Supersymmetric QFT’s are of long-standing interest for their high degree of solvability, phenomenological implications, and rich connections to mathematics. In my talk, I will describe how the holomorphic twist isolates the protected quantities which give SUSY QFTs their potency by restricting to the cohomology of one supercharge. I will briefly introduce infinite dimensional symmetry algebras, generalizing Virasoro and Kac-Moody symmetries, which emerge. Finally, I will explain a potential novel UV manifestation of confinement, dubbed “holomorphic confinement,” in the example of pure SU(N) super Yang-Mills. Based on arXiv:2207.14321 and 2 forthcoming works with Kasia Budzik, Davide Gaiotto, Brian Williams, Jingxiang Wu, and Matthew Yu.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/quantum-matter-seminar/

Quantitative reasoning tasks which can involve mathematics, science, and programming are often challenging for machine learning models in general and for language models in particular. We show that transformer-based language models obtain significantly better performance on math and science questions when trained in an unsupervised way on a large, math-focused dataset. Performance can be further improved using prompting and sampling techniques including chain-of-thought and majority voting. Minerva, a model that combines these techniques, achieves SOTA on several math and science benchmarks. I will describe the model, its capabilities and limitations.

For more information, please see: https://cmsa.fas.harvard.edu/tech-in-math/

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, October 5th.

Please see the seminar page for more details: https://math.harvard.edu/~ctm/sem/.

The Sachdev-Ye-Kitaev model was introduced as a toy model of interacting fermions without any particle-like excitations. I will describe how this toy model yields the universal low energy quantum theory of generic charged black holes in asymptotically 3+1 dimensional Minkowski space. I will also discuss how extensions of the SYK model yield a realistic theory of the strange metal phase of correlated electron systems.

Math is more than just a way to describe the world, and it is more than just a set of skills, like doing arithmetic and factoring a quadratic. Math is a deeply human enterprise that fulfills basic human longings, such as for beauty and truth. When properly engaged, it builds virtues like persistence, creativity, and a competence to solve new problems. These virtues will serve you well no matter what you do in life. It was an incarcerated man–now his friend–that helped distinguished mathematician Francis Su see this more clearly than ever before.

In this talk, we construct special Lagrangian fibrations for log Calabi–Yau surfaces and scattering diagrams from Lagrangian Floer theory of the fibers. These scattering diagrams recover the algebro-geometric scattering diagrams of Gross–Pandharipande–Siebert and Gross–Hacking–Keel. The argument relies on a holomorphic/tropical disc correspondence to control the behavior of holomorphic discs, allowing us to relate open Gromov–Witten invariants to log Gromov-Witten invariants. This talk is based on joint work with Man-Wai Mandy Cheung, Hansol Hong, and Yu-Shen Lin.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/algebraic-geometry-in-string-theory/

We will talk about our recent construction of the smallest closed exotic 4-manifolds with signature zero known to date. Our novel examples are derived from fairly special small Lefschetz fibrations we build, with spin and non-spin monodromies. This is joint work with N. Hamada.

The minimal model program is an ambitious program that aims to understand the geometry of complex projective varieties (eg. manifolds defined by polynomial equations).  In this talk we will discuss some recent results and challenges encountered trying to extend the minimal model program to the context of Kähler varieties.

We propose an externally imposed superlattice potential as a platform for manipulating topological phases, which has both advantages and disadvantages compared to a moire superlattice. In the first example, we apply the superlattice potential to the 2D surface of a 3D topological insulator. The superlattice potential creates tunable van Hove singularities, which, when combined with strong spin-orbit coupling and Coulomb repulsion give rise to a topological meron lattice spin texture. Thus, the superlattice potential provides a new route to the long sought-after goal of realizing spontaneous magnetic order on the surface of a 3D TI. In the second example, we show that a superlattice potential applied to Bernal-stacked bilayer graphene can generate flat Chern bands, similar to in twisted bilayer graphene, whose bandwidth can be as small as a few meV. The superlattice potential offers flexibility in both lattice size and geometry, making it a promising alternative to achieve designer flat bands without a moire heterostructure.

The classical infinite Ramsey theorem states that if we colour pairs of natural numbers using two colours, there is an infinite set all of whose pairs get the same colour. This is the beginning of a rich theory, which touches on many areas of mathematics including graph theory, set theory and dynamics. I will give an overview of Ramsey theory, emphasising the diverse ideas which are at play in this area.

We study them, in particular showing on a smooth complex quasi-projective variety the existence of $\ell$-adic absolutely irreducible local systems for all $\ell$ the moment there is a complex irreducible topological local system. The proof is purely arithmetic.

