Mathematical Structures of Scattering Amplitudes
2024-11-25 16:30 - 17:30 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Anastasia Volovich, Brown University
Title: Mathematical Structures of Scattering Amplitudes
Abstract: Planar N=4 Yang-Mills scattering amplitudes have been computed to very high loop order. They have many remarkable properties that have sparked interest from mathematicians working on combinatorics, algebraic geometry, and number theory. At the same time, several methods that have been developed for N=4 Yang-Mills can often be applied to more general quantum field theories, including QCD. I will overview some of these exciting developments.

Computability on $mathbb R$ and other continuum-size structures
2024-12-02 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Russell Miller, CUNY
Title: Computability on $mathbb R$ and other continuum-size structures
Abstract: We begin by recalling the notion of a computable function on the real numbers $mathbb R$, developed independently by Gregorczyk and Lacombe over sixty years ago. Using this notion, we note that the real numbers that are themselves computable form a countable subfield of $mathbb R$ with exactly the same first-order properties as $mathbb R$ itself. (Logicians would therefore call it an emph{elementary subfield}.) So, in a first-order sense, everything that happens in $mathbb R$ is already exemplified in this much nicer subfield. However, even when one knows that an existential statement holds for all parameters, it may be impossible (both in $mathbb R$ and in the subfield) to give a computable procedure for producing witnesses. Similar results hold in $mathbb C$.
We will then turn to a different continuum-sized structure: the absolute Galois group $operatorname{Gal}(mathbb Q)$ of the rational numbers. Once again the computable elements of this group form a subgroup, but now it is an open problem whether the group and the subgroup have the same first-order theory, let alone whether this is an elementary subgroup. (If they do have the same theory, this would put nice upper bounds on the complexity of the theory of $operatorname{Gal}(mathbb Q)$.) However, using joint work with Kundu, we can show that once again there is no computable procedure for producing witnesses to the truth of (true) existential statements, either in the full group or in the subgroup.

Factorization Homology
2024-12-03 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speakers: Sunghyuk Park and Vasily Krylov, Harvard CMSA
Title: Factorization Homology

Can Transformers Reason Logically? A Study in SAT-Solving
2024-12-04 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Leyan Pan, Georgia Tech
Title: Can Transformers Reason Logically? A Study in SAT-Solving
Abstract: Transformer-based LLMs have apparently demonstrated capabilities that resembles human reasoning. In our recent work, we investigated the Boolean reasoning abilities of decoder-only Transformers equipped with Chain-of-Thought, establishing that a Transformer model can decide all 3-SAT instances up to a bounded size (i.e., number of variables and clauses). In this talk, I will first review recent studies that formally examine the expressiveness of Transformer models. Next, I will explain how we establish an equivalence between Chain-of-Thought reasoning and algorithm, in our case, the DPLL SAT-solving algorithm. I will then discuss how to encode 3-SAT formulas and partial assignments as vectors so that the high-level operations in DPLL can be represented as vector operations and implemented using attention mechanisms within Transformers. Finally, I will present experimental results that support our theoretical predictions. I will also address why standard Transformers can only solve reasoning problems of bounded length, leading to failures in length-generalization, and discuss potential solutions to overcome this limitation.

On the Coulomb branch scaling dimensions of 4d N=2 SCFTs
2024-12-06 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Robert Moscrop
Title: On the Coulomb branch scaling dimensions of 4d N=2 SCFTs
Abstract: To each four dimensional N=2 superconformal field theory of rank-r, one can associate an r-tuple of numbers given by the scaling dimensions of a special set of protected operators whose VEVs parameterise the Coulomb branch. In this talk, I will demonstrate how the special geometry of the Coulomb branch heavily constrains not only the values of the scaling dimensions, but also the possible r-tuples of such numbers that can occur.

A simple model for universal quantum computation
2024-12-06 16:00 - 17:00 | Virtual
Freedman CMSA Seminar
Speaker: Michael Freedman
Title: A simple model for universal quantum computation
Abstract: I’ll present joint (unpublished) work with Charlie Marcus on a surprisingly simple – and potentially practical (?)– model for universal quantum computation whose only quantum primitive is the ability to measure a pair of adjacent electrons into either singlet (spin=0) or triplet (spin=1) sectors according to the Born rule. The electrons are located on quantum dots arranged in a triangular lattice whose edges are tiny strips of s-wave superconductor.

