Research
Research
Research
Research
2,735+ open-access research outputs.
The classification of maximal algebras of square block Toeplitz matrices is a considerably more difficult problem and has received relatively little attention in the existing literature. In this work,…
In 2014, Michal Lewicki and Andrzej Olbry\'s proved that if a real valued function $f$ defined on the real line satisfies the conditional functional equation \[ f(tx + (1-t)y) = t f(x) + (1-t) f(y),\q…
A graph is Schur-positive if its chromatic symmetric function expands non-negatively in the Schur basis. We determine a full Schur-positivity classification for complete multipartite graphs by showing…
Christoph, Dragani\'{c}, Gir\~{a}o, Hurley, Michel, and M\"{u}yesser conjectured that, when $d\mid n$, the expected number of cycles in a uniformly random cycle-factor of a directed $d$-regular graph …
Genomic Schur functions were introduced by Pechenik and Yong in connection with the $K$-theory of Grassmannians. Pechenik proved that genomic Schur functions admit a positive expansion in the basis of…
We study modular analogues of Schur numbers for systems of linear equations. We show that these only depend on the number of equations, not their coefficients and in the case of one equation show stro…
One of the central open problems in both algebraic combinatorics and representation theory is to find a positive combinatorial rule for Kronecker coefficients $ g_{\lambda \, \mu \, \nu}$. A notable a…
We prove that on a semisimple Lie algebra $\mathfrak{g}$ over a finite field of large characteristic, if a complex-valued invariant function $f$ and its Fourier transform $\hat f$ are both supported i…
We study the edge-expansion of the graph of a random $0/1$ polytope $P^d_p$, defined as the convex hull of a random subset of the points in $\{0,1\}^d$ where every point is retained independently and …
The extent to which the geometry of an object is determined by some associated spectral data is a longstanding problem. We investigate this problem in the context of the Steklov spectrum, focusing on …
For finite graphs $G$ and $H$, let $\RR(G,H)$ denote the isomorphism classes of Ramsey-minimal graphs for $(G,H)$. We prove two 1981 conjectures of Burr, Erd\H{o}s, Faudree, Rousseau, and Schelp: Rams…
Biochemical signalling cascades transduce extracellular stimuli into cellular responses through sequences of discrete, node-to-node activations. While signal fidelity depends critically on local inter…
Many engineering failures thermal hotspot concentration, Hertz contact fatigue localization, boundary-layer loss, mixing dead zones are geometric failure modes: changing the material delays the failur…
For an integer $t \geq 3$, let $\mathcal{L}(t)$ denote the linear equation $x_1 + x_2 + \cdots + x_{t-1} = x_t,$ where all variables are positive integers. For integers $k \geq 1$ and $t_0,t_1,\dots,t…
A 0/1-polytope is the convex hull of a subset $V\subseteq \{0,1\}^n$. A celebrated conjecture of Mihail and Vazirani asserts that the graph of every 0/1-polytope has edge-expansion at least 1. In this…
For a set of $n$ points $V \subseteq \mathbb{R}$ let $G(V, p)$ be the random graph on $V$ where each possible edge is present independently with probability $p$. We call a subset $U \subseteq V$ {\emp…
We investigate infinite-level Shimura varieties within the framework of analytic stacks of Clausen-Scholze, developing their smooth, completed, locally analytic, and de Rham realizations. We formulate…
We establish two identities for Lambert series and double Lambert series, thereby resolving conjectures of Andrews, Dixit, Schultz and Yee (Acta Arith.~181:253--286, 2017), as well as Amdeberhan, Andr…
Graph theory provides powerful tools for modeling concepts in number theory, leading to the introduction of graphs derived from arithmetic properties. One such structure is the divisor prime graph, $G…
We show that for every group $G$ there exists an embedding $G \to H$ such that a non-trivial endomorphism of $G$ extends to $H$ if and only if it is an inner automorphism. This strengthens a theorem o…
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