Research Papers

https://doi.org/10.5281/zenodo.21400100

Abstract:
This paper extends a recent article that appears in the 2026 Annals of Mathematics — Lue Pan’s second paper on locally analytic vectors in the completed cohomology of modular curves, whose program includes a classicality theorem for p-adic modular forms of weight one — from modular curves to Hilbert modular surfaces. We work over a real quadratic field in which a prime p, at least 5, splits, and we study ordinary Hilbert modular forms of partial weight (1, k): weight one at one of the two embeddings and a regular weight k at the other. At such weights the usual weight-raising differential operator disappears in the weight-one direction, the two Hodge-Tate weights at the corresponding prime collide, and the standard routes to classicality and modularity break down.

The paper separates what can be proved outright from what cannot yet be proved. Unconditionally, we show that the correct classical differential operator in the singular direction is the second-order Bol operator, and that no first-order operator exists on the flag line, even after factoring through an auxiliary algebraic vector bundle; any first-order singular operator is therefore forced to live on infinite-level period sheaves of the kind Pan uses, with the Bol operator as its classical shadow. We compute the relevant local analytic kernels in both first-order and second-order form; we construct the derived “old” quotient at Iwahori level and prove erasure criteria for its ordinary part, resting on a Cartier trace theorem that tolerates unit distortions, extra variables, and matrix coefficients, and on a mixed filtration-adic contraction theorem; we prove a diagonal compactness theorem showing that, in defect-one patching, the patched complex, the patched action, and the augmentation already follow from finite-level data alone; and we establish seven local criteria forcing a finite flat Hecke algebra to equal the weight algebra, together with component-support theorems for the integral question.

The global conclusions are conditional on six precisely stated inputs, organized in three groups and each accompanied by evidence and by exact statements of what remains open. Assuming the branch-separation, patching, and transversality inputs, the deformation ring, the Hecke algebras, and the one-variable weight algebra at a fixed ordinary point all coincide, so the eigenvariety is smooth of dimension one there and the weight map is a local isomorphism. Assuming in addition the operator and comparison inputs, the classical forms of weight (1, k) are exactly the joint kernel of the two partial operators, in the spirit of Pan’s theorem. A final input on integral component support upgrades this to a deformation-to-Hecke isomorphism on every component, and hence to modularity of every lift satisfying the stated conditions. The paper closes with an analysis of why each open input is a natural and fruitful target for further research. Computer checks of the key finite formulas are included as reproducible sanity evidence only; no proof depends on them.

Keywords:
Hilbert modular surfaces, partial weight one, Bol operators, completed cohomology, Goren–Oort strata, Cartier trace, Taylor–Wiles patching

(Note: Appendix files can be found at this paper’s DOI.)

https://doi.org/10.5281/zenodo.21352063

Abstract:
This paper extends the theory introduced in a 2026 article that appears in the Journal of the American Mathematical Society by Peter Scholze. His theory attaches to every analytic base — archimedean or nonarchimedean, uniformly — a category of motives with spectrum coefficients, and it satisfies Tate cancellation over every base, a property which classically holds only over fields.

We enrich the coefficients of this theory from ordinary spectra to genuine C_2-equivariant spectra: objects carrying a symmetry of order two together with genuine fixed-point data. Such coefficients are the natural home of real algebraic K-theory, which binds together algebraic K-theory, Grothendieck-Witt theory, and L-theory.

Two general results drive the paper. The first is a fracture theorem: for every presentable stable coefficient category, genuine C_2-objects are equivalent to triples consisting of an object with C_2-action, an object with no action, and a gluing map through the Tate construction. The second is an orientation criterion: writing R for the Tate construction of the unit, the sought-after normalized Tate line — an equivariant refinement of the Tate twist whose geometric fixed points are trivial — exists exactly when the Tate twist becomes trivial as a module over R, and the space of choices is then a torsor under the units of R.

A canonical equivariant refinement of the Tate twist exists over every base and satisfies integral cancellation. The normalized refinement, by contrast, does not: over the real numbers, a theorem of Lin (the C_2 case of the Segal conjecture) identifies R with the 2-completed sphere, and the sign action of complex conjugation then yields a 2-adic obstruction — an orientation would force a 2-adic unit to equal its own negative. Consequently no normalized line exists over any base admitting a real point, including the integers with their archimedean norm; the obstruction is arithmetic and archimedean in nature. It vanishes over the complex numbers, after inverting 2, and at nonarchimedean geometric points of residue characteristic 2.

The failed universal statement is thereby replaced by a concrete program: compute the locus of bases over which the Tate twist is orientable, and determine there when a stable orientation descends to an effective one.

Keywords:
Berkovich motives, genuine equivariant spectra, Tate construction, cancellation, Picard obstruction

https://doi.org/10.5281/zenodo.20711000

Abstract:
Conditional value-at-risk (CVaR), or expected shortfall, leads to large-scale constrained quadratic programs in finance, energy, logistics, contrl, and learning. Recent algorithms for CVaR-constraint quadratic programs identify projection onto a top-k-sum sublevel set as the central computational primitive. This paper studies the complementary problem: how to differentiate this projection, and how to use it as a differentiable layer.

