Research
Research
Research
Research
273+ open-access research outputs.
In this paper, we study the Fu\v{c}ik spectrum for the operator with rapidly increasing weight, which is defined as a set $\Sigma$ comprising those $(\alpha, \beta) \in \mathbb{R}^2$ such that \begin{…
This thesis is an exposition of Ollivier-Ricci Curvature of metric spaces as introduced by Yann Ollivier, which is based upon the 1-Wasserstein Distance and optimal transport theory. We present some o…
The cyclic sieving phenomenon was introduced by Reiner, Stanton and White in 2004 as a generalization of Stembridge's $q=-1$ phenomenon. In a paper from 2008, Eu and Fu studied many occurrences of thi…
We introduce minimal Richardson orbits and pseudo-polarizations for nilpotent orbits in classical Lie algebras of types B, C, and D. For any nilpotent orbit, we classify all minimal Richardson orbits …
It is proved by M. Paun (1997, 2017) that the fundamental group of a compact Kahler manifold X is almost Abelian if the anti-canonical bundle -KX is nef. In this paper, we apply the recent geometric a…
For a cubic hypersurface $X$, work of Galkin--Shinder and Voisin shows the existence of a birational map relating the Hilbert scheme of two points $X^{[2]}$ with a certain projective bundle over $X$. …
Claesson and Linusson [Proc. Am. Math. Soc., 139 (2011), 435-449] observed that there are n! matchings on [2n] with no left-nestings. Inspired by this result, this paper is devoted to exploring a deep…
We study Fourier transforms induced by Markman's projectively hyperholomorphic bundles on products of hyper-K\"ahler varieties of $K3^{[n]}$-type. As applications, we prove the following. (a) Derived …
Let $D$ be a totally definite quaternion algebra over a totally real number field $F$, and $\mathcal{O}$ be an $O_F$-order (of full rank) in $D$. The type number $t(\mathcal{O})$ is an important arith…
In this note, we study Einstein manifolds whose curvature operator of the second kind $\mathring{R}$ satisfies the cone condition \[ \alpha^{-1}\big(\sum_{i=1}^{[\alpha]} \lambda_i+ (\alpha - [\alpha]…
In this paper, we investigate the Fu\v{c}\'{i}k spectrum $\Sigma_L$ associated with the logarithmic Laplacian. This spectrum is defined as the set of all pairs $(\alpha,\beta) \in \mathbb{R}^2$ for wh…
We prove that complete Riemannian manifolds of dimension $n\ge3$ with harmonic curvature and $\frac{n(n+2)}{2(n+1)}$-nonnegative curvature operator of the second kind must be Einstein. In particular, …
Using Bochner techniques, we prove that a compact Einstein manifold of dimension $n \ge 4$ has constant curvature provided that the curvature operator of the second kind satisfies a cone condition tha…
Convolution of valuations was introduced by the first named author and Fu for linear spaces, and later by Alesker and the first named author for compact Lie groups. In this paper we study the convolut…
We prove that under a mild condition, a multiset of tagged permissible arcs over a skew-tiling is uniquely determined by its intersection vector. As an application, it is proved that -- up to isomorph…
We prove a conjecture by Belmans, Fu and Krug concerning the Hochschild homology of the symmetric powers of a small dg category $\mathscr{C}$. More precisely, we show that these groups decompose into …
In recent years, there has been extensive work on inequalities among partition functions. In particular, Nicolas, and independently DeSalvo--Pak, proved that the partition function $p(n)$ is eventuall…
We provide exposition into the field of projection theory, which lies at the intersection of incidence geometry and geometric measure theory. We first give the necessary preliminaries in Chapter 2, fo…
Pipeline-parallel distributed optimization is essential for large-scale machine learning but is challenged by significant communication overhead from transmitting high-dimensional activations and grad…
Given a holomorphic or anti-holomorphic involution on a complex variety, the Smith inequality says that the total $\mathbb{F}_2$-Betti number of the fixed locus is no greater than the total $\mathbb{F…
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