Research
S. Mui, On the $L^{p}$ Aleksandrov Problem for negative $p$, Adv. Math, 408 (2022). link
We prove the existence of a solution to the $L^{p}$ Aleksandrov problem or the case of given even measures and $p ∈ (−1, 0)$. Furthermore, a sufficient measure concentration condition was provided for the case of $p ≤ −1$, again provided that the given measure is even.
S. Mui, On the $L^{p}$ dual Minkowski problem for $-1 < p < 0$, Calc. Var. Par. Diff. Eq., 64 (2023) link
We prove the existence of a solution to the $L^{p}$ dual Minkowski problem for the case of $q < p+1$, $-1 < p < 0$, and $p \neq q$ for even measures.
K. Boroczky, A. Kovacs, S. Mui, G. Zhang, Dual curvature density equation with group symmetry, J. Diff. Eq., 465 (2026). link
We prove that if the density function of the dual curvature measure is invariant under a closed subgroup of the orthogonal group, the geometric partial differential equation corresponding to the $L^p$ dual Minkowski problem for certain range of negative $p$ has a solution using a variational method.
M. Fernandez, G. Livshyts, S. Mui, On the smallest singular value of log-concave random matrices, arXiv:2508.17745, (2026). submitted. link
We prove sharp lower tail estimates for the smallest singular value of $A$, an isotropic log-concave random matrix, in the following cases: (1) when $N=n$ and $A$ is drawn from an unconditional distribution, with no independence assumption; (2) when the columns of $A$ are independent and $N\geq n$; (3) when $A$ is sufficiently tall, that is $N\geq (1+\lambda) n$ for any positive constant $\lambda$.
S. Mui, L. Qin, S. Wang, On the Minkowski problem with respect to a mixed Euclidean-Gaussian density, arXiv:2607.27618, (2026). submitted. link
We pose a new class of Minkowski problems corresponding to the mixed Euclidean-Gaussian volume. Using the flow method, we prove the existence of smooth normalized solutions for the corresponding Monge-Ampère-type equation in the symmetric case. Then, by approximation, we obtain the existence of origin-symmetric solutions to the mixed Euclidean-Gaussian Minkowski problem.
K. Boroczky, S. Mui, G. Zhang, The weighted dual Minkowski problem under group symmetry, arXiv:2609.20047, (2026). submitted. link
We generalize previous results on the existence of solutions to the dual Minkowski problem for origin-symmetric convex bodies. It is proved: (i) a full characterization of the G-invariant weighted dual curvature measure for q\in(0,1], (ii) a necessary and sufficient condition for the existence of solutions to the G-invariant weighted dual Minkowski problem for q\in(1,n), (iii) a sufficient condition for the G-invariant weighted dual curvature measure for q>1, and (iv) an extension of Henk-Pollehn’s dual curvature measure concentration property.
Undergraduate Research
Nash-Kuiper surfaces
You can find my work on Nash-Kuiper surfaces (2015-2017) here