Milen Yakimov’s research while affiliated with Northeastern University and other places

We describe a connection between the subjects of cluster algebras, polynomial identity algebras, and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac, and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we obtain an explicit formula for the discriminant of the integral form over of each quantum unipotent cell of De Concini, Kac, and Procesi for arbitrary symmetrizable Kac–Moody algebras, where is a root of unity.

We develop a theory of Wilson's adelic Grassmannian ${\mathrm{Gr}}^{\mathrm{ad}}(R)$ and Segal-Wilson's rational Grasssmannian ${\mathrm{Gr}}^ {\mathrm{rat}}(R)$ associated to an arbitrary finite dimensional complex algebra R. We provide several equivalent descriptions of the former in terms of the indecomposable projective modules of R and its primitive idempotents, and prove that it classifies the bispectral Darboux transformations of the R-valued exponential function. The rational Grasssmannian ${\mathrm{Gr}}^{\mathrm{rat}}(R)$ is defined by using certain free submodules of R(z) and it is proved that it can be alternatively defined via Wilson type conditions imposed in a representation theoretic settings. A canonical embedding ${\mathrm{Gr}}^{\mathrm{ad}}(R) \hookrightarrow {\mathrm{Gr}}^{\mathrm{rat}}(R)$ is constructed based on a perfect pairing between the R-bimodule of quasiexponentials with values in R and the R-bimodule R[z].

We investigate the restriction of the discrete Fourier transform $F_N : L^2(\mathbb{Z}/N \mathbb{Z}) \to L^2(\mathbb{Z}/N \mathbb{Z})$ to the space $\mathcal C_a$ of functions with support on the discrete interval $[-a,a]$, whose transforms are supported inside the same interval. A periodically tridiagonal matrix J on $L^2(\mathbb{Z}/N \mathbb{Z})$ is constructed having the three properties that it commutes with $F_N$, has eigenspaces of dimensions 1 and 2 only, and the span of its eigenspaces of dimension 1 is precisely $\mathcal C_a$. The simple eigenspaces of J provide an orthonormal eigenbasis of the restriction of $F_N$ to $\mathcal C_a$. The dimension 2 eigenspaces of J have canonical basis elements supported on $[-a,a]$ and its complement. These bases give an interpolation formula for reconstructing $f(x)\in L^2(\mathbb{Z}/N\mathbb{Z})$ from the values of f(x) and $\widehat f(x)$ on $[-a,a]$, i.e., an explicit Fourier uniqueness pair interpolation formula. The coefficients of the interpolation formula are expressed in terms of theta functions. Lastly, we construct an explicit basis of $\mathcal C_a$ having extremal support and leverage it to obtain explicit formulas for eigenfunctions of $F_N$ in $C_a$ when $\dim \mathcal C_a \leq 4$.

Classical prolate spheroidal functions play an important role in the study of time-band limiting, scaling limits of random matrices, and the distribution of the zeros of the Riemann zeta function. We establish an intrinsic relationship between discrete–continuous bispectral functions and the prolate spheroidal phenomenon. The former functions form a vast class, parametrized by an infinite dimensional manifold, and are constructed by Darboux transformations from classical bispectral functions associated to orthogonal polynomials. Special cases include spherical functions. We prove that all such Darboux transformations which are self-adjoint in a certain sense give rise to integral operators possessing commuting differential operators and to discrete integral operators possessing commuting shift operators. One particularly striking implication of this is the correspondence between discrete and continuous pairs of commuting operators. Moreover, all results are proved in the setting of matrix valued functions, which provides further advantages for applications. Our methods rely on the use of noncommutative matrix valued Fourier algebras associated to discrete–continuous bispectral functions. We produce the commuting differential and shift operators in a constructive way with explicit upper bounds on their orders and bandwidths, which is illustrated with many concrete examples.