Quantum Effects Inside Black Holes
2024-12-10 09:30 - 11:00 | Black Hole Initiative, 20 Garden Street, Cambridge MA, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Joint CMSA/BHI Foundation Seminar
Location: BHI seminar room

Speaker: Noa Zilberman, Princeton
Title: Quantum Effects Inside Black Holes

Abstract: Astrophysical black holes are known to be rotating. Within classical General Relativity, the simplest spacetime solution (the Kerr solution) describing a rotating black hole reveals a traversable passage through an inner horizon – which in turn may lead to another external universe. But does this remain the case when taking quantum effects into account?
Answering this question, along others, requires one to understand the manner in which quantum energy fluxes affect the internal geometry of a black hole. It has been widely anticipated, yet inconclusive (till this work), that such effects would diverge at the inner horizon of a spinning black hole. This divergence, if indeed takes place, may drastically affect the internal black hole geometry, potentially preventing the inner horizon traversability. Clarifying this issue requires the computation of the quantum energy fluxes in black hole interiors. However, this has been a serious challenge for decades.
Using a combination of old and new methods, we have managed to compute the quantum energy fluxes at the inner horizon of a spinning black hole, in a vacuum state corresponding to an evaporating black hole. We found that these fluxes are either positive or negative, depending on the black hole spin (and polar angle). The sign of these fluxes may be crucial to the nature of their backreaction on the geometry (as should be dictated by the semiclassical Einstein equation).
In this seminar, we shall briefly describe the basic framework of semiclassical general relativity and the renormalization procedure, and then present our novel results for the quantum fluxes at the inner horizon of a rotating black hole, briefly mentioning possible implications for the inner horizon traversability.

Nonlinear Bosonization of (Non-)Fermi Liquids
2024-12-13 10:00 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Yi-Hsien Du (MIT)
Title: Nonlinear Bosonization of (Non-)Fermi Liquids
Abstract: Fermi liquid theory is a cornerstone of condensed matter physics. I will show how to formulate Fermi liquid theory as an effective field theory. In this approach, the space of low-energy states of a Fermi liquid is identified with a coadjoint orbit of the group of canonical transformations. The method naturally leads to a nonlinear bosonized description of the Fermi liquid with nonlinear corrections fixed by the geometry of the Fermi surface. I will present that the resulting local effective field theory captures both linear and nonlinear effects in Landau’s Fermi liquid theory. The approach can be extended to encompass non-Fermi liquids, which correspond to strongly interacting fixed points obtained by deforming Fermi liquids with relevant interactions. I will also discuss how Berry curvature can be captured in the effective field theory approach.

CMSA Q&A Seminar: Edgar Shaghoulian
2025-01-22 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Edgar Shaghoulian, UC Santa Cruz
Topic: What are the main questions in quantum gravity today?

Joint BHI/CMSA Foundation Seminar: Quantum Criticality in Black Hole Dynamics
2025-01-27 09:30 - 10:30 | Black Hole Initiative, 20 Garden Street, Cambridge MA, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Joint BHI/CMSA Foundation Seminar
Location: BHI Seminar Room
Speaker: Uri Kol, Harvard CMSA

Title: Quantum Criticality in Black Hole Dynamics
Abstract: Black hole perturbation theory captures a few important effects in the dynamics of binary mergers, such as tidal deformations and the decay of ringdown modes, as well as the physics of the photon ring. However, deriving qualitative results that lead to robust predictions in this theory remains a challenging problem, despite a rich scientific history.
Motivated by the immense experimental progress made in the field over the past decade, in this talk I will describe a new critical phenomenon emerging in the theory of perturbations around Kerr black holes. A critical point is identified at zero black hole temperature, giving rise to a wide regime at finite temperatures which is dominated by critical perturbations. As in quantum many-body systems, the physics in the critical regime is described exclusively by a set of critical exponents, therefore leading to robust predictions that are unique to the Kerr metric.

