Search

Generative Artificial Intelligence (GenAI) offers numerous opportunities to revolutionise teaching and learning in Computing Education (CE). However, educators have expressed concerns that students may over-rely on GenAI and use these tools to generate solutions without engaging in the learning process. While substantial research has explored GenAI use in CE, and many Computer Science (CS) educators have expressed their opinions and suggestions on the subject, there remains little consensus on implementing curricula and assessment changes. In this paper, we describe our experiences with using GenAI in CS-focused educational settings and the changes we have implemented accordingly in our teaching in recent years since the popularisation of GenAI. From our experiences, we propose two primary actions for the CE community: 1) redesign take-home assignments to incorporate GenAI use and assess students on their process of using GenAI to solve a task rather than simply on the final product; 2) redefine the role of educators to emphasise metacognitive aspects of learning, such as critical thinking and self-evaluation. This paper presents and discusses these stances and outlines several practical methods to implement these strategies in CS classrooms. Then, we advocate for more research addressing the concrete impacts of GenAI on CE, especially those evaluating the validity and effectiveness of new teaching practices., Comment: Accepted to ACE'25

CG (Computer Graphics) is a popular field of CS (Computer Science), but many students find this topic difficult due to it requiring a large number of skills, such as mathematics, programming, geometric reasoning, and creativity. Over the past few years, researchers have investigated ways to harness the power of GenAI (Generative Artificial Intelligence) to improve teaching. In CS, much of the research has focused on introductory computing. A recent study evaluating the performance of an LLM (Large Language Model), GPT-4 (text-only), on CG questions, indicated poor performance and reliance on detailed descriptions of image content, which often required considerable insight from the user to return reasonable results. So far, no studies have investigated the abilities of LMMs (Large Multimodal Models), or multimodal LLMs, to solve CG questions and how these abilities can be used to improve teaching. In this study, we construct two datasets of CG questions requiring varying degrees of visual perception skills and geometric reasoning skills, and evaluate the current state-of-the-art LMM, GPT-4o, on the two datasets. We find that although GPT-4o exhibits great potential in solving questions with visual information independently, major limitations still exist to the accuracy and quality of the generated results. We propose several novel approaches for CG educators to incorporate GenAI into CG teaching despite these limitations. We hope that our guidelines further encourage learning and engagement in CG classrooms., Comment: 8 pages, 8 figures, 1 table, to be published in SIGGRAPH Asia 2024 Educator's Forum

We survey several recent examples of derived structures emerging in connection with the Langlands correspondence. Cases studies include derived Galois deformation rings, derived Hecke algebras, derived Hitchin stacks, and derived special cycles. We also highlight some open problems that we expect to be important for future progress., Comment: Written for the proceedings of the 2022 IHES summer school. Comments and corrections appreciated!

Higher theta series on moduli spaces of Hermitian shtukas were constructed by Feng--Yun--Zhang and conjectured to be modular, parallel to classical conjectures in the Kudla program. In this paper we prove the modularity of higher theta series after restriction to the generic locus. The proof is an upgrade, using motivic homotopy theory, of earlier work of Feng--Yun--Zhang which established generic modularity of $\ell$-adic realizations. In the process, we develop some general tools of broader utility. One such is the "motivic sheaf-cycle correspondence", a categorical trace formalism for extracting computations in the Chow group from computations in Voevodsky's derived category of motives. Another new tool is the "derived homogeneous Fourier transform", which we use to implement a form of Fourier analysis for motives., Comment: Corrected some statements in section 7.5. This paper absorbs arXiv:2311.13270 as section 8

We study conjectures of Ben-Zvi--Sakellaridis--Venkatesh that categorify the relationship between automorphic periods and $L$-functions in the context of the Geometric Langlands equivalence. We provide evidence for these conjectures in some low-rank examples, by using derived Fourier analysis and the theory of chiral algebras to categorify the Rankin-Selberg unfolding method., Comment: Some corrections

We develop a theory of Smith-Treumann localization and relative parity sheaves in the context of Fargues-Scholze's Geometrization of the Local Langlands Correspondence. We then apply this theory to prove some conjectures of Treumann-Venkatesh concerning mod-$\ell$ Local Langlands functoriality between a reductive group $G$ and its fixed subgroup under an order $\ell$ automorphism. As another application, we calculate the Fargues-Scholze parameters of mod-$\ell$ Howe-unramified toral representations., Comment: Substantive corrections to Section 5, plus minor revisions elsewhere

