Research

Geometry/Topology
(Leaders: ,
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Monday, April 15, 2024

Title: Khovanov skein lasagna module detects exotic 4-manifolds
Speaker: Qiuyu Ren, UC Berkeley
Time: 2:00pm–3:00pm
Place: CMC 108

Abstract

We present new calculations of the Khovanov-Rozansky \(\mathrm{gl}_2\) skein lasagna modules defined by Morrison-Walker-Wedrich, generalizing several previous works. In particular, our calculation shows that the \(-1\) traces on the knots \(-5_2\) and \(P(3,-3,-8)\) have non-isomorphic skein lasagna modules, thus are non-diffeomorphic (while they are homeomorphic by Kirby moves \(+\) Freedman's result). This leads to the first gauge/Floer-theory-free proof of the existence of compact exotic 4-manifolds. Time permitting, we sketch some proofs or present some other results in our work. This is joint work with Michael Willis.

Monday, April 8, 2024

Title: Graphs, Groups, and Polynomials: Relationships in Mechanical Movements
Speaker: Craig Lusk, USF Department of Mechanical Engineering
Time: 2:00pm–3:00pm
Place: CMC 108

Abstract

The design and analysis of simple mechanical devices is perhaps surprisingly rich in its mathematical content. In recent research, I have been exploring connections between graph theory, wallpaper groups, and polynomials. These have emerged as consequences of approaches to simplify the teaching of vector methods for mechanism analysis and design. Using colored graphs to represent the motion of linkages has yielded fascinating results about cognate linkages (mechanical devices that trace the same polynomial curves), and how to use Penrose tilings to generate shape morphing mechanical arrays.

Monday, April 1, 2024

Title: Relative Strengths of Knot Invariants by Experiment
Speaker: Ryan Maguire, Dartmouth College
Time: 2:00pm–3:00pm
Place: CMC 108

Abstract

Four knot polynomials have been well studied by topologists, graph theorists, and algebraists alike: The Alexander, Jones, HOMFLY-PT, and Khovanov polynomials. It is known that the Khovanov polynomial is “stronger” than the Jones polynomial and similarly one may state that HOMFLY-PT is stronger than both the Alexander and Jones polynomials. No comparison can be made between the Jones and Alexander polynomials since there are families of knots with identical Alexander polynomials but distinct Jones polynomials, and vice-versa, but experiment tells us the Jones polynomial is stronger, on average, at distinguishing knots. We have tabulated the Alexander, Jones, and HOMFLY-PT polynomials for all knots up to 19 crossings, and the Khovanov polynomial for up to 17 crossings. Using this we can experiment on the relative strengths of these knot invariants and generate statistics on them.

Monday, March 25, 2024

Title: Explicit Generators for the Stabilizers of Rational Points in Thompson's Group \(F\)
Speaker: Dima Savcuk
Time: 2:00pm–3:00pm
Place: CMC 108

Abstract

We construct explicit finite generating sets for the stabilizers in Thompson's group \(F\) of rational points of a unit interval or a Cantor set. Our technique is based on the Reidemeister-Schreier procedure in the context of Schreier graphs of such stabilizers in \(F\). It is well known that the stabilizers of dyadic rational points are isomorphic to \(F\times F\) and can thus be generated by 4 explicit elements. We show that the stabilizer of every non-dyadic rational point \(b\in (0,1)\) is generated by 5 elements that are explicitly calculated as words in generators \(x_0, x_1\) of \(F\) that depend on the binary expansion of \(b\). We also provide an alternative simple proof that the stabilizers of all rational points are finitely presented. This is a joint work with Krystofer Baker.

Monday, March 18, 2024

Title: Dichotomies in four-dimensional topology
Speaker: Inanç Baykur, University of Massachusetts Amherst
Time: 2:00pm–3:00pm
Place: Zoom Meeting (ID: 867 2354 6486 Passcode: 393108)

Abstract

I am going to describe a few major problems on smooth, symplectic, and complex structures on four-manifolds, along with certain dichotomies suggested by mounting evidence --- some by our recent work. Despite the more philosophical context, the talk will aim to explain various tool sets for constructing novel four-manifolds.

