## news

##### Morgan Opie–Graduating Ph.D. Recipient–Profiled in Harvard Gazette

Morgan Opie, graduating mathematics Ph.D. recipient, is profiled in the Harvard Gazette as part of their coverage of Harvard's 2021 Graduation on Thursday, May 27....

##### Special Colloquium Series–Late Spring 2021

Spring 2021 Special Colloquium Series: Talks-Videos-Registration All talks on Zoom 3-4pm • Lydia Bieri -- University of Michigan New Structures in Gravitational Waves May 25,...

Below are undergraduate prizes awarded in Mathematics for the 2020-2021 academic year. Alexander Prize • Hanna Mularczyk Friends Prize • Kenz Kallal • Lux Zhao...

##### Melanie Matchett Wood wins $1M NSF Waterman Award Professor Melanie Matchett Wood has received the National Science Foundation's annual Alan T. Waterman award, which recognizes an outstanding early-career U.S. science or engineering researcher... Read more ##### 2021 Friends of Math Thesis Prize: Kenz Kallal and Lux Zhao The Friends of Harvard Mathematics Undergraduate Thesis Prize is awarded annually for one or two exceptional undergraduate Senior Honors theses. In 2021 the Prize was... Read more See Older News ## announcements ##### The Azat Miftakhov Day June 16, 2021 10:00 am In solidarity with Azat Miftakhov *Starting at 4pm Central European Summer Time (UTC+2)* It will be organized online by the Azat Miftakhov committee in solidarity with Azat... Read more ##### Conference: Mathematics of statistical mechanics and quantum physics August 1, 2022 - August 4, 2022 Conference: Mathematics of statistical mechanics and quantum physics Dates: August 1-4, 2022 More information to follow Read more See Older Announcements ## upcoming events • June 14, 2021 3:00 pm Special Colloquium COLLOQUIUMS Speaker: Michael Douglas - Harvard CMSA & Stony Brook University Title: Special Colloquium • June 15, 2021 10:00 am via Zoom Video Conferencing MATHEMATICAL PICTURE LANGUAGE SEMINAR Speaker: Jeongwan Haah - Microsoft Title: TBD < 2020 > June « » Sun Mon Tue Wed Thu Fri Sat May 1 2 • CMSA EVENT: CMSA Geometry Seminar: Equivariant Floer theory and SYZ mirror symmetry ##### CMSA EVENTCMSA Geometry Seminar: Equivariant Floer theory and SYZ mirror symmetry 9:30 AM-10:30 AM June 2, 2020 In this talk, we will first review a symplectic realization of the SYZ program and some of its applications. Then I will explain some recent works on equivariant Lagrangian Floer theory and disc potentials of immersed SYZ fibers. They are joint works with Hansol Hong, Yoosik Kim and Xiao Zheng. • MATHEMATICAL PICTURE LANGUAGE SEMINAR ##### MATHEMATICAL PICTURE LANGUAGE SEMINARKitaev’s finite group model as an error correcting code 10:00 AM-11:00 AM June 2, 2020 Kitaev’s quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually, a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. This implies that Kitaev’s model satisfies both TQO1 and TQO2 conditions of Bravyi-Hastings-Michalakis, and so it is a topological order in the sense of B-H-M. We note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of this result. We also note that more recently Q. Yang and Z. Wang proved the same result for the more general class of Levin-Wen models, but their method of proof is very different. via Zoom: https://harvard.zoom.us/j/779283357 3 • CMSA EVENT: CMSA Quantum Matter/Quantum Field Theory Seminar: Magnetic Black Holes ##### CMSA EVENTCMSA Quantum Matter/Quantum Field Theory Seminar: Magnetic Black Holes 10:30 AM-12:00 PM June 3, 2020 We discuss properties of magnetically charged black holes in the Standard Model. We will discuss how the electroweak symmetry is restored around the black hole. In addition, the Hawking evaporation rate is greatly enhanced by a factor of the charge of the black hole. These provide interesting candidates for primordial black holes which can have a relatively low mass. via Zoom Video Conferencing: https://harvard.zoom.us/s/977347126 • INFORMAL GEOMETRY AND DYNAMICS SEMINAR ##### INFORMAL GEOMETRY AND DYNAMICS SEMINARSlices of Thurston’s Master Teapot 4:00 PM-5:30 PM June 3, 2020 Thurston’s Master Teapot is the closure of the set of all points$(z,\lambda) \in \mathbb{C} \times \mathbb{R}$such that$\lambda$is the growth rate of a critically periodic unimodal self-map of an interval and$z$is a Galois conjugate of$\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a “restricted iterated function system.” An application of this characterization is that the Master Teapot is not invariant under the map$(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu. 4 • CMSA EVENT: CMSA Condensed Matter/Math Seminar: Coupled layers, p-string condensate, and exactly solvable fracton models ##### CMSA EVENTCMSA Condensed Matter/Math Seminar: Coupled layers, p-string condensate, and exactly solvable fracton models 10:30 AM-12:00 PM June 4, 2020 In this talk, I will introduce a class of gapped fracton models, dubbed “cage-net fracton models”. I will first review the coupled layer construction for exactly solvable fracton models. This construction leads to a general mechanism to obtain cage-net fracton models through a “p-string condensation”, where the extended one-dimensional particle strings built out of pointlike excitations are condensed. This p-string condensation generalizes the concept of anyon condensation in a conventional topological order and allows us to establish the properties of the fracton phase, such as its ground-state wave function, the spectrum and the mobility of excitations. To illustrate the main idea, I will focus on a simple example: doubled-Ising cage-net model. I will show that there are intrinsic non-Abelian excitations with restricted-mobility in this model and they cannot be understood as bound states among two-dimensional non-Abelian anyons and Abelian particles. If time permits, I will discuss another class of exactly solvable fracton models based on a generalization of the twisted gauge theory. via Zoom Video Conferencing: https://harvard.zoom.us/s/977347126 5 6 7 8 • CMSA EVENT: Legendrian graphs and their invariants ##### CMSA EVENTLegendrian graphs and their invariants 9:30 PM-10:30 PM June 8, 2020 Legendrian graphs naturally appear in the study of Weinstein manifolds with a singular Lagrangian skeleton, and a tangle decomposition of Legendrian submanifolds. I will introduce various invariant of Legendrian graphs including DGA type, polynomial type, sheaf theoretic one, and their relationship. This is joint work with Byunghee An, and partially with Tamas Kalman and Tao Su. 9 • MATHEMATICAL PICTURE LANGUAGE SEMINAR ##### MATHEMATICAL PICTURE LANGUAGE SEMINARThe sign problem and its relation to the spectral gap of quantum many-body systems 10:00 AM-11:00 AM June 9, 2020 The partition function of a quantum system without a sign problem can be represented by a path integral in which