This is work in progress with Johan de Jong, relying in part on earlier work with Michael Groechenig.

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, October 12th.

Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

In 2016, Ellenberg and Gijswijt made a breakthrough on the famous cap-set problem, which asks about the maximum size of a subset of (F_3)^n not containing a three-term arithmetic progression. They proved that any such set has size at most 2.756^n. Their proof was later reformulated by Tao, introducing what is now called the slice rank polynomial method. This talk will explain Tao’s proof of the Ellenberg-Gijswijt bound for the cap-set problem, and discuss some related problems.

For more information, please see: https://people.math.harvard.edu/~demarco/AlgebraicDynamics/

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/algebraic-geometry-in-string-theory/

Solutions of the algebraic Bethe ansatz for quantum magnet chains are, generally, multivalued functions of the parameters of the integrable system. I will explain how to compute some monodromies of solutions of Bethe ansatz for the Gaudin magnet chain. Namely, the Bethe eigenvectors in the Gaudin model can be regarded as a covering of the Deligne-Mumford moduli space of stable rational curves, which is unramified over the real locus of the Deligne-Mumford space. The monodromy action of the fundamental group of this space (called cactus group) on the eigenlines can be described very explicitly in purely combinatorial terms of Kashiwara crystals — i.e. combinatorial objects modeling the tensor category of finite-dimensional representations of a semisimple Lie algebra g. More specifically, this monodromy action is naturally equivalent to the action of the same group by commutors (i.e. combinatorial analog of a braiding) on a tensor product of Kashiwara crystals. This is joint work with Iva Halacheva, Joel Kamnitzer, and Alex Weekes.

Representations of 3-manifold groups into groups such as SU(2) and SL(2,C) have been actively studied for decades.  Many topological invariants are defined by considering these representations, such as the Casson invariant, the Casson-Lin invariant, and the A polynomial.  In 2010, Kronheimer-Mrowka showed that the fundamental group of every non-trivial knot in S^3 admits an irreducible representation in SU(2) such that the image of the meridian is traceless, which answered a conjecture of Cooper.  In this talk, I will present a result that generalizes Kronheimer-Mrowka’s theorem to the case of links.  We show that for every link L that is not the unknot, the Hopf link, or a connected sum of Hops links, its fundamental group admits an irreducible SU(2) representation such that the image of every meridian is traceless.  The proof is based on an excision formula of singular instanton Floer homology.  This is joint work with Yi Xie.

I will explain a new type of holographic dualities between n+1D topological orders with a chosen boundary condition and nD (potentially gapless) quantum liquids. It is based on the idea of topological Wick rotation, a notion which was first used in arXiv:1705.01087 and was named, emphasized and generalized later in arXiv:1905.04924. Examples of these holographic dualities include the duality between 2+1D toric code model and 1+1D Ising chain and its finite-group generalizations (independently discovered by many others); those between 2+1D topological orders and 1+1D rational conformal field theories; and those between n+1D finite gauge theories with a gapped boundary and nD gapped quantum liquids. I will also briefly discuss some generalizations of this holographic duality and its relation to AdS/CFT duality.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event/quantum-matter-in-mathematics-and-physics/

Jameel Al-Aidroos was a treasured colleague and a master of a myriad of things. He could simplify a mathematical idea and present a question or comment in a way that could bring his students to view concepts from new perspectives. Likewise, when he worked with graduate students around pedagogy and with school teachers around the joys of engaging with mathematics, the thoughtful way he approached teaching and mentorship made a big difference. Watching Jameel teach motivated and inspired both novice and veteran teachers to become better instructors and to bring their best selves forward. His superpower as a teacher was asking just the right questions to focus students’ attention on the core of the mathematical idea.

Through a number of seminars and workshops, Jameel shared his question-asking approach and pedagogical skills with graduate students and faculty. An astute and careful listener, a masterful communicator, and a deep thinker, he forged impactful connections. As a mentor and colleague, Jameel was without equal: he applied himself wholeheartedly to hone his craft in the classroom and as a mentor. He was quick to volunteer to do whatever was needed to promote team projects and through this, we discovered his enormous talents as an interviewer, film editor, and voice-over artist, among other things. It seemed there were no tasks he did not choose to rise to and polish, while consistently taking a step out of the limelight to let his students and colleagues shine.

The grace and generosity of spirit he extended to his students and colleagues are an indelible part of his legacy. Jameel carried this grace and generosity throughout his long battle with cancer. We honor his contributions and dedication to teaching and learning at Harvard via this speaker series as a small way to remember Jameel’s extraordinary warmth of character and pedagogical skills. He motivated and inspired his students and colleagues; through this series, we hope to celebrate and keep alive that legacy by bringing speakers who share new perspectives on mathematics and pedagogy, and motivate us to reflect on our professional roles.

