Topological to Gravitational Leptogenesis, via Ultra Unification
2025-02-03 15:00 - 16:00 (America/New_York) | 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).

The Combinatorics of the Amplituhedron – Tiles, Tilings, and Cluster Algebras
2025-02-28 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Matteo Parisi
Title: The Combinatorics of the Amplituhedron – Tiles, Tilings, and Cluster Algebras
Abstract: The amplituhedron is the image of the positive Grassmannian—the region of the Grassmannian where all Plücker coordinates are nonnegative—under a totally positive linear map. It is a far-reaching generalization of cyclic polytopes and hyperplane arrangements, and the positive Grassmannian itself. The “volume” of the amplituhedron encodes probabilities of particle interactions in the quantum field theory N=4 super Yang-Mills, and calculating this volume involves decomposing (or tiling) the amplituhedron into smaller pieces (or tiles) and summing their volumes. This talk will delve into the rich combinatorics of these tiles and tilings, presenting recent results on some of the central conjectures in this area, including the magic number, BCFW tiling, and cluster adjacency conjectures.

Quantum Field Theory and Physical Mathematics
2025-03-03 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Kai Xu, Harvard
Title: Finite Landscape of 6d N=(1,0) Supergravity
Abstract: We present a bottom-up argument showing that the number of massless fields in six-dimensional quantum gravitational theories with eight supercharges is uniformly bounded. Specifically, we show that the number of tensor multiplets is bounded by T≤193, and the rank of the gauge group is restricted to r(V)≤480. Given that F-theory compactifications on elliptic CY 3-folds are a subset, this provides a bound on the Hodge numbers of elliptic CY 3-folds: h1,1(CY3)≤491, h1,1(Base)≤194 which are saturated by special elliptic CY 3-folds. This establishes that our bounds are sharp and also provides further evidence for the string lamppost principle.

Large value estimates in number theory and computer science
2025-03-03 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Larry Guth, MIT
Title: Large value estimates in number theory and computer science
Abstract: A large value estimate for a matrix M is a simple type of estimate in quantitative linear algebra. Estimates of this type appear in many parts of math, both pure and applied. One example is the large value problem for Dirichlet polynomials from analytic number theory, which is related to estimates about the zeroes of the Riemann zeta function. We will also give some examples from computer science. Many large value problems are difficult. On the pure math side, the sharp conjecture about large values of Dirichlet polynomials has been open for a long time and is out of reach of current methods. On the computer science side, we don’t know any efficient algorithm to approximately solve the large value problem for a given matrix M. Many experts think that such an algorithm does not exist. In this talk we will survey how large value estimates come up, the known methods for working on them, and some of the obstacles to fully understanding them.

CMSA Q&A Seminar: Puskar Mondal
2025-03-05 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Puskar Mondal, Harvard CMSA
Topic: What is the positive energy theorem?

Machine Learning G2 Geometry
2025-03-05 14:00 - 15:00 | Hybrid
New Technologies in Mathematics Seminar
Speaker: Elli Heyes, Imperial College
Title: Machine Learning G2 Geometry
Abstract: Compact Ricci-flat Calabi-Yau and holonomy G2 manifolds appear in string and M-theory respectively as descriptions of the extra spatial dimensions that arise in the theories. Since 2017 machine-learning techniques have been applied extensively to study Calabi-Yau manifolds but until 2024 no similar work had been carried out on holonomy G2 manifolds. In this talk, I will firstly show how topological properties of these manifolds can be learnt using neural networks. I will then discuss how one could try to numerically learn metrics on compact holonomy G2 manifolds using machine-learning and why these approximations would be useful in M-theory.

A Tetrahedral Approach to Calabi-Yau Geometry
2025-03-07 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Charles Doran, CMSA
Title: A Tetrahedral Approach to Calabi-Yau Geometry
Abstract: We will open with a quick introduction to the what and why of Calabi-Yau geometry. Following this, we will consider the problem of deforming tetrahedra while preserving the areas of their faces, following our noses to discover a beautiful path to elliptic curves, K3 surfaces, and beyond. Time permitting, we will also discuss motivations and applications across physics. The talk should be broadly accessible.

Comments on Non-Invertible Symmetries in K3 CFTs and the Conway Moonshine Module
2025-03-10 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Sarah Harrison, Northeastern
Title: Comments on Non-Invertible Symmetries in K3 CFTs and the Conway Moonshine Module
Abstract: There is an established connection between discrete symmetry groups of K3 non-linear sigma models and a distinguished N=1 chiral SCFT called the Conway moonshine module. More specifically, all symmetry groups of K3 NLSMs preserving the N=4 superconformal algebra can be obtained as subgroups of “Conway zero”, the group of symmetries of the Conway module, and their explicit action on the BPS spectrum can (almost always) be obtained via traces in the Conway module. A natural question is whether this relation extends to fusion category symmetry of these theories. I will discuss positive evidence in this direction, by exploring examples of non-invertible topological defect lines in K3 NLSMs and the Conway module. This is based on work in progress with R. Angius, S. Giaccari, and R. Volpato.