every amplitude is efficiently computable and nonnegative, which is a substantial simplification from the interference of complex amplitudes in the general quantum case. In quantum computing the presence of a sign problem has been recast as a virtue, because it helps to increase the complexity of the quantum system beyond the range of classical simulation. This is particularly important for quantum adiabatic algorithms based on ground states, where the run time depends on the scaling of the spectral gap above the ground state. This motivates us to study the relation of the sign problem to the spectral gap, using methods such as random matrix theory and spectral graph theory. The latter relates the discrete geometry of ground states (in a world where vertices are basis elements and edges are Hamiltonian matrix elements) to the level spacings in the low energy spectrum using the higher-order signed Cheeger inequalities. This talk will include analytical results from 1703.10133 and 2004.07681. via Zoom: https://harvard.zoom.us/j/779283357 10 11 • CMSA EVENT: CMSA Condensed Matter/Math Seminar: Symmetry enriched U(1) quantum spin liquids and beyond ##### CMSA EVENTCMSA Condensed Matter/Math Seminar: Symmetry enriched U(1) quantum spin liquids and beyond 10:30 AM-12:00 PM June 11, 2020 I will present our characterization and classification of 3+1 dimensional U(1) quantum spin liquids (QSLs) enriched by symmetries. These QSLs are spin system described by a deconfined U(1) gauge theory at low energies, and we assume that the only gapless degree of freedom is the photon. I will mostly focus on the example where the symmetry includes SO(3) spin rotation and time reversal, from which I will summarize our general scheme for the characterization and classification. The characterization and classification are based on the properties of the matters coupled to the emergent U(1) gauge field, although they are gapped. I will also discuss some applications of the ideas developed here to topological phases protected by crystalline symmetries. In order to avoid potential misunderstanding due to the difference in the ideology and language between physicists with different background, I will give an overview of these differences at the beginning of the talk. via Zoom Video Conferencing: https://harvard.zoom.us/s/977347126 12 13 14 15 • CMSA EVENT: CMSA Quantum Matter/Quantum Field Theory Seminar: Traversable wormholes in four and two dimensions ##### CMSA EVENTCMSA Quantum Matter/Quantum Field Theory Seminar: Traversable wormholes in four and two dimensions 8:30 PM-10:00 PM June 15, 2020 In my talk I discuss traversable wormholes in four and two dimensions. In four dimensions I present a solution based on two magnetically charged black holes. It is a solution of classical Einstein gravity which requires U(1) gauge field and massless fermions only and it does not need exotic matter or boundary conditions. It is a long wormhole that does not lead to causality violations in the ambient space. Very similar wormholes in two dimensional Jackiw–Teitelboim(JT) gravity can be constructed in Sachdev–Ye–Kitaev(SYK) model, where one can study the real-time formation of the wormhole numerically. I will explain similarities and differences between these four- and two-dimensional solutions and argue that in SYK the formation of the wormhole is smooth and takes time independent of N in the large N limit. Based on arXiv: 1807.04726 and 1912.03276 via Zoom Video Conferencing: https://harvard.zoom.us/s/977347126 16 • CMSA EVENT: CMSA Geometry and Physics seminar: Loops in hypertoric varieties and symplectic duality ##### CMSA EVENTCMSA Geometry and Physics seminar: Loops in hypertoric varieties and symplectic duality 9:30 AM-10:30 AM June 16, 2020 Hypertoric varieties are algebraic symplectic varieties associated to graphs, or more generally certain hyperplane arrangements. They make many appearances in modern geometric representation theory. I will discuss certain infinite dimensional or infinite type generalizations of hypertoric varieties which occur in the study of enumerative invariants, focusing on some elementary examples. Joint work with Artan Sheshmani and Shing-Tung Yau. • MATHEMATICAL PICTURE LANGUAGE SEMINAR ##### MATHEMATICAL PICTURE LANGUAGE SEMINARThe power of adiabatic quantum computation with no sign problem 10:00 AM-11:00 AM June 16, 2020 Interference is an essential part of quantum mechanics. However, an important class of Hamiltonians considered are those with “no sign problem”, where all off-diagonal matrix elements of the Hamiltonian are non-negative. This means that the ground state wave function can be chosen to have all amplitudes real and positive. In a sense, no destructive interference is possible for these Hamiltonians so that they are “almost classical”, and there are several simulation algorithms which work well in practice on classical computers today. In this talk, I’ll discuss what happens when one considers adiabatic evolution of such Hamiltonians, and show that they still have some power that cannot be efficiently simulated on a classical computer; to be precise and formal, I’ll show this “relative to an oracle”, which I will explain. I’ll discuss implications for simulation of these problems and open questions. 17 • CMSA EVENT: CMSA Quantum Matter/Quantum Field Theory Seminar: QCD(adj) and deformed Yang-Mills: From weak coupling confinement to adiabatic continuity ##### CMSA EVENTCMSA Quantum Matter/Quantum Field Theory Seminar: QCD(adj) and deformed Yang-Mills: From weak coupling confinement to adiabatic continuity 10:30 AM-12:00 PM June 17, 2020 More than 10 years ago, I showed that non-supersymmetric QCD with adjoint fermions admits a (non-thermal) compactification on$R^3 \times S^1$where non-perturbative gauge dynamics becomes calculable. The mass gap, linear confinement and discrete chiral symmetry breaking are sourced by magnetic bions, which are correlated monopole-instanton anti-instanton pairs which have non-vanishing magnetic charge but have zero topological charge. This construction led to the idea of the double-trace deformation of pure Yang-Mills theory on$R^3 x S^1$, which is continuously connected to pure YM on$R^4$in the sense of continuity of all gauge invariant order parameters. It turns out that all qualitative non-perturbative properties of deformed YM theory are in agreement with all of our non-perturbative expectations concerning pure YM theory. Over the last year, numerical simulations have shown that the topological susceptibility of deformed YM on small$S^1 \times R^3$are in precise agreement with the numerical results on large$S^1 \times R^3\$. Therefore, it is very likely that there is more truth in deformed YM construction than that meets the eye. I will give a lecture style talk on these topics.