Jameel Al-Aidroos Mathematical Pedagogy Lecture Series

When: October 17, 2022

Where: Hall E, Science Center, 1 Oxford Street, Cambridge, MA, 02138

Register for the In-Person Event.

Register for the Online Event.

Download a detailed PDF schedule of lectures and events.

Organizers

• Robin Gottlieb | Harvard Professor of the Practice in Teaching of Mathematics, Director of the Emerging Scholars Program
• Brendan Kelly | Harvard Department of Mathematics Senior Preceptor, Director of Introductory Mathematics

I will present a review about recent progress in chartin non-invertible symmetries for four-dimensional quantum field theorie that have a six-dimensional origin. These include in particular N= supersymmetric Yang-Mills theories, and also a large class of N= supersymmetric theories which are conformal and do not have  conventional Lagrangian description (the so-called theories of “clas S”). Among the main results, I will explain criteria for identifyin examples of systems with intrinsic and non-intrinsic non-invertibl symmetries, as well as explore their higher dimensional origin. Thi seminar is based on joint works with Vladimir Bashmakov, Azeem Hasan and Justin Kaidi.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/quantum-matter-seminar/

I will explain how to reinterpret invariant distributions on p-adic GL_n and its Lie algebra using symmetric functions. This reinterpretation, combined with some old results of Waldspurger, allows us to compute the Shalika germs of tamely ramified regular semisimple elements in GL_n and for example point-counts of compactified Jacobians on plane curve singularities. I will exposit the method and some interesting consequences such as how to t-deform many of the results using K-theory of Hilbert schemes of points on C^2. Time permitting, I will discuss the relation to triply graded knot homology. Based on joint work with Cheng-Chiang Tsai.

Webpage: https://sites.google.com/view/harvardmitag

Lévy matrices are symmetric random matrices whose entry distributions lie in the domain of attraction of an alpha-stable law; such distributions have infinite variance when alpha is less than 2. Due to the ubiquity of heavy-tailed randomness, these models have been broadly applied in physics, finance, and statistics. When the entries have infinite mean, Lévy matrices are predicted to exhibit a phase transition separating a region of delocalized eigenvectors from one with localized eigenvectors. We will discuss the physical context for this conjecture, and describe a result establishing it for values of alpha close to zero and one. This is joint work with Amol Aggarwal and Charles Bordenave.

Language models are showing impressive performance on many natural language tasks, including question-answering. However, language models – like most deep learning models – are black boxes. We cannot be sure how they obtain their answers. Do they reason over relevant knowledge to construct an answer or do they rely on prior knowledge – baked into their weights – which may be biased? An alternative approach is to develop models whose output is a human interpretable, faithful reasoning trace leading to an answer. In this talk we will characterise faithful reasoning in terms of logically valid reasoning and demonstrate where current reasoning models fall short. Following this, we will introduce Selection-Inference, a faithful reasoning model, whose causal structure mirrors the requirements for valid reasoning. We will show that our model not only produces more accurate reasoning traces but also improves final answer accuracy.

For more information, please see: https://cmsa.fas.harvard.edu/event_category/new-technologies-in-mathematics-seminar-series/

I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier work of Thorne. This gives a uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves. I will also mention some applications to rational points on certain families of curves.

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, October 19th.

Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

Symmetric mass generation (SMG) is a novel mechanism for massless fermions to acquire a mass via a strong-coupling non-perturbative interaction effect. In contrast to the conventional Higgs mechanism for fermion mass generation, the SMG mechanism does not condense any fermion bilinear coupling and preserves the full symmetry. It is connected to a broad range of topics, including anomaly cancellation, topological phase classification, and chiral fermion regularization. In this talk, I will introduce SMG through toy models, and review the current understanding of the SMG transition. I will also mention recent numerical efforts to investigate the SMG phenomenon. I will conclude the talk with remarks on future directions.

Some of the most dramatic events during human development is the axial elongation of the embryo with concomitant changes in the geometry and composition of the underlying tissues. The posterior part of the embryo gives rise to the spinal cord, vertebral column, ribcage, back muscles, and dermis.  In this talk, I will present our attempts at coaxing human embryonic stem cells to form these structures of the early human embryo that closely recapitulate the geometry, relative arrangements, composition, and dynamics of development of the early spinal cord flanked progenitors of the musculoskeletal system. Our goal was to do so, such that we could build hundreds of these organoids at a time. I will also present preliminary results for the use of this system to understand key events during early human development through imaging and genetic perturbations.