Monday, March 4, 2024

Title: Topological Bounds for the chromatic number of Random Geometric Graphs
Speaker: Francisco Martinez-Figueroa
Time: 2:00pm–3:00pm
Place: CMC 108

Abstract

In his 1978 proof of Kneser's Conjecture, Lovász pioneered the study of topological obstructions to the chromatic number of a graph. These techniques leverage the connectivity of specific cellular complexes and are linked to graphs through Borsuk-Ulam-like theorems. A pertinent question concerns the efficacy of these methodologies in bounding the chromatic number of random graphs. While Kahle’s work demonstrated their inefficiency for the Erdös-Renyi random graph model, it remains interesting to evaluate their performance with other random graph models. In this presentation, we explore the effectiveness of topological bounds for various random geometric graph models. We demonstrate their significant efficacy for Random G-Borsuk graphs, in contrast to their limited effectiveness for random \(\epsilon\)-distance graphs on spheres.

Monday, February 26, 2024

Title: Instantons and Khovanov homology in \(RP^3\)
Speaker: Hongjian Yang, Stanford University
Time: 2:00pm–3:00pm
Place: CMC 108

Abstract

Following Kronheimer and Mrowka’s approach, we show that Khovanov homology detects the unknot and the projective unknot in \(RP^3\). I’ll explain the idea of the proof. Time permitting, I’ll discuss potential further detection results.

Monday, February 19, 2024

Title: Grid diagram and complex numbers
Speaker: Zhenkun Li
Time: 2:00pm–3:00pm
Place: CMC 108

Abstract

Knots in \(S^3\) can be represented by their grid diagrams. Based on grid diagrams, grid homology was constructed and shown to be invariant of knots as a combination of works of Manolescu, Ozsvath, Sarkar, Szabo, Thurston. While the classical construction used \(Z_2\) coefficients, we study the construction of grid homology using complex coefficients. This problem arises when we work on a project aiming at relating instanton Floer homology with Heegaard Floer homology. The difficulty is that, the coefficients in the differential of the relevant chain complex were previously restricted to \(0,1\), but now can be an arbitrary complex numbers, while we still need to show that the homology constructed is independent of all the choice of these coefficients and hence still a knot invariant. This is a join work with John Baldwin, Steven Sivek and Fan Ye.

Monday, February 12, 2024

Title: Parametrizing Spaces of Geometric Graphs
Speaker: Greg McColm
Time: 2:00pm–3:00pm
Place: CMC 108

Abstract

Considering a graph as a combinatorial object, it can be realized by embedding it in a geometric space. Different embeddings will preserve different automorphisms of the original graph in that the preserved automorphisms are mapped to symmetries of the realized graph. We may parametrize an ensemble of embeddings so that the space of parameters producing graphs of (at least) a particular symmetry form vector space. We look at one method for constructing and representing such graphs, and we will look at several examples.

Monday, February 5, 2024

Title: Bar-Natan homology for null-homologous links in \(RP^3\) and genus bound
Speaker: Daren Chen, Caltech
Time: 2:00pm–3:00pm
Place: CMC 108

Abstract

We will briefly review Khovnaov homology and its Bar-Natan deformation for links in \(S^3\), and describe how to obtain a slice genus bound using them. Then, we will introduce a Bar-Natan homology for null homologous links in \(RP^3\). As in the case for the usual Bar-Natan homology, this gives rise to a certain genus bound. More explicitly, it gives a genus bound for equivariant slice surface bounding the lift of the knot in S^3, such that the involution reverses the orientation on the surface.

Monday, January 8, 2024

Title: Rational homology spheres and \(\mathrm{SL}(2,C)\) representations
Speaker: Sudipta Ghosh, University of Notre Dame
Time: 2:00pm–3:00pm
Place: CMC 116

Abstract

Building on non-vanishing theorems of Kronheimer and Mrowka in instanton Floer homology, Zentner proved that if \(Y\) is a homology \(3\)-sphere other than \(S^3\), then its fundamental group admits a homomorphism to \(\mathrm{SL}(2,C)\) with non-abelian image. In this talk, I will discuss how to generalize this to any \(Y\) whose first homology is \(2\)-torsion or \(3\)-torsion, other than \(\#^n\) \(RP^3\) for any \(n\) or lens spaces of order \(3\). This is joint work with Steven Sivek and Raphael Zentner.