Finite tensor categories (FTCs) T${\textbf{T}}$ are important generalizations of the categories of finite dimensional modules of finite dimensional Hopf algebras, which play a key role in many areas of mathematics and mathematical physics. There are two fundamentally different support theories for them: a cohomological one and a universal one based on the noncommutative Balmer spectra of their stable (triangulated) categories T̲${\underline{{\textbf{T}}}}$. In this paper we introduce the key notion of the categorical center CT̲∙$C^\bullet _{{\underline{{\textbf{T}}}}}$ of the cohomology ring RT̲∙$R^\bullet _{{\underline{{\textbf{T}}}}}$ of an FTC, T${\textbf{T}}$. This enables us to put forward a complete and detailed program to investigate the relationship between the two support theories, based on CT̲∙$C^\bullet _{{\underline{{\textbf{T}}}}}$ of the cohomology ring RT̲∙$R^\bullet _{{\underline{{\textbf{T}}}}}$ of an FTC, T${\textbf{T}}$. Our main result is the construction of a continuous map from the noncommutative Balmer spectrum of an arbitrary FTC, T${\textbf{T}}$, to the Proj${{\,\textrm{Proj}\,}}$ of the categorical center CT̲∙$C^\bullet _{{\underline{{\textbf{T}}}}}$ and a theorem that this map is surjective under a weaker finite generation assumption for T${\textbf{T}}$ than the one conjectured by Etingof–Ostrik. We conjecture that, for all FTCs, (i) the map is a homeomorphism and (ii) the two-sided thick ideals of T̲${\underline{{\textbf{T}}}}$ are classified by the specialization closed subsets of ProjCT̲∙${{\,\textrm{Proj}\,}}C^\bullet _{{\underline{{\textbf{T}}}}}$. We verify parts of the conjecture under stronger assumptions on the category T${\textbf{T}}$. Many examples are presented that demonstrate how in important cases CT̲∙$C^\bullet _{{\underline{{\textbf{T}}}}}$ arises as a fixed point subring of RT̲∙$R^\bullet _{{\underline{{\textbf{T}}}}}$ and how the two-sided thick ideals of T̲${\underline{{\textbf{T}}}}$ are determined in a uniform fashion (while previous methods dealt on a case-by-case basis with case specific methods). The majority of our results are proved in the greater generality of monoidal triangulated categories and versions of them for Tate cohomology are also presented.

Given a braided monoidal category $\mathcal{C}$ and $\mathcal{C}$-module category $\mathcal{M}$, we introduce a version of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of $\mathcal{C}$ adapted for $\mathcal{M}$; we refer to this category as the reflective center $\mathcal{E}_\mathcal{C}(\mathcal{M})$ of $\mathcal{M}$. Just like $\mathcal{Z}(\mathcal{C})$ is a canonical braided monoidal category attached to $\mathcal{C}$, we show that $\mathcal{E}_\mathcal{C}(\mathcal{M})$ is a canonical braided module category attached to $\mathcal{M}$. We also study when $\mathcal{E}_\mathcal{C}(\mathcal{M})$ possesses nice properties such as being abelian, finite, and semisimple. Our second goal pertains to when $\mathcal{C}$ is the category of modules over a quasitriangular Hopf algebra H, and $\mathcal{M}$ is the category of modules over an H-comodule algebra A. We show that $\mathcal{E}_\mathcal{C}(\mathcal{M})$ here is equivalent to a category of modules over (or, is represented by) an explicit algebra, denoted by $R_H(A)$, which we call the reflective algebra of A. This result is akin to $\mathcal{Z}(\mathcal{C})$ being represented by the Drinfeld double $\text{Drin}(H)$ of H. Our third set of results is also in the Hopf setting above. We show that reflective algebras are quasitriangular H-comodule algebras, and examine their corresponding quantum K-matrices. We also establish that the reflective algebra $R_H(\Bbbk)$ is an initial object in the category of quasitriangular H-comodule algebras, where $\Bbbk$ is the ground field. The case when H is the Drinfeld double of a finite group is illustrated. Lastly, we study the reflective center $\mathcal{E}_\mathcal{C}(\mathcal{M})$ as a module category over $\mathcal{Z}(\mathcal{C})$ in the Hopf setting. This action gives the reflective algebra $R_H(A)$ the structure of a $\text{Drin}(H)$-comodule algebra.

Beginning with the work of Landau, Pollak and Slepian in the 1960s on time‐band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point W of Wilson's infinite dimensional adelic Grassmannian gives rise to an integral operator , acting on for a contour , which reflects a differential operator with rational coefficients in the sense that on a dense subset of . By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function . The exact size of this algebra with respect to a bifiltration is in turn determined using algebro‐geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time‐band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions always reflect a differential operator. A 90° rotation argument is used to prove that the time‐band limited operators of the generalized Fourier transforms with kernels admit a commuting differential operator. These methods produce vast collections of integral operators with prolate‐spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.

The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two approaches leading to the notion of Poisson trace orders. It is proved that all regular and reduced traces are always compatible with any Poisson order structure. The modified discriminant ideals of all Poisson trace orders are proved to be Poisson ideals and the zero loci of discriminant ideals are shown to be unions of symplectic cores, under natural assumptions (maximal orders and Cayley–Hamilton algebras). A base change theorem for Poisson trace orders is proved. A broad range of Poisson trace orders are constructed based on the proved theorems: quantized universal enveloping algebras, quantum Schubert cell algebras and quantum function algebras at roots of unity, symplectic reflection algebras, 3D and 4D Sklyanin algebras, Drinfeld doubles of pre-Nichols algebras of diagonal type, and root of unity quantum cluster algebras.