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Researcher profile

Sanghyuk Lee

Sanghyuk Lee contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.CAmath.APArtificial IntelligenceComputation and LanguageMachine Learningmath.FA

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Published work

14 published item(s)

preprint2026arXiv

MeMo: Memory as a Model

Large language models (LLMs) achieve strong performance across a wide range of tasks, but remain frozen after pretraining until subsequent updates. Many real-world applications require timely, domain-specific information, motivating the need for efficient mechanisms to incorporate new knowledge. In this paper, we introduce MeMo (Memory as a Model), a modular framework that encodes new knowledge into a dedicated memory model while keeping the LLM parameters unchanged. Compared to existing methods, MeMo offers several advantages: (a) it captures complex cross-document relationships, (b) it is robust to retrieval noise, (c) it avoids catastrophic forgetting in the LLM, (d) it does not require access to the LLM's weights or output logits, enabling plug-and-play integration with both open and proprietary closed-source LLMs, and (e) its retrieval cost is independent of corpus size at inference time. Our experimental results on three benchmarks, BrowseComp-Plus, NarrativeQA, and MuSiQue, show that MeMo achieves strong performance compared to existing methods across diverse settings.

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preprint2023arXiv

Endpoint eigenfunction bounds for the Hermite operator

We establish the optimal $L^p$, $p=2(d+3)/(d+1),$ eigenfunction bound for the Hermite operator $\mathcal H=-Δ+|x|^2$ on $\mathbb R^d$. Let $Π_λ$ denote the projection operator to the vector space spanned by the eigenfunctions of $\mathcal H$ with eigenvalue $λ$. The optimal $L^2$--$L^p$ bounds on $Π_λ$, $2\le p\le \infty$, have been known by the works of Karadzhov and Koch-Tataru except $p=2(d+3)/(d+1)$. For $d\ge 3$, we prove the optimal bound for the missing endpoint case. Our result is built on a new phenomenon: improvement of the bound due to asymmetric localization near the sphere $\sqrtλ\mathbb S^{d-1}$.

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preprint2022arXiv

Carleman inequalities and unique continuation for the polyharmonic operators

We obtain a complete characterization of $L^p-L^q$ Carleman estimates with weight $e^{v\cdot x}$ for the polyharmonic operators. Our result extends the Carleman inequalities for the Laplacian due to Kenig--Ruiz--Sogge. Consequently, we obtain new unique continuation properties of higher order Schrödinger equations relaxing the integrability assumption on the solution spaces.

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preprint2022arXiv

Pointwise convergence of sequential Schrödinger means

We study pointwise convergence of the fractional Schrödinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function $\sup_{n} |e^{it_n(-Δ)^{α/2}} f| $ can be deduced from those on $\sup_{0<t\le 1} |e^{it(-Δ)^{α/2}} f|$ when $\{t_n\}$ is contained in the Lorentz space $\ell^{r,\infty}$. Consequently, our results provide seemingly optimal results in higher dimensions, which extend the recent work of Dimou-Seeger, and Li-Wang-Yan to higher dimensions. Our approach based on a localization argument also works for other dispersive equations and provides alternative proofs of previous results on sequential convergence.

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preprint2022arXiv

Remarks on dimension of unions of curves

We study an analogue of Marstrand&#39;s circle packing problem for curves in higher dimensions. We consider collections of curves which are generated by translation and dilation of a curve $γ$ in $\mathbb R^d$, i.e., $ x + t γ$, $(x,t) \in \mathbb R^d \times (0,\infty)$. For a Borel set $F \subset \mathbb R^d\times (0,\infty)$, we show the unions of curves $\bigcup_{(x,t) \in F} ( x+tγ)$ has Hausdorff dimension at least $α+1$ whenever $F$ has Hausdorff dimension bigger than $α$, $α\in (0, d-1)$. We also obtain results for unions of curves generated by multi-parameter dilation of $γ$. One of the main ingredients is a local smoothing type estimate (for averages over curves) relative to fractal measures.

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preprint2022arXiv

Sharp smoothing properties of averages over curves

We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $γ$ in $\mathbb R^d$, $d\ge 3$. Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal $L^p$ Sobolev regularity estimates, which settle the conjecture raised by Beltran-Guo-Hickman-Seeger. Besides, we show the sharp local smoothing estimates for every $d$. As a result, we establish, for the first time, nontrivial $L^p$ boundedness of the maximal average over dilations of $γ$ for $d\ge 4$.

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preprint2022arXiv

Strichartz and uniform Sobolev inequalities for the elastic wave equation

We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are deduced. For the purpose we diagonalize the symbols of the Lamé operator and its semigroup, which also gives an alternative and simpler proofs of the previous results on perturbed elastic wave equations. Furthermore, we obtain uniform Sobolev inequalities for the elastic wave operator.

