A q-deformed real number, or “q-real”, was defined by Morier-Genoud and the second author. When x∈ℝ such that x≥0, the q-analogue [x]_q is a power series with integer coefficients in one formal variable q. In general a q-real is a formal Laurent series. The main goal of this paper is to study the coefficients of q-reals as functions on ℝ and give a combinatorial interpretation of these coefficients. This allows us to prove a conjecture studied by several authors stating that the q-deformed golden ratio has the smallest radius of convergence among the radii of the q-reals associated with positive real numbers. This is a q-analogue of the classical Hurwitz theorem. Our approach is combinatorial. We prove that for every real number x in the interval (1,2) the absolute value of each coefficient of the power series representing the q-real [x]_q is dominated by the absolute value of the corresponding coefficient of the q-deformed golden ratio. The main notion is a certain collection of ordered rooted trees associated with a q-real. We prove that the golden ratio corresponds to a universal class of trees.

Large language models have made substantial progress on mathematical reasoning, but existing benchmarks typically evaluate well-specified problems with final answers, step-by-step solutions, or complete proofs. They do not capture collaborative open-problem solving: a setting in which participants propose partial arguments, identify gaps or errors in prior steps, repair flawed reasoning, and gradually synthesize incremental contributions into a proof. We introduce CrowdMath, a dataset of 164 expert-annotated progress chains from the MIT PRIMES–Art of Problem Solving (AoPS) CrowdMath program (2016-2025), a collaborative research initiative whose discussions have led to peer-reviewed publications. Each chain traces a multi-participant forum discussion from an open-problem statement to a completed proof. Posts are labeled by their functional roles in the evolving solution process, including partial progress, proof completion, erroneous reasoning, and error identification. We define evaluation tasks and benchmark six frontier models. Models achieve 83-88

Let k be an algebraically closed field of characteristic p≥ 5, and let Ver_p^+ be the even part of the Verlinde fusion category Ver_p, the semisimplification of Rep_k(ℤ/p). Let 𝔤 be a linearly reductive Lie algebra in Ver_p^+, i.e., one whose finite-dimensional representations are semisimple. A basic class of examples is obtained by semisimplifying a simple Lie algebra over k equipped with the action of ℤ/p by a principal unipotent element, when p exceeds its Coxeter number. We prove that 𝔤 is invariantless, i.e., that the unit object is not a summand of 𝔤. For odd m with 3≤ m≤ p-2, set 𝔤_m:=Hom_Ver_p^+(L_m,𝔤) and E_𝔤:=⊕_3≤ m≤ p-2, m odd𝔤_m^(1)[m], where (1) denotes Frobenius twist. Our main result is an isomorphism of graded algebras H^∙_CE(𝔤)≅⋀^∙ E_𝔤^*. We also identify this algebra with the de Rham cohomology H^∙_dR(G) of the group scheme G=exp(𝔤) and show that the induced graded Hopf algebra structure agrees with the standard one on the exterior algebra. Moreover, if V is a simple 𝔤-module on which 𝔤 acts nontrivially, then H^∙_CE(𝔤,V)=0. Hence for every finite-dimensional 𝔤-module V one has H^∙_CE(𝔤,V)≅⋀^∙ E_𝔤^*⊗ V^𝔤. This recovers the theorem of Borel and Chevalley on the cohomology of complex semisimple Lie algebras and its analogue in sufficiently large positive characteristic.

Let 𝒞 be a symmetric tensor category over an algebraically closed field 𝐤 of characteristic 2. We study Clifford and Weyl algebras of objects of 𝒞 with a (skew-)symmetric bilinear form. When the form is non-degenerate, we establish simplicity and the Azumaya property for such algebras under suitable assumptions. We also compute Clifford and Weyl algebras in the Verlinde category Ver_p and use them to prove that if 𝒞 is Frobenius exact then the Weyl algebra of a symplectic object of 𝒞 with finite symmetric algebra is Azumaya. Using this, we introduce the symplectic Witt group 𝒮𝒲(𝒞), the subgroup of the Brauer group Br(𝒞) consisting of Morita classes of such Azumaya algebras, and when 𝒞= Rep(G)⊠ sVec for a finite group G of order coprime to char(𝐤), express 𝒮𝒲(𝒞) in terms of second Stiefel-Whitney classes of orthogonal representations of G.

Recently, Harman and the second author introduced a new construction of pre-Tannakian tensor categories based on oligomorphic groups. We develop tools for analyzing the Drinfeld centers of these categories, and compute the center explicitly in a number of cases. In particular, we find several finitely tensor-generated pre-Tannakian categories (including the Delannoy category) that are identified with their own center via the canonical functor; prior to this work, we knew no such examples besides the category of vector spaces.