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event/active-matter-seminar-6/

Ramsey showed that any graph on N nodes contains a clique or independent set of size (log N)/2.  Erdos showed that there exist graphs on N nodes with no clique or independent set of size 2 log N, and asked for an explicit construction of such graphs.  This turns out to relate to the question of extracting high-quality randomness from two independent low-quality sources.  I’ll explain this connection and our recent exponential improvement in constructing two-source extractors.

This seminar will begin at 11:00am and lunch will follow at 12:00pm.

Hyperfiniteness via Borel asymptotic dimension, following https://arxiv.org/abs/2009.06721.

In two dimensions, one can study quantum field theories on unorientable manifolds by introducing crosscaps. This defines a class of states called crosscap states which share a few similarities with the notion of boundary states. In this talk, I will show that integrable theories remain integrable in the presence of crosscaps, and this allows to exactly determine the crosscap state.

Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra.  Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear.  In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions.  These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.

Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher.

The notion of topological phases has dramatically changed our understanding of insulators. There is much to learn about a band insulator beyond the assertion that it has a gap separating the valence bands from the conduction bands. In the particular case of two dimensions, the occupied bands may have a nontrivial topological twist determining what is called a Chern insulator. This topological twist is not just a mathematical observation, it has observable consequences—the transverse Hall conductivity is quantized and proportional to the 1st Chern number of the vector bundle of occupied states over the Brillouin zone. Finer properties of band insulators refer not just to the topology, but also to their geometry. Of particular interest is the momentum-space quantum metric and the Berry curvature. The latter is the curvature of a connection on the vector bundle of occupied states. The study of the geometry of band insulators can also be used to probe whether the material may host stable fractional topological phases. In particular, for a Chern band to have an algebra of projected density operators which is isomorphic to the W∞ algebra found by Girvin, MacDonald and Platzman—the GMP algebra—in the context of the fractional quantum Hall effect, certain geometric constrains, associated with the holomorphic character of the Bloch wave functions, are naturally found and they enforce a compatibility relation between the quantum metric and the Berry curvature of the band. The Brillouin zone is then endowed with a Kähler structure which, in this case, is also translation-invariant (flat). Motivated by the above, we will provide an overview of the geometry of Chern insulators from the perspective of Kähler geometry, introducing the notion of a  Kähler band which shares properties with the well-known ideal case of the lowest Landau level. Furthermore, we will provide a prescription, borrowing ideas from geometric quantization, to generate a flat Kähler band in some appropriate asymptotic limit. Such flat Kähler bands are potential candidates to host and realize fractional Chern insulating phases. Using geometric quantization arguments, we then provide a natural generalization of the theory to all even dimensions.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/new-technologies-in-mathematics-seminar-series/

The usual language of algebraic geometry is not appropriate for arithmetical geometry: addition is singular at the real prime. We developed two languages that overcome this problem: one replace s rings by the collection of “vectors” or by bi-operads, and another based on “matrices” or props. Once one understands the delicate commutativity condition one can proceed following Grothendieck’s footsteps exactly. The props, when viewed up to conjugation, give us new commutative rings with Frobenius endomorphisms.

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, October 26th.

Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

Elliptic curves are ubiquitous in number theory, algebraic geometry, complex analysis, cryptography, physics, and beyond. They were present in Diophantus’ Arithmetica (3rd century AD) and, nowadays, they are more relevant than ever as a key ingredient in the algorithms that, for instance, secure Bitcoin transactions or encrypt WhatsApp messages. In this talk, we will introduce elliptic curves, explain their central role in mathematics, and discuss related open problems and applications.

Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction.  We use topological techniques – adapting tools used to study mapping class groups – to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a topological branched cover is equivalent to.  This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.

For more information, please see: https://people.math.harvard.edu/~demarco/AlgebraicDynamics/

I will outline the construction of a 2-category associated to a hyperKahler moment map. The construction is based on partial differential equations in one, two and three dimensions, combined with a three-dimensional version of the Gaiotto–Moore–Witten web formalism.

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event/algebraic-geometry-in-string-theory/

We discuss recent joint work with Kyle Binder on defining the unipotent
fundamental group of tropical varieties. This fundamental group arises from the
Tannakian formalism using tropical vector bundles with integrable connection. By
employing the Orlik–Solomon theorem, we prove that this computes the unipotent
completion of the fundamental group of algebraic varieties with smooth
tropicalization.

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http://math.mit.edu/seminars/combin/

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