A homotopy of 2d SCFTs and an implication for Topological Modular Forms
2025-01-27 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Vivek Saxena (YITP Stony Brook and NHETC Rutgers)
Title: A homotopy of 2d SCFTs and an implication for Topological Modular Forms
Abstract: The Segal-Stolz-Teichner conjecture states that there exists an isomorphism between deformation classes of two-dimensional N=(0,1) superconformal field theories (SCFTs) and generalized cohomology classes known as Topological Modular Forms (TMFs). Such 2d N=(0,1) SCFTs arise naturally in physics as worldsheet theories of (possibly compactified) heterotic strings. Recently, this connection was used to prove the absence of global anomalies in heterotic string theories and make predictions about topological terms in their low-energy effective actions, among other things.
In this talk, after giving a brief overview of these ideas, I will describe a physics ”proof” (using methods from 2d CFT) of a mathematical conjecture of Tachikawa and Yamashita about TMF classes in degree 31. Specifically, by examining the two worldsheet theories corresponding to two T-dual nine-dimensional spacetime non-supersymmetric heterotic string theories (namely the $(E_8)_1 times (E_8)_1$ theory and the $(E_8)_2$ theory), I will argue that the $(E_8)_2$ theory corresponds to the unique nontrivial torsion element $[(E_8)_2]$ of TMF$^{31}$ with zero mod-2 elliptic genus.

Review of the excision property in factorization homology
2025-01-28 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Keyou Zeng, Harvard CMSA
Title: Review of the excision property in factorization homology

Smooth projective fibrations over the projective line and their sections
2025-01-31 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Iacopo Brivio
Title: Smooth projective fibrations over the projective line and their sections
Abstract: Suppose $fcolon Xto mathbf{CP}^1$ is a smooth projective fibration, is it then true that $f$ has a section? This deceptively simple result was established by Seidel and McDuff using deep methods of symplectic geometry. Alex Pieloch recently generalized this to morphisms with at most one singular fibers and furthermore showed that $X$ uniruled, that is covered by images of $mathbf{P}^1$. In my talk I will explain how to recover Pieloch’s result from the Good Minimal Model Conjecture.

Topological to Gravitational Leptogenesis, via Ultra Unification
2025-02-03 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Juven Wang (LIMS, Royal Institution)
Title: Topological to Gravitational Leptogenesis, via Ultra Unification
Abstract: We propose a new mechanism for leptogenesis, named Topological Leptogenesis, in which dark matter consists of topological order, so gapped anyon excitations decay to generate the Standard Model lepton asymmetry. Topological Leptogenesis is in contrast with the Majorana Leptogenesis and Gravitational Leptogenesis. In the standard Majorana Leptogenesis, the baryon asymmetry of the present universe is attributed to the leptogenesis from the sterile right-handed neutrino with heavy Majorana fermion mass decaying into the Standard Model leptons in the very early universe, while the electroweak sphaleron causes baryogenesis at a later time. Gravitational leptogenesis can be regarded as an intermediate step between Majorana particle leptogenesis and topological non-particle leptogenesis. Topological Leptogenesis is derived out of Ultra Unification constructed from a mod 16 nonperturbative global anomaly cancellation that adds a symmetry-extended anomalous gapped topological quantum field theory (TQFT) to the Standard Model. Thus (1) the Beyond-the-Standard-Model (BSM) Dark Matter partly consists of Topological Order with low energy TQFT, while there are anyon string excitations above the energy gap. (2) The 5th force is a Topological Discrete Gauge Force of (B-L) that mediates between the Standard Model particles, the BSM Topological Order, and gapped anyon string non-particle excitations.
Based on https://arxiv.org/abs/2501.00607, https://arxiv.org/abs/2012.15860, https://arxiv.org/abs/2302.14862 and https://arxiv.org/abs/2411.05786.

Rational approximation and the AAA algorithm
2025-02-03 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Nick Trefethen, Harvard University
Title: Rational approximation and the AAA algorithm
Approximation by rational functions used to be mainly a theoretical subject, but with the introduction of the AAA algorithm in 2018, it became computationally practical and indeed easy. The implications for what we can do numerically are enormous. This talk will outline the algorithm and demonstrate its application to a collection of problems. We can also use it to demonstrate the potential theory that underlies the theory of rational approximation, a topic that goes back to Joseph Walsh here at Harvard a century ago.

Blob homology
2025-02-04 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Sunghyuk Park, Harvard CMSA
Title: Blob homology

CMSA Q&A Seminar: Subir Sachdev
2025-02-05 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar

Speaker: Subir Sachdev, Harvard University
Topic: What is a quantum critical region?