The Breuil-M\'{e}zard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" in the moduli space of mod $p$ Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_q/\mathbb{Q}_q)$ that should govern congruences between mod $p$ automorphic forms. For generic parameters, we propose a construction of Breuil-M\'{e}zard cycles in arbitrary rank, and verify that they satisfy the Breuil-M\'{e}zard Conjecture for all sufficiently generic tame types and small Hodge-Tate weights. Our method is purely local and group-theoretic, and completely distinct from previous approaches to the Breuil-M\'ezard Conjecture. In particular, we leverage new connections between the Breuil-M\'ezard Conjecture and phenomena occurring in homological mirror symmetry and geometric representation theory., Comment: The earlier version contained an error in what was formerly Section 5, which has now be excised. Several corrections, clarifications, and explanations added to Appendix B. Some minor notational adjustments to Section 10. A few typos corrected throughout, especially in Section 10

In a previous paper we constructed $\textit{higher}$ theta series for unitary groups over function fields, and conjectured their modularity properties. Here we prove the generic modularity of the $\ell$-adic realization of higher theta series in cohomology. The proof debuts a new type of Fourier transform, occurring on the Borel-Moore homology of moduli spaces for shtuka-type objects, that we call the $\textit{arithmetic Fourier transform}$. Another novelty in the argument is a $\textit{sheaf-cycle correspondence}$ extending the classical sheaf-function correspondence, which facilitates the deployment of sheaf-theoretic methods to analyze algebraic cycles. Although the modularity property is a statement within classical algebraic geometry, the proof relies on derived algebraic geometry, especially a nascent theory of $\textit{derived Fourier analysis}$ on derived vector bundles, which we develop.

We sketch the construction of a derived enhancement of the reciprocity isomorphism of class field theory. Details will appear in a forthcoming joint paper of the authors with A. Raksit., Comment: In memory of John Coates

Brou\'{e}'s Abelian Defect Conjecture predicts interesting derived equivalences between derived categories of modular representations of finite groups. We investigate a generalization of Brou\'{e}'s Conjecture to ring spectrum coefficients and prove this generalization in the cyclic defect case, following an argument of Rouquier., Comment: 34 pages, comments welcome!

Let $G$ be a split semi-simple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on modularity lifting theorems, together with a Smith theory argument to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of representations called toral supercuspidal representations., Comment: Minor revisions

In previous work, we defined certain virtual fundamental classes for special cycles on the moduli stack of Hermitian shtukas, and related them to the higher derivatives of non-singular Fourier coefficients of Siegel-Eisenstein series. In the present article, we construct virtual fundamental classes in greater generality, including those expected to relate to the higher derivatives of singular Fourier coefficients. We assemble these classes into "higher" theta series, which we conjecture to be modular. Two types of evidence are presented: structural properties affirming that the cycle classes behave as conjectured under certain natural operations such as intersection products, and verification of modularity in several special situations. One innovation underlying these results is a new approach to special cycles in terms of derived algebraic geometry., Comment: To appear in Annales scientifiques de l'\'Ecole normale sup\'erieure

We construct special cycles on the moduli stack of unitary shtukas. We prove an identity between (1) the r-th central derivative of non-singular Fourier coefficients of a normalized Siegel--Eisenstein series, and (2) the degree of special cycles of "virtual dimension 0" on the moduli stack of unitary shtukas with r legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series., Comment: Minor revisions

Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $\mathbb{Z}/p\mathbb{Z}$-extensions of global function fields, we prove the existence of base change for mod $p$ automorphic forms on arbitrary reductive groups. For $\mathbb{Z}/p\mathbb{Z}$-extensions of local function fields, we construct a base change homomorphism for the mod $p$ Bernstein center of any reductive group. We then use this to prove existence of local base change for mod $p$ irreducible representation along $\mathbb{Z}/p\mathbb{Z}$-extensions for all large enough $p$, and that Tate cohomology realizes descent along base change, verifying a function field version of a conjecture of Treumann-Venkatesh. The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for mod $p$ spherical Hecke algebras, in a joint appendix with Gus Lonergan., Comment: Substantial revisions

