Open research questions in Advanced Algebra and Geometry

What the literature leaves open

• The existing literature on the cone conjecture has a gap in the case of log Calabi-Yau mirrors of Fano 3-folds. The paper claims to fill this gap by proving a special case of the cone conjecture for certain types of log Calabi-Yau 3-folds.

  On the cone conjecture for log Calabi-Yau mirrors of Fano 3-folds · 2026 · DOI

• The lack of systematic results on the decomposition of VK+ for Iwahori-spherical representations of GLn. The need for a combinatorial algorithm to compute the multiplicities of each πμ in St(⟨a⟩)K+. The absence of a connection between the decomposition of VK+ and the combinatorics of partitions.

  The $$K_+$$-fixed vectors of Iwahori-spherical $$GL_n$$-representations: connections with Zelevinsky’s segments · 2026 · DOI

• The existing literature does not provide a clear understanding of the normalization step in the construction of SK(G, X). There is a lack of knowledge about the moduli interpretation of SK(G, X). The relationship between SK(G, X) and the Siegel modular variety SK(GSp, S±) is not well understood.

  Normalization in integral models of Shimura varieties of abelian type · 2026 · DOI

• The height zeta function method has been applied to rational points and integral points, but not to Campana points. The paper aims to fill this gap by proving a variant of Manin's conjecture for Campana points.

  Campana points on wonderful compactifications · 2026 · DOI

• The lack of understanding of the spectral gap absorption principle for unitary representations of semisimple algebraic groups. The need for a proof that the unitary dual of simple algebraic groups is filtered by the integrability parameter of matrix coefficients.

  A spectral gap absorption principle · 2026 · DOI

• The gap identified by this paper is in understanding when nice Hodge-theoretic properties hold on compact complex manifolds with semistable degenerations. There are few results addressing these properties for such manifolds.

  On Hodge structures of compact complex manifolds with semistable degenerations · 2026 · DOI

• The structure of the ring of Weyl invariant E8 Jacobi forms has remained unclarified since Wirthmüller's study in 1992. The ring of W(E8)-invariant weak Jacobi forms was not known to be finitely generated.

  The ring of Weyl invariant $$E_8$$ Jacobi forms · 2026 · DOI

• The lack of a proof of the Trinh-Xue conjecture for exceptional type groups. The need for a conceptual proof and explicit computations to fill this gap. The limitation of prior work to specific cases, rather than a general proof.

  Intersections of blocks of cyclotomic Hecke algebras · 2026 · DOI

• The lack of a simple and explicit description of parabolic Mirković-Vilonen intersections using alcove walks. The need for a new combinatorial description of branching to Levi subgroups of irreducible highest weight representations.

  Alcove Walk Models for Parabolic Mirković-Vilonen Intersections and Branching to Levi Subgroups · 2026 · DOI

• The sup-norm problem for Hecke-Maaß forms remains open for n ≥ 2. Prior work has only achieved a bound for n = 2. The paper aims to fill this gap by extending the result to n ≥ 2.

  The sup-norm problem for newforms of large level on $\operatorname{PGL}(n)$ · 2026 · DOI

• The boundedness conjecture for the a-function was previously open. There was a lack of techniques for studying the a-function in general Coxeter groups.

  Boundedness of Lusztig’s $\boldsymbol{a}$-function for Coxeter groups of finite rank · 2026 · DOI

• Prior work has used gigantic lower bounds, which limits the applicability of the results. The paper identifies a need for stronger results for special periods. The paper aims to fill this gap by considering special periods and constructing matrices from Fourier coefficients.

  Periods and symmetric square periods of modular forms · 2026 · DOI

• The properties of the Rumin differentials in irreducible unitary representations are not well understood. There is a need to compute the spectrum and the zeta regularized determinant of these differentials. The alternating product of the Rumin differentials in the generic representations needs to be evaluated.

  Regularized Determinants of the Rumin Complex in Irreducible Unitary Representations of the (2,3,5) Nilpotent Lie Group · 2026 · DOI

• It must adapt previous work to Levi subgroups of G. The paper also needs to show that the endoscopic lifting coincides with Langlands-Shelstad transfer.

  Proof of Vogan’s conjecture on Arthur packets for $$\mathop {GL}_n$$ over p-adic fields · 2025 · DOI

• Investigate the relation between A-stability and other properties of representations. Study the dynamics of the action of Out./ on S A.; G/. Explore the applications of A-stable representations to other areas of mathematics.

  Commutator-stable representations of hyperbolic groups · 2026 · DOI

• Investigate the case where the metric is not CAT(1). Explore the applications of the result to other areas of mathematics and physics.

  Rigidity of compact rank one symmetric spaces · 2026 · DOI

• Prior work has focused on nonpositively curved symmetric spaces or higher-rank symmetric spaces.