We show that the top-k-sum projection is piecewise affine and that an active-face certificate from the forward projection determines its backward pass. On each fixed face, the Jacobian is the orthogonal projector onto the corresponding tangent space, giving exact vector-Jacobian products in O(m) time once the certificate is available. The method handles degenerate tied plateaus and provides adjoints with respect to both the projection input and the CVaR risk budget, without forming a deterministic-equivalent epigraph quadratic program. For full CVaR-constrained quadratic programs, accurate solves are differentiated through a reduced active-face KKT system, while early-stopped solves are differentiated by unrolling.

Numerical tests validate the adjoints against finite differences and an independent solver, including degenerate cases. Batched CPU/GPU implementations scale to 200 million scenarios, differentiating a 200-million-scenario projection in under 0.12 seconds on a single GPU, while deterministic-equivalent differentiation becomes impractical near 100,000 scenarios. Pre-specified reproducible experiments show that this scale is decision-relevant: hard CVaR constraints place realized out-of-sample tail risk on budget as the scenario count grows, including under distribution shift, whereas calibrated penalty methods drift or retain substantial per-instance dispersion. Code and a one-command reproduction are provided.

Keywords:
conditional value-at-risk (CVaR); top-k-sum projection; differentiable optimization; vector-Jacobian product; ADMM; risk-constrained quadratic programming

https://doi.org/10.5281/zenodo.19947043

In this monograph, I carried the central insight of my earlier paper (“Beyond Words: Mindfulness in the Diamond Sutra”) out of contemplative inquiry and into analytic philosophical argument.
(This, perhaps, is not a move that the writers of the Diamond Sutra would have made — but I wanted to test whether analytic philosophy can carry the result. This book is that test.)

(The essay can be viewed or downloaded below.)

Abstract
Some research literature have suggested that stories can influence the self-concept. We want to test whether a narrative does influence people’s notions of self in their memory or imagination. Our hypothesis is that being exposed to a story makes one reconstructs one’s self in one’s memory or imagination to be more aligned with the story’s main character. With an experiment, we exposed participants (n=129) to two conditions (control: report of facts; treatment: the facts in a story form), and then asked the Likert-scale questions to see whether they report selves in their memory or imagination that resemble the story’s main character. Using independent sample t-tests, we found no statistically significant results. Therefore, we failed to reject our Null hypothesis — there was not enough evidence to reject it.
Keywords: story, narrative, self-concept, mental time travel, memory reconstruction

Abstract
This is a planned study, despite its language. It was part of a class. In this proposed study, we examine the relation between vividness of one’s visual imagery and one’s recollection of autobiographical memories. Despite the thousands of citations of David Marks’ 1973 foundational paper of assessing one’s visual imagery vividness, only a few had looked directly at visual imagery in connection with autobiographical memories. This study seeks to fill that gap. We are the first to utilize both the Vividness of Visual Imagery Questionnaire and the Autobiographical Recollection Test in a correlational study. We studied 100 college students (49 male, 48 female, 3 non-binary or prefer-not-to-say). We found that scores of VVIQ of Marks correlates to high scores of ART. Moreover, the correlation extends to all aspects of Autobiographical Recollection as modeled by Berntsen et al. T-tests confirm the findings by showing that the effect size is noteworthy: high scorers in VVIQ have higher means of ART scores — and all 7 of ART subscales — than low scorers of VVIQ. These findings suggest, inter alia, further possible explorations into the directionality of the relation of the two constructs, into  connection between the processes underlying the generation of vivid imageries and the recollection of autobiographical memories, and into the development of clinical treatments to improve wellbeing.
Keywords: visual imagery, autobiographical memory, vividness, memory recollection, narrative coherence, narrative relevance

Abstract
Inspired by prior groundbreaking studies on maximizing by Simon and other researchers, we sought to see whether the findings extend to lower-stakes decision-making. The particular lower-stakes context we chose to study is class selection by college students. We examined the relationships between maximizing tendency and four variables: information-seeking behavior, efforting behavior, regret affect, satisfaction affect. We had 130 different participants (64% female, 32% male, 4% other) from three semesters of a particular Pscyhology class at a university. The participants took an online questionnaire, which measured their maximizing tendency, behaviors, and affects related directly to our investigation. Through correlation tests, we found that maximizing was positively correlated with information-seeking (r = .34, p < .001), with efforting (r = .37, p < .001), and with regret (r = .23, p = .01); maximizing was negatively correlated on the outcome satisfaction variable (r = -.19, p = .03). With a Median Split and independent samples t-tests, we found that maximizers score higher in every area except on outcome satisfaction, where we did not find any result statistically significant. Our findings suggest there is a relationship between maximizing and the aforesaid behaviors and affects.
Keywords: maximizing, satisficing