Is every knot isotopic to the unknot?
2025-02-07 14:00 - 15:45 | Virtual
Freedman CMSA Seminar
*via Zoom*
Speaker: Sergey Melikhov, Steklov Math Institute
Title: Is every knot isotopic to the unknot?
Abstract: The following problem was stated by D. Rolfsen in his 1974 paper; according to R. Daverman it was being discussed since the mid-60s. Is every knot in $S^3$ isotopic (=homotopic through embeddings) to a PL knot — or, equivalently, to the unknot? In particular, is the Bing sling isotopic to a PL knot? We show that the Bing sling $B$ is not isotopic to any PL knot by an isotopy which extends to an isotopy of any 2-component link obtained from $B$ by adding a disjoint component $Q$ such that $lk(B,Q)=1$. Moreover, the assertion remains true if the additional component is allowed to self-intersect, and even to get replaced by a new one at any time instant $t$, as long as it remains disjoint from the original component $K_t$ and represents the same conjugacy class as the old one in $G/[G’,G”]$, where $G=pi_1(S^3setminus K_t)$. The are examples showing that the latter result cannot be improved in certain ways. I plan to present a sketch of the proof, modulo some ingredients. The details can be found in arXiv:2406.09365 and the main ingredients in arXiv:2406.09331 and arXiv:math/0312007v3.

Joint BHI/CMSA Foundation Seminar: Quantum Effects Inside Black Holes
2025-02-10 09:30 - 10:30 | Black Hole Initiative, 20 Garden Street, Cambridge MA, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Joint BHI/CMSA Foundation Seminar
Location: BHI seminar room
Speaker: Noa Zilberman (Princeton University)
Title: Quantum Effects Inside Black Holes
Abstract: Astrophysical black holes are known to be rotating. Within classical General Relativity, the simplest spacetime solution (the Kerr solution) describing a rotating black hole reveals a traversable passage through an inner horizon – which in turn may lead to another external universe. But does this remain the case when taking quantum effects into account?
Answering this question, along others, requires one to understand the manner in which quantum energy fluxes affect the internal geometry of a black hole. It has been widely anticipated, yet inconclusive (till this work), that such effects would diverge at the inner horizon of a spinning black hole. This divergence, if indeed takes place, may drastically affect the internal black hole geometry, potentially preventing the inner horizon traversability. Clarifying this issue requires the computation of the quantum energy fluxes in black hole interiors. However, this has been a serious challenge for decades.
Using a combination of old and new methods, we have managed to compute the quantum energy fluxes at the inner horizon of a spinning black hole, in a vacuum state corresponding to an evaporating black hole. We found that these fluxes are either positive or negative, depending on the black hole spin (and polar angle). The sign of these fluxes may be crucial to the nature of their backreaction on the geometry (as should be dictated by the semiclassical Einstein equation).
In this seminar, we shall briefly describe the basic framework of semiclassical general relativity and the renormalization procedure, and then present our novel results for the quantum fluxes at the inner horizon of a rotating black hole, briefly mentioning possible implications for the inner horizon traversability.

On 5d conformal matter
2025-02-10 11:00 - 12:00 | Virtual
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Mario De Marco (UL Brussels)
Title: On 5d conformal matter
Abstract: SCFTs with 8 supercharges lie at the sweet spot of the classification program of susy scale-invariant theories: with this amount of susy, the classification is in principle achievable, despite being non-trivial. In this talk, I will present the geometric approach to the classification of 5d SCFTs, based on geometric engineering on non-compact Calabi-Yau manifolds. The strategy is to imagine a 5d SCFT as a non-Lagrangian generalization of a quiver, and to build this generalised quiver by gauging together indecomposable blocks, called “5d conformal matter theories”. In this talk, I will start with a recap of the geometric construction of [2311.04984]. I will then present some forthcoming results, concerning the (p,q) web realization of 5d conformal matter theories, their reduction to 4d, their magnetic quiver, and the discovery of exotic “trifundamental” and “quadrifundamental” 5d indecomposable blocks (with flavor group at least SO(4k+2)^3 or SO(4k)^4).
In collaboration with M. Del Zotto, M. Graffeo, J. Grimminger, and A. Sangiovanni.

AI in math and theoretical physics: Status and prospects
2025-02-10 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Michael Douglas, Harvard CMSA
Title: AI in math and theoretical physics: status and prospects
Abstract: AI is making great progress and has the potential to change how we work in unprecedented ways. In this talk I will survey a few recent works which illustrate the state of the art, some from my own research, some developed at the CMSA’s recent program on Mathematics and Machine Learning. I will then report on current developments in AI and speculate on how they will affect our work in the next few years.

Discussion of Integrating quantum groups over surfaces
2025-02-11 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Ben Gammage, Harvard University
Title: Discussion of Integrating quantum groups over surfaces
https://arxiv.org/abs/1501.04652

CMSA Q&A Seminar: Javier Gomez-Serrano
2025-02-12 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Javier Gomez-Serrano, Brown University
Topic: Please tell us about the Millennium prize problem for Navier-Stokes and segue from that into ML?