We study a symplectic variant of algebraic $K$-theory of the integers, which comes equipped with a canonical action of the absolute Galois group of $\mathbf{Q}$. We compute this action explicitly. The representations we see are extensions of Tate twists $\mathbf{Z}_p(2k-1)$ by a trivial representation, and we characterize them by a universal property among such extensions. The key tool in the proof is the theory of complex multiplication for abelian varieties., Comment: 64 pages, final accepted version

Fix a positive integer $n$ and a finite field $\mathbb F_q$. We study the joint distribution of the rank of $E$, the $n$-Selmer group of $E$, and the $n$-torsion in the Tate-Shafarevich group of $E$ as $E$ varies over elliptic curves of fixed height $d \geq 2$ over $\mathbb F_q(t)$. We compute this joint distribution in the large $q$ limit. We also show that the "large $q$, then large height" limit of this distribution agrees with the one predicted by Bhargava-Kane-Lenstra-Poonen-Rains.

We introduce a derived enhancement of local Galois deformation rings that we call the "spectral Hecke algebra", in analogy to a construction in the Geometric Langlands program. This is a Hecke algebra that acts on the spectral side of the Langlands correspondence, i.e. on moduli spaces of Galois representations. We verify the simplest form of derived local-global compatibility between the action of the spectral Hecke algebra on the derived Galois deformation ring of Galatius-Venkatesh, and the action of Venkatesh's derived Hecke algebra on the cohomology of arithmetic groups., Comment: Results reformulated in terms of the derived Hecke algebra

Reeder and Yu have recently given a new construction of a class of supercuspidal representations called epipelagic representations. We explicitly calculate the Local Langlands Correspondence for certain families of epipelagic representations of unitary groups, following the general construction of Kaletha [Kal15]. The interesting feature of our computation is that we find simplifications within L-packets of the two novel invariants introduced in [Kal15], the toral invariant and the admissible L-embedding., Comment: Final version, to appear in International Journal of Number Theory

Using the Langlands-Kottwitz paradigm, we compute the trace of Frobenius composed with Hecke operators on the cohomology of nearby cycles, at places of parahoric reduction, of perverse sheaves on certain moduli stacks of shtukas. Following an argument of Ngo, we then use this to give a geometric proof of a base change fundamental lemma for parahoric Hecke algebras for GL_n over local function fields. This generalizes a theorem of Ngo, who proved the base change fundamental lemma for spherical Hecke algebras for GL_n over local function fields, and extends to positive characteristic (for GL_n) a fundamental lemma originally introduced and proved by Haines for p-adic local fields., Comment: Final version, to appear in Selecta Mathematica

Using the calculus of Steenrod operations in \'etale cohomology developed in [Feng17], we prove that the analogue of Tate's pairing on higher Brauer groups is alternating on 2-torsion. This improves upon a result of Jahn [Jahn15, Math. Annalen]., Comment: This paper is being withdrawn because it has been completely subsumed by (version 2 of) arXiv:1706.00151

We prove a 1966 conjecture of Tate concerning the Artin-Tate pairing on the Brauer group of a surface over a finite field, which is the analogue of the Cassels-Tate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of Poonen-Stoll on the Cassels-Tate pairing. Our method is based on studying a connection between the Artin-Tate pairing and (generalizations of) Steenrod operations in \'{e}tale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate the relevant \'{e}tale Steenrod operations in terms of characteristic classes., Comment: Final version. To appear in Compositio

We completely classify the possible extensions between semistable vector bundles on the Fargues-Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder-Narasimhan polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze's language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the euclidean geometry of Harder-Narasimhan polygons., Comment: 41 pages, 17 figures: Final version; to appear in J Inst. Math. Jussieu

In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL_2(\hat{Z}). In Greicius' thesis, he develops necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius' methods in several directions. First, we consider the analogous problem for elliptic curves with full 2-torsion. Following Greicius, we obtain necessary and sufficient conditions for the associated adelic representation to be maximal and also develop a battery of computationally effective tests that can be used to verify these conditions. We are able to use our tests to construct an infinite family of curves over Q(alpha) with maximal image, where alpha is the real root of x^3 + x + 1. Next, we extend Greicius' tests to more general settings, such as non-semistable elliptic curves over arbitrary cubic number fields. Finally, we give a general discussion concerning such problems for arbitrary torsion subgroups., Comment: 23 pages, this version incorporates the suggestions of the referee