Introduction to the probabilistic approach to Louville theory
2025-03-11 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Leon Liu, Harvard
Title: Introduction to the probabilistic approach to Louville theory
Abstract: I will give an introduction to the probabilistic approach to Louville theory, following Hairer’s notes.

CMSA Q&A Seminar: Dan Freed
2025-03-12 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Dan Freed, Harvard University
Topic: What are spectra (in homotopy theory)?

Discovery in Mathematics with Automated Conjecturing
2025-03-12 14:00 - 15:00 | Hybrid – G10
New Technologies in Mathematics Seminar
Speaker: Randy Davila, RelationalAI and Rice University
Title: Discovery in Mathematics with Automated Conjecturing
Abstract: Automated conjecturing is a form of artificial intelligence that applies heuristic-driven methods to mathematical discovery. Since the late 1980s, systems such as Fajtlowicz’s Graffiti, DeLaViña’s Graffiti.pc, and TxGraffiti have collectively contributed to over 130 publications in mathematical journals. In this talk, we outline the evolution of automated conjecturing, focusing on TxGraffiti, a program that employs linear optimization methods and several distinct heuristics to generate mathematically meaningful conjectures. We will then introduce GraphMind, a dueling framework where the Optimist proposes conjectures while the Pessimist seeks counterexamples, fostering a feedback loop that strengthens automated reasoning. Finally, we will present GraffitiAI, a Python package that extends automated conjecturing across various mathematical domains.
Bio: Randy R. Davila is a Lecturer in the Department of Computational Applied Mathematics & Operations Research at Rice University and a Library Engineer at RelationalAI, specializing in relational knowledge graph systems for intelligent data management. He earned his PhD in Mathematics from the University of Johannesburg in 2019, with research focused on graph theory and combinatorial optimization. His work explores artificial intelligence in mathematical conjecture generation, graph theory, and neural network applications to combinatorial problems. As the creator of TxGraffiti, he has developed AI-driven systems that have contributed to numerous mathematical publications. His recent projects include GraphMind, a dueling agent-based framework that pairs conjecture generation with counterexample discovery, and GraffitiAI, a Python package for automated conjecturing across mathematical disciplines.

Freedman CMSA Seminar: Michael Freedman (CMSA) & Elia Portnoy (MIT)
2025-03-12 15:00 - 17:00 | Hybrid – G10
Freedman CMSA Seminar
Speaker: Michael Freedman, Harvard CMSA (3:00–4:00 pm ET)
Title: How many links can you fit in a box?
Abstract: I’ll discuss a “made up” problem on the interface of topology and packing, which may well be classified as “recreational math”. Here is the first question suppose you have a unit box, how many unlinked (split) copies of the Hopf link (c_1,i,c_2,i) and be embedded so that for each copy the two components c_1,i and c_2,i maintain a distance of at least some fixed epsilon >0. Is this number even finite?

Speaker: Elia Portnoy, MIT (4:00–5:00 pm ET)
Title: An explicit packing of links in a box and some progress in quantitative embeddings
Abstract: Following Freedman’s talk, I’ll begin by showing how to pack a large number of links in a box with certain geometric and topological constraints (joint with Fedya Manin). If time permits, I’ll also discuss some progress and open questions for the following quantitative embedding problem: given a simplicial complex X, what is the smallest size of a map from X to R^n so that the preimage of each unit ball intersects a small constant number of simplices?

The Andersen-Kashaev volume conjecture for FAMED geometric triangulations
2025-03-24 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Ka Ho Wong (Yale)
Title: The Andersen-Kashaev volume conjecture for FAMED geometric triangulations
Abstract: In the early 2010s, Andersen and Kashaev defined a TQFT based on quantum Teichmuller theory. In particular, they define a partition function for every ordered ideal triangulation of hyperbolic knot complement in $mathbb{S}^3$ equipped with an angle structure. The Andersen-Kashaev volume conjecture suggests that the partition function can be expressed in terms of a Jones function of the knot which, in its semi-classical limit, decay exponentially with decay rate the hyperbolic volume of the knot complement. In this talk, we will introduce a purely combinatorial condition on triangulations which, together with the geometricity of the triangulations, imply the Andersen-Kashaev volume conjecture and its generalization. This talk is based on the joint work with Fathi Ben Aribi.

The Toda Lattice as a Soliton Gas
2025-03-24 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Amol Aggarwal, Columbia University
Title: The Toda Lattice as a Soliton Gas
Abstract: A basic tenet of integrable systems is that, under sufficiently irregular initial data, they can be thought of as dense collections of many solitons, or “soliton gases.” In this talk we focus on the Toda lattice, which is an archetypal example of an integrable Hamiltonian dynamical system. We explain how the system, under certain random initial data, can be interpreted through solitons, and provide a framework for studying how these solitons asymptotically evolve in time. The arguments use ideas from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices.
