In 2020, Naisse and Putyra gave the first extension of odd Khovanov homology to tangles. To answer this relatively longstanding open question, they introduced new algebraic objects called grading categories. Recently, we have shown that grading categories, while slightly unfriendly to their user, admit flexible generalizations which allow for other interesting constructions. These include categorified Jones-Wenzl projectors and, ultimately, an "odd" colored link homology theory. In this talk, I’ll introduce grading categories and aim to discuss ongoing work regarding the Hochschild (co)homology of algebras graded by grading categories, and its application to link homology.
We study the scaling limit of the dimer model on ‘critical’ graphs. We establish a connection between the dimer model and the free Dirac Fermion quantum field theory in various ways by studying them in a background ‘gauge field’. We speculate on how to leverage the fundamental nature of the Dirac Fermion in 2D CFT to study general CFTs from the dimer point-of-view.
Topological and piecewise linear manifolds require extremely different methods, but the global structures are almost the same. A (compact, connected) topological manifold M has an invariant ks[M] in H^4(M;Z/2), and for dimensions greater than 4, M has a PL structure if and only if km[M]=0. In dimension 4 this reduces to an invariant in Z/2. A PL structure implies the invariant is trivial, but the converse is dramatically false. The question here is: when does the homotopy type of M determine the Kirby-Siebenmann invariant, and in these cases how can it be calculated? The answer is known for oriented manifolds; we speculate on the unorientable case.
A real form of the exceptional simple Lie group of rank 2 can be obtained from the local symmetries of a pair of spheres rolling along each other, but only when the ratio of the radii of the spheres is 1:3 or 3:1. While this result might seem miraculous, we will attempt to provide a fairly simple visual explanation for why such a ratio of radii is needed, using a bit of basic Lie theory and a lot of pictures.
The volume conjecture connects a large $n$ limit of the colored Jones polynomial $J_n(K)$ of a knot $K$ to the hyperbolic volume of its complement. In recent work, Agarwal, Lee, Gang, and Romo proposed the length conjecture, which connects the large $n$ limit of a colored Jones polynomial of the link $K\cup L$ to the ``length" of $L$ in the complement of $K$. A careful statement of this conjecture requires many ingredients, the most important of which are a state integral model for perturbative $\mathrm{SL}(2,\mathbb{C})$ Chern-Simons theory and a $3$d quantum trace map. In this talk, we will begin with an overview of the necessary ingredients to define the volume conjecture. Then, we will discuss the modifications required to extend the volume conjecture to the length conjecture, along the way giving a working definition of the aforementioned state integral and 3d quantum trace. Time permitting, I will discuss ongoing work, joint with Mauricio Romo, on proving this conjecture for twist knots.
The Sageev construction is a method for building group actions on CAT(0) cube complexes using codimension-one subgroups. It has been especially useful in the study of hyperbolic and relatively hyperbolic groups, thanks to the work of Agol and Wise. In this talk, we will describe how to construct a group action on a CAT(0) cube complex from a countable group acting on a sufficiently nice topological space M. When M is the Gromov boundary of a hyperbolic (relatively hyperbolic) group, our more general approach retrieves Sageev’s construction. This is joint work with Jason Manning.
We consider the homogeneous mean-field Bose gas at temperatures proportional to the critical temperature of its Bose--Einstein condensation phase transition. Our main result is a trace-norm approximation of the grand canonical Gibbs state in terms of a reference state, which is given by a convex combination of products of coherent states and Gibbs states associated with certain temperature-dependent Bogoliubov Hamiltonians. The convex combination is expressed as an integral over a Gibbs distribution of a one-mode Φ4-theory describing the condensate. We interpret this result as a justification of Bogoliubov theory at positive temperature. Further results derived from the norm approximation include various limiting distributions for the number of particles in the condensate, as well as precise formulas for the one- and two-particle density matrices of the Gibbs state. Key ingredients of our proof, which are of independent interest, include two novel abstract correlation inequalities. The proof of one of them is based on an application of an infinite-dimensional version of Stahl's theorem. This is joint work with Phan Thành Nam, Marcin Napiórkowski.
Error-correcting codes are crucial for ensuring reliable data transmission and storage in the presence of errors or data corruption. Designing efficient, high-performance codes requires understanding their structural and combinatorial properties. A key aspect of classical coding theory is determining bounds on coding parameters and identifying invariants that describe code structures. One powerful tool for analyzing these properties and their performance is the theory of anticodes. Within the 'quantum revolution,' quantum communication and computation are transforming information processing through the principles of quantum mechanics. However, quantum systems introduce complex error models, making it challenging to directly apply classical coding theory techniques.In this talk, we present a new framework for analyzing stabilizer codes from an anticode perspective. We extend the notion of generalized distance to quantum anticodes and derive the McWilliams identities. This approach leads to a powerful generalization of the quantum cleaning lemma, which can be interpreted as the quantum counterpart of a classical duality result. The new results in this talk are joint work with C. Cao and B. Lackey.
The log etale topology is a natural analogue of the etale topology for log schemes. Unfortunately, very few things satisfy log etale descent -- not even vector bundles or the structure sheaf. We introduce a new rhizomic topology that sits in between the usual and log etale topologies and show most things do satisfy rhizomic descent! As a case study, we look at tropical abelian varieties and give some exotic examples.
Vaughan Jones showed how to associate links in the $3$-sphere to elements of Thompson’s group $F$ and proved that $F$ gives rise to all link types. This talk will discuss two recent extensions of Jones’ work– the first is a method of building annular links from Thompson’s group $T$, which contains $F$ as a subgroup, and the second is a method of building $(n,n)$-tangles from $F$ . Annular links from $T$ arise from Jones’s unitary representations of the Thompson groups, and tangles from $F$ give rise to an action of $F$ on Khovanov’s chain complexes. This talk includes joint work with Slava Krushkal and Yangxiao Luo