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Changho Han

Publications and source records attributed to Changho Han.

12 recordsLinked to original sources

Surrogate modeling for interpreting black-box LLMs in medical predictions

Large language models (LLMs), trained on vast datasets, encode extensive real-world knowledge within their parameters, yet their black-box nature obscures the mechanisms and extent of this encoding. Surrogate modeling, which uses simplified models to approximate complex systems, can offer a path toward better interpretability of black-box models. We propose a surrogate modeling framework that quantitatively explains LLM-encoded knowledge. For a specific hypothesis derived from domain knowledge, this framework approximates the latent LLM knowledge space using observable elements (input-output pairs) through extensive prompting across a comprehensive range of simulated scenarios. Through proof-of-concept experiments in medical predictions, we demonstrate our framework's effectiveness in revealing the extent to which LLMs "perceive" each input variable in relation to the output. Particularly, given concerns that LLMs may perpetuate inaccuracies and societal biases embedded in their training data, our experiments using this framework quantitatively revealed both associations that contradict established medical knowledge and the persistence of scientifically refuted racial assumptions within LLM-encoded knowledge. By disclosing these issues, our framework can act as a red-flag indicator to support the safe and reliable application of these models.

cs.CL

Secant rank and syzygies of projections of elliptic normal curves

We study the syzygies of projections of elliptic normal curves. Let $C \subset \mathbb{P}^{d-1}$ be an elliptic normal curve of degree $d \ge 5$, and let $C_q$ denote the projection of $C$ from a point $q$. We obtain sharp bounds for the Green--Lazarsfeld index of $C_q$ in terms of the secant rank of $q$. More precisely, if $q \in C^s \setminus C^2$, where $C^s$ is the $s$-th secant variety of $C$, then $\mathrm{index}(C_q) \le s-3$, and equality holds for a general point $q$ of $C^s$. In particular, $\mathrm{index}(C_q) = \lceil \frac{d}{2} \rceil - 3$ for a general point $q$ in $\mathbb{P}^{d-1}$. The proof realizes projected elliptic curves as hyperplane sections of elliptic ruled surface scrolls and exploits the known syzygetic properties of these scrolls.

math.AG

The stacky Batyrev-Manin conjecture and modular curves

Let $\mathcal{X}_0(N)$ be the Deligne--Rapoport modular stack of elliptic curves endowed with a cyclic rational $N$-isogeny over a number field $F$. Let $N\in\{1,2,3,4,5,6,7,8,9,10,12,13,16,18,25\},$ which are precisely the values for which the coarse moduli space of $\mathcal{X}_0(N)$ is isomorphic to $\mathbb{P}^1$. We show that the stacky Batyrev--Manin conjecture [DY24] holds for the naive height on $\mathcal{X}_0(N)$ when $F=\mathbb{Q}$. In the process, we give a concrete description of $\mathcal{X}_0(N)$ as a square root stack over a stacky curve.

math.NT

Counting 5-isogenies of elliptic curves over $\mathbb{Q}$

We show that the number of $5$-isogenies of elliptic curves defined over $\mathbb{Q}$ with naive height bounded by $H > 0$ is asymptotic to $C_5\cdot H^{1/6} (\log H)^2$ for some explicitly computable constant $C_5 > 0$. This settles the asymptotic count of rational points on the genus zero modular curves $X_0(m)$. We leverage an explicit $\mathbb{Q}$-isomorphism between the stack $\mathscr{X}_0(5)$ and the generalized Fermat equation $x^2 + y^2 = z^4$ with $\mathbb{G}_m$-action of weights $(4, 4, 2)$.

math.NT

Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism

We describe Baily-Borel, toroidal, and geometric -- using the KSBA stable pairs -- compactifications of some moduli spaces of K3 surfaces with a nonsymplectic automorphism of order $3$ and $4$ for which the fixed locus of the automorphism contains a curve of genus $\ge2$. For order $3$, we treat all the maximal-dimensional such families. We show that the toroidal and the KSBA compactifications in these cases admit simple descriptions in terms of certain $ADE$ root lattices.

math.AG

Extending the Torelli map to alternative compactifications of the moduli space of curves

Determining the limiting behaviour of the Jacobian as the underlying curve degenerates has been the subject of much interest. For nodal singularities, there are beautiful constructions of Caporaso as well as Pandharipande of compactified universal Jacobians over the moduli space of stable curves. Alexeev later obtained a canonical such compactification by extending the Torelli map out of the Deligne-Mumford compactification of $\mathcal{M}_{g,n}$. In contrast, Alexeev and Brunyate proved that the Torelli map does not extend over the cuspidal locus in Schubert's alternative compactification of pseudostable curves. In this paper, we consider curves with singularities that locally look like the axes in $m$-space, which we call axis-like singularities. We construct an alternative compactification of $\mathcal{M}_{g,n}$ consisting of curves with such singularities and prove that the Torelli map extends out of this compactification. Furthermore, for every alternative compactification in the sense of Smyth, we identify an axis-like locus over which the Torelli map extends.