We consider the Knizhnik–Zamolodchikov equations in Deligne Categories in the context of (𝔤𝔩_m,𝔤𝔩_n) and (𝔰𝔬_m,𝔰𝔬_2n) dualities. We derive integral formulas for the solutions in the first case and compute monodromy in both cases.

The Delannoy category is an interesting pre-Tannakian category associated to the oligomorphic group 𝔾 of automorphisms of the totally ordered set (R, <) . By construction, it admits some obvious simple commutative algebras, corresponding to certain transitive 𝔾 -sets. We show that these account for all of the simple commutative algebras in the Delannoy category. Previous results of this kind have been limited to interpolation categories; since the Delannoy category cannot be obtained by interpolation, new methods are required.

Let k be a field, and let C be a Cauchy complete k-linear braided category with finite-dimensional morphism spaces and End(1) = k. We call an indecomposable object X of C non-negligible if there exists Y is an element of C such that 1 is a direct summand of Y.circle times X. We prove that every non-negligible object X is an element of C such that dim End(X-circle times n) < n! for some n is automatically rigid. In particular, if C is semisimple of moderate growth and weakly rigid, then C is rigid. As applications, we simplify Huang's proof of rigidity of representation categories of certain vertex operator algebras, and we get that for a finite semisimple monoidal category C, the data of a C-modular functor is equivalent to a modular fusion category structure on C, answering a question of Bakalov and Kirillov. Furthermore, we show that if C is rigid and has moderate growth, then the quantum trace of any nilpotent endomorphism in C is zero. Hence C admits a semisimplification, which is a semisimple braided tensor category of moderate growth. Finally, we discuss rigidity in braided r-categories which are not semisimple, which arise in logarithmic conformal field theory. These results allow us to simplify a number of arguments of Kazhdan and Lusztig.

We discuss the classification of twisted Deligne products of two semisimple tensor categories 𝒞,𝒟, i.e., categorifications of the tensor product of their Grothendieck rings in which the factors are categorified by 𝒞 and 𝒟. In particular, we show that if both factors have no non-trivial gradings, or if one factor has neither non-trivial gradings nor tensor structures on the identity functor, then the only twisted Deligne product is the ordinary one. Using the work arXiv:2405.10207 by Müller, Peña Pollastri and Plavnik, this gives, in principle, a group-theoretical classification of twisted Deligne products and, more generally, exact factorizations of arbitrary fusion categories. In the Appendix we introduce the notion of categorical n-cocycles for n=2,3,4 and show that they are all pullbacks of group n-cocycles from the universal grading group of the underlying based ring. In the case of 4-cocycles, this answers a question of Johnson-Freyd, Ostrik and Yu from arXiv:2601.09060.

We define the notion of a Lie superalgebra over a field k of characteristic 2 which unifies the two pre-existing ones – ℤ/2 -graded Lie algebras with a squaring map and Lie algebras in the Verlinde category Ver_4^+(k) , and prove the PBW theorem for this notion. We also do the same for the restricted version. Finally, we discuss mixed characteristic deformation theory of such Lie superalgebras (for perfect k), introducing and studying a natural lift of our notion of Lie superalgebra to characteristic zero – the notion of a mixed Lie superalgebra over a ramified quadratic extension R of the ring of Witt vectors W(k).

M. Kontsevich conjectured and T. Bitoun proved that if M is a nonzero holonomic D-module then the p-support of a generic reduction of M to characteristic p>0 is Lagrangian. We provide a new elementary proof of this theorem and also generalize it to q-D-modules. The proofs are based on Bernstein's theorem that any holonomic D-module can be transformed by an element of the symplectic group into a vector bundle with a flat connection, and a q-analog of this theorem. We also discuss potential applications to quantizations of symplectic singularities and to quantum cluster algebras.

We introduce a new notion of a periodic pencil of flat connections on a smooth algebraic variety X. This is a family ∇(s_1,...,s_n) of flat connections on a trivial vector bundle on X depending linearly on parameters s_1,...,s_n and generically invariant, up to isomorphism, under the shifts s_i↦ s_i+1 for all i. If in addition ∇ has regular singularities, we call it a quasi-motivic pencil. We use tools from complex analysis to establish various remarkable properties of such pencils over ℂ. For example, we show that the monodromy of a quasi-motivic pencil is defined over the field of algebraic functions in e^2π is_j, and that its singularities are constrained to an arrangement of hyperplanes with integer normal vectors. Then we show that many important examples of families of flat connections, such as Knizhnik-Zamolodchikov, Dunkl, and Casimir connections, are quasi-motivic and thus periodic pencils. Besides being interesting in its own right, the periodic property of a pencil of flat connections turns out to be very useful in computing the eigenvalues of the p-curvature of its reduction to positive characteristic. This will be done in our forthcoming paper.

We study the number of indecomposable summands in tensor powers of the vector representation of SL2. Our main focus is on positive characteristic where this sequence of numbers and its generating function show fractal behavior akin to Mahler functions.