Discovering Data Structures: Nearest Neighbor Search and Beyond
2025-02-12 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Omar Salemohamed, Mila
Title: Discovering Data Structures: Nearest Neighbor Search and Beyond
Abstract: As neural networks learn increasingly sophisticated tasks—from image recognition to mastering the game of Go—we ask: can deep learning discover data structures entirely from scratch? We introduce a general framework for data structure discovery, which adapts to the underlying data distribution and provides fine-grained control over query and space complexity. For nearest neighbor (NN) search, our model (re)discovers classic algorithms like binary search in one dimension and learns structures reminiscent of k-d trees and locality-sensitive hashing in higher dimensions. Additionally, the model learns useful representations of high-dimensional data such as images and exploits them to design effective data structures. Beyond NN search, we believe the framework could be a powerful tool for data structure discovery for other problems and adapt our framework to the problem of estimating frequencies over a data stream. To encourage future work in this direction, we conclude with a discussion on some of the opportunities and remaining challenges of learning data structures end-to-end.

Jack polynomials and enumeration of non-orientable maps
2025-02-14 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Houcine Ben Dali, Harvard CMSA
Title: Jack polynomials and enumeration of non-orientable maps
Abstract: A map is a graph embedded on a surface, which may be orientable or not. The representation theory of the symmetric group can be used to write the generating series of maps on orientable surfaces using Schur symmetric functions.
Several conjectures suggest that Jack polynomials—a one-parameter deformation of Schur functions—are related to the enumeration of non-orientable maps counted with a “non-orientability” weight. In this talk, I will discuss some of these conjectures and present recent progress in this direction.

Algebras of "symmetries" in QFT and applications to solitons (1)
2025-02-18 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Dan Freed, Harvard CMSA and Math
Title: Algebras of “symmetries” in QFT and applications to solitons (1)
https://arxiv.org/abs/2403.08883

CMSA Q&A Seminar: Phillip Matchett Wood
2025-02-19 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Phillip Matchett Wood, Harvard University
Topic: Info session on the CMSA/Mathematics Summer REU Program (Research Experience for Undergraduates)

Toward constructing a large-scale quantum computer based on TQFT
2025-02-24 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Liyuan Chen ( Harvard)
Title: Toward constructing a large-scale quantum computer based on TQFT
Abstract: Topological quantum computation, motivated by topological quantum field theory (TQFT), offers a promising path toward fault-tolerant universal quantum computation. However, the practical realization of such a system remains challenging due to the difficulty of finding suitable topological materials. In this work, we provide a comprehensive blueprint for constructing a fault-tolerant universal quantum computer based on the quantum double model $mathcal{D}(S_3)$, a specific non-Abelian topological order. We implement logical computations using quantum circuits on qubits and qutrits, including a single non-Clifford gate, compatible with near-term quantum devices. This work bridges the gap between abstract mathematical frameworks and noise-resilient quantum computation on near-term devices. Our proposal offers a promising path to realize a non-Abelian anyon-based large scale quantum computer.

Algebras of "symmetries" in QFT and applications to solitons (2)
2025-02-25 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Dan Freed, Harvard CMSA and Math
Title: Algebras of “symmetries” in QFT and applications to solitons (2)
https://arxiv.org/abs/2403.08883

Datasets for Math: From AIMO Competitions to Math Copilots for Research
2025-02-26 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Simon Frieder, Oxford
Title: Datasets for Math: From AIMO Competitions to Math Copilots for Research
Abstract: This talk begins with a brief exposition of the AI Mathematical Olympiad (AIMO) on Kaggle, now in its second iteration, outlining datasets and models available to contestants. Taking a broader perspective, I then examine 1) the overarching issues the current datasets suffer from—such as binary evaluation or constrained sets of use cases— and 2) the trajectory they set for competition-style mathematical problem-solving, which is different from mathematical research practice. I argue for a fundamental shift in dataset structure and composition, both for training and evaluation, and introduce the idea of mapping mathematical workflows to data, a key example underscoring the need for this shift. I touch upon new thinking LLMs and their role in redefining LLM math evaluation, highlighting their implications for dataset design. Finally, I propose general improvements to the current state of mathematical datasets, including mathematical adaptations of dataset documentation (e.g., datasheets).