  Rigidity of compact rank one symmetric spaces · 2026 · DOI

• Computing the first moment of cubic Hecke L-functions over Q(√(-3)) is challenging. The error term needs careful handling. The paper requires a deep understanding of the cubic large sieve and its optimality.

  Hecke L-functions away from the central line · 2026 · DOI

• Investigate the Hodge-theoretic properties in higher dimensions beyond 3-folds. Explore the implications for non-SNC varieties or those with more complex intersection loci.

  On Hodge structures of compact complex manifolds with semistable degenerations · 2026 · DOI

• Further study of the decomposition of the K+-fixed subspace VK+ into irreducible K/K+-modules - Investigation of the properties of the Kostka number Kµ′λ′ - Extension of the results to a larger setting, beyond…

  The $$K_+$$-fixed vectors of Iwahori-spherical $$GL_n$$-representations: connections with Zelevinsky’s segments · 2026 · DOI

• Refining the function field conjecture of David and Meisner for higher order Hecke L-functions. Investigating the behaviour of higher order Hecke L-functions attached to nth order residue symbols.

  Hecke L-functions away from the central line · 2026 · DOI

• Further study of the sup-norm problem for Hecke-Maaß forms on X n (N) for general N, - Investigation of the properties of the lattice and the Iwasawa coordinates, - Application of the results to other problems in number…

  The sup-norm problem for newforms of large level on $\operatorname{PGL}(n)$ · 2026 · DOI

• The results are limited to Hensel semi-global fields, - The technique is based on the field patching method, - The study assumes that K is algebraically closed in F, - The results are based on the assumption that m is coprime to the characteristic…

  Patching for étale algebras and the period-index problem for higher degree Galois cohomology groups over Hensel semi-global fields · 2026 · DOI

• Extending the field patching technique to more general fields, - Applying the technique to other areas of arithmetic geometry, - Studying the period-index problem for higher degree Galois cohomology groups over other types…

  Patching for étale algebras and the period-index problem for higher degree Galois cohomology groups over Hensel semi-global fields · 2026 · DOI

• Investigate the linear independence of the kernel functions of the L-functions or symmetric square L-functions, - Study the matrices formed by the Fourier coefficients, - Improve the bounds…

  Periods and symmetric square periods of modular forms · 2026 · DOI

Most-cited papers in Advanced Algebra and Geometry

• In Defence of Representations · Journal for the Theory of Social Behaviour · 1996 · 78 citations
• Der Einfluß der Ligatur auf die Friktion zwischen Bracket und Bogen · Journal of Orofacial Orthopedics / Fortschritte der Kieferorthopädie · 1990 · 41 citations
• Multiparticle Factorization and the Rigidity of String Theory · Physical Review Letters · 2024 · 31 citations
• Alternative Representations · ˜The œPsychology of learning and motivation/˜The œpsychology of learning and motivation · 1988 · 27 citations
• Yule’s Characteristic K Revisited · Computers and the Humanities · 2005 · 23 citations
• Sampled-Data Connectivity-Preserving Consensus for Multiple Heterogeneous Euler-Lagrange Systems · IEEE Transactions on Automation Science and Engineering · 2024 · 16 citations
• A Dual Characterization of Subdirectly Irreducible BAOs · Studia Logica · 2004 · 11 citations
• On Drinfeld modular forms of higher rank II · Journal of Number Theory · 2022 · 11 citations
• Jump Liars and Jourdain’s Card via the Relativized T-scheme · Studia Logica · 2009 · 10 citations
• A Heterotic Hermitian–Yang–Mills Equivalence · Communications in Mathematical Physics · 2025 · 10 citations

Most recent work

• Geometrization of the local Langlands correspondence, motivically · Journal für die reine und angewandte Mathematik (Crelles Journal) · 2026
• Derivation of the Tree-Level Weinberg Angle and the SU(2) Gauge Structure from the PDL Axioms C1–C4 (Projective Dynamic Logo Framework — Document D57) · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Inductive Feit and Galois-McKay conditions for some small-rank simple groups of Lie Type · Journal of Algebra · 2026
• A Jacobian criterion for artin v-stacks · Selecta Mathematica · 2026
• Springer theory for symplectic Galois groups · Geometry & Topology · 2026
• Special Orthogonal, Special Unitary, and Symplectic Groups as Products of Grassmannians · SIAM Journal on Matrix Analysis and Applications · 2026
• Traces of Hecke operators on Drinfeld modular forms for GL2(Fq[T]) · Mathematika · 2026
• Some remarks on strong multiplicity one for paramodular forms · Mathematika · 2026
• A motivic fundamental lemma · Geometry & Topology · 2026
• A note on a cluster structure of the coordinate ring of a simple algebraic group · Proceedings of the American Mathematical Society · 2026

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