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UC Davis is situated in the heart of California. Founded in 1933, the Department of Mathematics stays relevant in mathmatics research both through continuing research and in discovering new talent. Research is encouraged across all levels, from undergraduate through graduate, as well as through outreach programs.

Cover page of A class of two-dimensional AKLT models with a gap

A class of two-dimensional AKLT models with a gap

(2021)

The AKLT spin chain is the prototypical example of a frustration-free quantum spin system with a spectral gap above its ground state. Affleck, Kennedy, Lieb, and Tasaki also conjectured that the two-dimensional version of their model on the hexagonal lattice exhibits a spectral gap. In this paper, we introduce a family of variants of the two-dimensional AKLT model depending on a positive integer n, which is defined by decorating the edges of the hexagonal lattice with one-dimensional AKLT spin chains of length $n$. We prove that these decorated models are gapped for all $n\geq 3$.

Natural Graph Wavelet Packet Dictionaries

(2021)

We introduce a set of novel multiscale basis transforms for signals on graphs that utilize their “dual” domains by incorporating the “natural” distances between graph Laplacian eigenvectors, rather than simply using the eigenvalue ordering. These basis dictionaries can be seen as generalizations of the classical Shannon wavelet packet dictionary to arbitrary graphs, and do not rely on the frequency interpretation of Laplacian eigenvalues. We describe the algorithms (involving either vector rotations or orthogonalizations) to construct these basis dictionaries, use them to efficiently approximate graph signals through the best basis search, and demonstrate the strengths of these basis dictionaries for graph signals measured on sunflower graphs and street networks.

Cover page of Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology

Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology

(2021)

We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors studied by Abel, Elias and Hogancamp are idempotents in the category of Soergel bimodules, and they correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme. As a consequence, we conjecture that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. We define a family of differentials dN on these dg algebras and conjecture that their homology matches that of the glN projectors, generalizing earlier conjectures of the first and third authors with Oblomkov and Shende.

Cover page of Markov Chains Through Semigroup Graph Expansions (A Survey)

Markov Chains Through Semigroup Graph Expansions (A Survey)

(2021)

We review the recent approach to Markov chains using the Karnofksy–Rhodes and McCammond expansions in semigroup theory by the authors and illustrate them by two examples.

Cover page of Al'brekht's Method in Infinite Dimensions

Al'brekht's Method in Infinite Dimensions

(2020)

In 1961 E. G. Albrekht presented a method for the optimal stabilization of smooth, nonlinear, finite dimensional, continuous time control systems. This method has been extended to similar systems in discrete time and to some stochastic systems in continuous and discrete time. In this paper we extend Albrekht's method to the optimal stabilization of some smooth, nonlinear, infinite dimensional, continuous time control systems whose nonlinearities are described by Fredholm integral operators.

Cover page of Algebraic Weaves and Braid Varieties

Algebraic Weaves and Braid Varieties

(2020)

In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting stratifications of braid varieties and their quotients. It is shown that the maximal charts of these stratifications are exponential Darboux charts for the holomorphic symplectic structures, and we relate these strata to exact Lagrangian fillings of Legendrian links.

Cover page of A Note on Complex-Hyperbolic Kleinian Groups

A Note on Complex-Hyperbolic Kleinian Groups

(2020)

Let Γ be a discrete group of isometries acting on the complex hyperbolic n-space HCn. In this note, we prove that if Γ is convex-cocompact, torsion-free, and the critical exponent δ(Γ) is strictly lesser than 2, then the complex manifold HCn/Γ is Stein. We also discuss several related conjectures.

Cover page of Hausdorff dimension of non-conical limit sets

Hausdorff dimension of non-conical limit sets

(2020)

Geometrically infinite Kleinian groups have non-conical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the non-conical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove that the Hausdorff dimension of the non-conical limit set is positive.

Cover page of An insertion algorithm on multiset partitions with applications to diagram algebras

An insertion algorithm on multiset partitions with applications to diagram algebras

(2020)

We generalize the Robinson–Schensted–Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length k with entries in [n] and pairs of tableaux of the same shape with one being a standard Young tableau of size n and the other being a standard multiset tableau of content [k]. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson [15].