with Mihai Munteanu) showing that certain n-torus families of symplectic embeddings between 2n-d ellipsoids become homologically essential if certain quantitative invariants are close enough. We will also discuss work in progress in which we use similar methods to study Lagrangian embeddings.
Nathan Dowlin : A spectral sequence from Khovanov homology to knot Floer homology
- Geometry and Topology ( 222 Views )Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will give a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related.
Jonathan Hanselman : The cosmetic surgery conjecture and Heegaard Floer homology
- Geometry and Topology ( 159 Views )The cosmetic surgery conjecture states that no two surgeries on a given knot produce the same 3-manifold (up to orientation preserving diffeomorphism). Floer homology has proved to be a powerful tool for approaching this problem; I will survey partial results that are known and then show that these results can be improved significantly. If a knot in S^3 admits purely cosmetic surgeries, then the surgery slopes are +/- 2 or +/- 1/q, and for any given knot we can give an upper bound for q in terms of the Heegaard Floer thickness. In particular, for any knot there are at most finitely many potential pairs of cosmetic surgery slopes. With the aid of computer computation we show that the conjecture holds for all knots with at most 15 crossings.
Lisa Traynor : Legendrian Torus Links
- Geometry and Topology ( 147 Views )Legendrian torus knots were classified by Etnyre and Honda. In particular, for any smooth torus knot we know the mountain range that lists all of its Legendrian representatives. I will discuss the classification of Legendrian torus links. In this classification, a natural realization question arises: what n-tuples of points on the mountain range of a (p,q)-torus knot can occur in a Legendrian (np, nq)-torus link? Another part of the classification is to understand how many different ways an n-tuple on the mountain range can be realized as an ordered link. In particular, for Legendrian representatives of an (np,nq)-torus link is it possible to do invariant preserving permutations of the components? This is joint work with Jennifer Dalton and John Etnyre.
Mark A. Stern : A coarse Cheeger inequality for 1-forms
- Geometry and Topology ( 127 Views )Motivated by questions about the spectral geometry of hyperbolic manifolds, arising in number theory, we seek a lower bound for the first nonzero eigenvalue of the Laplace Beltrami operator on 1-forms on hyperbolic manifolds. We prove an analog of Cheeger's inequality. Joint work with Michael Lipnowski.
Siqi He : Classification of Nahm Pole Solutions to the KW Equations on $S^1\times\Sigma\times R^+$
- Geometry and Topology ( 227 Views )We will discuss Witten’s gauge theory approach to Jones polynomial by counting solutions to the Kapustin-Witten (KW) equations with singular boundary conditions over 4-manifolds. We will give a classification of solutions to the KW equations over $S^1\times\Sigma\times R^+$. We prove that all solutions to the KW equations over $S^1\times\Sigma\times R^+$ are $S^1$ direction invariant and we give a classification of the KW monopole over $\Sigma\times R^+$ based on the Hermitian-Yang-Mills type structure of KW monopole equation. This is based on joint works with Rafe Mazzeo.
Simon Brendle : Singularity formation in geometric flows
- Geometry and Topology ( 292 Views )Geometric evolution equations like the Ricci flow and the mean curvature flow play a central role in differential geometry. The main problem is to understand singularity formation. In this talk, I will discuss recent results which give a complete picture of all the possible limit flows in 2D mean curvature flow with positive mean curvature, and in 3D Ricci flow.
Bulent Tosun : Fillability of contact surgeries and Lagrangian discs
- Geometry and Topology ( 150 Views )It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties of a contact structure are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact (r) surgery is symplectically/Stein fillable for r in (0,1]. This is joint work with James Conway and John Etnyre.