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preprint2022arXiv

Unique continuation for the heat operator with potentials in weak spaces

We prove strong unique continuation property for the differential inequality $|(\partial_t +Δ)u(x,t)|\le V(x,t)|u(x,t)|$ with $V$ contained in weak spaces. In particular, we establish the strong unique continuation property for $V\in L^\infty_t L^{d/2,\infty}_x$, which has been left open since the works of Escauriaza [6] and Escauriaza-Vega [8]. Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces.

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preprint2021arXiv

Dimension of divergence set of the wave equation

We consider the Hausdorff dimension of the divergence set on which the pointwise convergence $\lim_{t\rightarrow 0} e^{it\sqrt{-Δ}} f(x) = f(x)$ fails when $f \in H^s(\mathbb R^d)$. We especially prove the conjecture raised by Barceló, Bennett, Carbery and Rogers \cite{BBCR} for $d=3$, and improve the previous results in higher dimensions $d\ge4$. We also show that a Strichartz type estimate for $f\to e^{it\sqrt{-Δ}} f$ with the measure $ dt\,dμ(x)$ is essentially equivalent to the estimate for the spherical average of $\widehat μ$ which has been extensively studied for the Falconer distance set problem. The equivalence provides shortcuts to the recent results due to B. Liu and K. Rogers.

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preprint2020arXiv

Almost everywhere convergence of Bochner-Riesz means for the Hermite operators

Let $H = -Δ+ |x|^2$ be the Hermite operator in ${\mathbb R}^n$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with $H$ which is defined by $S_R^λ(H)f(x) = \sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^λ P_k f(x).$ Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le p<\infty$, we prove that $$ \lim\limits_{R\to \infty} S_R^λ(H) f=f \ \ \ \text{a.e.} $$ for all $f\in L^p(\mathbb R^n)$ provided that $λ> λ(p)/2$ and $λ(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely, we also show the convergence generally fails if $λ< λ(p)/2$ in the sense that there is an $f\in L^p(\mathbb R^n)$ for $2n/(n-1)\le p$ such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical Bochner-Riesz means for the Laplacian. For $n\geq 2$ and $p\ge 2$ our result tells that the critical summability index for a.e. convergence for $S_R^λ(H)$ is as small as only the \emph{half} of the critical index for a.e. convergence of the classical Bochner-Riesz means. When $n = 1$, we show a.e. convergence holds for $f\in L^p({\mathbb R})$ with $ p\geq 2$ whenever $λ>0$. Compared with the classical result due to Askey and Wainger who showed the optimal $L^p$ convergence for $S_R^λ(H)$ on ${\mathbb R}$ we only need smaller summability index for a.e. convergence.

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preprint2020arXiv

Sharp $L^p$-$L^q$ estimate for the spectral projection associated with the twisted Laplacian

In this note we are concerned with estimates for the spectral projection operator $\mathcal{P}_μ$ associated with the twisted Laplacian $L$. We completely characterize the optimal bounds on the operator norm of $\mathcal{P}_μ$ from $L^p$ to $L^q$ when $1\le p\le 2\le q\le \infty$. As an application, we obtain uniform resolvent estimate for $L$.

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preprint2020arXiv

Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein-Gordon and fractional Schrödinger equations. Our estimates extend those of Frank-Sabin in the case of the wave and Klein-Gordon equations, and generalize work of Frank-Lewin-Lieb-Seiringer and Frank-Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results. The main novelty of this paper is the use of estimates for weighted oscillatory integrals which we combine with an approach due to Frank and Sabin. This strategy also leads us to proving new estimates for weighted oscillatory integrals with optimal decay exponents which we believe to be of wider independent interest. Applications to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces are also provided.

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preprint2020arXiv

Uniqueness in the Calderón problem and bilinear restriction estimates

Uniqueness in the Calderón problem in dimension bigger than two was usually studied under the assumption that conductivity has bounded gradient. For conductivities with unbounded gradients uniqueness results have not been known until recent years. The latest result due to Haberman basically relies on the optimal $L^2$ restriction estimate for hypersurface which is known as the Tomas-Stein restriction theorem. In the course of developments of the Fourier restriction problem bilinear and multilinear generalizations of the (adjoint) restriction estimates under suitable transversality condition between surfaces have played important roles. Since such advanced machineries usually provide strengthened estimates, it seems natural to attempt to utilize these estimates to improve the known results. In this paper, we make use of the sharp bilinear restriction estimates, which is due to Tao, and relax the regularity assumption on conductivity. We also consider the inverse problem for the Schrödinger operator with potentials contained in the Sobolev spaces of negative orders.

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