We investigate objects in symmetric tensor categories that have simultaneously finite symmetric and finite exterior algebra. This forces the characteristic of the base field to be p>0, and the maximal degree of non-vanishing symmetric and exterior powers to add up to a multiple of p. We give a complete classification of objects in tensor categories for which this sum equals p. All resulting tensor categories are Verlinde categories of reductive groups and we fill in some gaps in the literature on these categories.

We initiate a study of tensor ideals in linear rigid monoidal categories that are kernels of linear monoidal functors to abelian monoidal categories. We develop general methods and apply them to the category of tilting modules over quantum groups as well as to some representation categories of finite groups. In an appendix on Duflo involutions in monoidal categories, we make a connection between Duflo involutions in the affine Weyl group and tensor ideals for quantum groups, and prove some of Lusztig's conjectures for arbitrary Coxeter groups, at equal parameters, without invoking the boundedness hypothesis.

In arXiv:1812.00170, S. Morier-Genoud and V. Ovsienko introduced the notion of the q-rational number [x]_q, x∈ Q, a rational function specializing to x at q=1, obtained by q-deforming the continued fraction expansion of x. In arXiv:1908.04365 they introduced q-real numbers [x]_q, x∈ R - a Laurent series in q converging to the rational function [x]_q when x∈ Q. In arXiv:2102.00891 it is proved that if x∈ Q_>1 then the series [x]_q converges for |q|<3-2√(2)≈ 0.17 and conjectured that for all x∈ R_>1 this series converges in some disk centered in the origin, with the expected common radius of convergence R_*=3-√(5)/2≈ 0.38, achieved when x=1+√(5)/2 is the golden ratio. This was proved for rational x in arXiv:2405.15970 using the theory of Kleinian groups. In this paper we (partially) prove this conjecture by showing that for all x∈ R_>1, the series [x]_q converges in the disk |q|<3-2√(2) to a nonvanishing holomorphic function. This is achieved by giving an expansion of 1/[x]_q into a q-adically convergent series of rational functions converging absolutely and uniformly on compact sets in an explicit region D containing this disk. We also show that this expansion converges to a positive analytic function on the interval (-3-√(5)/2,1), giving a definition of [x]_q for q from this interval. Moreover, we show that the result of arXiv:2405.15970 implies convergence of [x]_q for |q|<2-√(3)≈ 0.27. We also give examples of explicit computation of [x]_q for transcendental numbers x, e.g. x= cotan(1). Finally, we propose a definition of the q-complex number [τ]_q, a meromorphic function of τ∈ C_+ which expresses via hypergeometric functions of modular functions of τ.

In Liu, Palcoux, and Wu’s article [Adv. Math. 390 (2021), Paper No. 107905, 63], they proved a remarkable necessary condition for a fusion ring to admit a unitary categorification, by constructing invariants of the fusion ring that have to be positive if it is unitarily categorifiable. The main goal of this note is to provide a somewhat more direct proof of this result. In the last subsection we discuss integrality properties of the Liu–Palcoux–Wu invariants.

Kac's ten-dimensional simple Jordan superalgebra over a field of characteristic 5 is obtained from a process of semisimplification, via tensor categories, from the exceptional simple Jordan algebra (or Albert algebra), together with a suitable order 5 automorphism. This explains McCrimmon's 'bizarre result' asserting that, in characteristic 5, Kac's superalgebra is a sort of 'degree 3 Jordan superalgebra'. As an outcome, the exceptional simple Lie superalgebra el(5;5), specific of characteristic 5, is obtained from the simple Lie algebra of type $E_8$ and an order 5 automorphism. In the process, precise recipes to obtain superalgebras from algebras in the category of representations of the cyclic group $C_p$, over a field of characteristic $p>2$, are given.

An important function attached to a complex simple Lie group G is its asymptotic character X(?, x) (where ?, x are real (co)weights of G) - the Fourier transform in x of its Duistermaat-Heckman function DH?( p) (continuous limit of weight multiplicities). It is shown in Garibaldi et al. that the best ?independent upper bound -c(G) for infx ReX(?, x) for fixed ? is strictly negative. We quantify this result by providing a lower bound for c(G) in terms of dim G. We also provide upper and lower bounds for DH?(0) when |?| = 1. This allows us to show that |X(?, x)| <= C(G)|?|-1|x|-1 for some constant C(G) depending only on G, which implies the conjecture in Remark 17.16 of Garibaldi et al. We also show that c(SLn) <= (pi 4 2 )n-2. Finally, in the appendix we prove Conjecture 1 in Coquereaux and Zuber (2018) about Mittag-Leffler type sums for G. (c) 2025 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

These are expanded notes of a course on basics of quantum field theory for mathematicians given by the author at MIT. The latest version (405 pages) is https://math.mit.edu/ etingof/gsm254.pdf (AMS book, https://bookstore.ams.org/GSM/254)