math.AG

Compact moduli of K3 surfaces with a nonsymplectic automorphism

We construct a modular compactification via stable slc pairs for the moduli spaces of K3 surfaces with a nonsymplectic group of automorphisms under the assumption that some combination of the fixed loci of automorphisms defines an effective big divisor, and prove that it is semitoroidal.

math.AG

Arithmetic inflection of superelliptic curves

In this paper, we explore the inflectionary behavior of linear series on superelliptic curves $X$ over fields of arbitrary characteristic. Here we give a precise description of the inflection of linear series over the ramification locus of the superelliptic projection; and we initiate a study of those inflectionary varieties that parameterize the inflection points of linear series on $X$ supported away from the superelliptic ramification locus that is predicated on the behavior of their Newton polytopes.

math.AG

Arithmetic inflection formulae for linear series on hyperelliptic curves

Over the complex numbers, Pl\"ucker's formula computes the number of inflection points of a linear series of projective dimension $r$ and degree $d$ on a curve of genus $g$. Here we explore the geometric meaning of a natural analogue of Pl\"ucker's formula in $\mathbb{A}^1$-homotopy theory for certain linear series on hyperelliptic curves defined over an arbitrary field.

math.AG

Enumerating odd-degree hyperelliptic curves and abelian surfaces over $\mathbb{P}^1$

Given asymptotic counts in number theory, a question of Venkatesh asks what is the topological nature of lower order terms. We consider the arithmetic aspect of the inertia stack of an algebraic stack over finite fields to partially answer this question. Subsequently, we acquire new sharp enumerations on quasi-admissible odd-degree hyperelliptic curves over $\mathbb{F}_q(t)$ ordered by bounded discriminant height.

math.AG

Stable log surfaces, admissible covers, and canonical curves of genus 4

We explicitly describe the KSBA/Hacking compactification of a moduli space of log surfaces of Picard rank 2. The space parametrizes log pairs $(S, D)$ where $S$ is a degeneration of $\mathbb{P}^1 \times \mathbb{P}^1$ and $D \subset S$ is a degeneration of a curve of class $(3,3)$. We prove that the compactified moduli space is a smooth Deligne--Mumford stack with 4 boundary components. We relate it to the moduli space of genus 4 curves; we show that it compactifies the blow-up of the hyperelliptic locus. We also relate it to a compactification of the Hurwitz space of triple coverings of $\mathbb{P}^1$ by genus 4 curves.

math.AG

Arithmetic of the moduli of semistable elliptic surfaces

We prove a new sharp asymptotic with the lower order term of zeroth order on $\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})$ for counting the semistable elliptic curves over $\mathbb{F}_q(t)$ by the bounded height of discriminant $\Delta(X)$. The precise count is acquired by considering the moduli of nonsingular semistable elliptic fibrations over $\mathbb{P}^{1}$, also known as semistable elliptic surfaces, with $12n$ nodal singular fibers and a distinguished section. We establish a bijection of $K$-points between the moduli functor of semistable elliptic surfaces and the stack of morphisms $\mathcal{L}_{1,12n} \cong \mathrm{Hom}_n(\mathbb{P}^{1}, \overline{\mathcal{M}}_{1,1})$ where $\overline{\mathcal{M}}_{1,1}$ is the Deligne-Mumford stack of stable elliptic curves and $K$ is any field of characteristic $\neq 2,3$. For $\mathrm{char}(K)=0$, we show that the class of $\mathrm{Hom}_n(\mathbb{P}^1,\mathcal{P}(a,b))$ in the Grothendieck ring of $K$-stacks, where $\mathcal{P}(a,b)$ is a 1-dimensional $(a,b)$ weighted projective stack, is equal to $\mathbb{L}^{(a+b)n+1}-\mathbb{L}^{(a+b)n-1}$. Consequently, we find that the motive of the moduli $\mathcal{L}_{1,12n}$ is $\mathbb{L}^{10n + 1}-\mathbb{L}^{10n - 1}$ and the cardinality of the set of weighted $\mathbb{F}_q$-points to be $\#_q(\mathcal{L}_{1,12n}) = q^{10n + 1}-q^{10n - 1}$. In the end, we formulate an analogous heuristic on $\mathcal{Z}_{\mathbb{Q}}(\mathcal{B})$ for counting the semistable elliptic curves over $\mathbb{Q}$ by the bounded height of discriminant $\Delta$ through the global fields analogy.

math.AG