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• CMSA EVENT: CMSA Condensed Matter/Math Seminar: Foliated QFT and Topological Defect Networks of Fracton Order

##### CMSA EVENTCMSA Condensed Matter/Math Seminar: Foliated QFT and Topological Defect Networks of Fracton Order

10:30 AM-12:00 PM
June 18, 2020

I will describe two new descriptions of gapped fracton topological order. These descriptions are generic and make minimal modifications to ordinary TQFT in order to obtain fracton physics. The first description constructs a fracton order by embedding a network of topological defects (aka interfaces) within an ordinary topological order, which results in the restricted fracton mobility. The second description takes the continuum limit by viewing the defect layers as infinitesimally separated. This is done by coupling a TQFT, such as BF theory, to a new kind of foliated gauge field. The first description is based on arXiv:2002.05166, while the second is based on a forthcoming work.

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• CMSA EVENT: CMSA Geometry and Physics Seminar – The geometry of Maurer–Cartan equation near degenerate Calabi–Yau varieties

##### CMSA EVENTCMSA Geometry and Physics Seminar – The geometry of Maurer–Cartan equation near degenerate Calabi–Yau varieties

9:30 AM-10:30 AM
June 22, 2020

In this talk, we construct a dgBV algebra PV*(X) associated to a possibly degenerate Calabi–Yau variety X equipped with local thickening data. This gives a version of the Kodaira–Spencer dgLa which is applicable to degenerated spaces including both log smooth or maximally degenerated Calabi–Yau. We use this to prove an unobstructedness result about the smoothing of degenerated Log Calabi–Yau varieties X satisfying Hodge–deRham degeneracy property for cohomology of X, in the spirit of Kontsevich–Katzarkov–Pantev. This is a joint work with Kwokwai Chan and Naichung Conan Leung.

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• MATHEMATICAL PICTURE LANGUAGE SEMINAR

##### MATHEMATICAL PICTURE LANGUAGE SEMINARQuantum Fourier analysis

10:00 AM-11:00 AM
June 23, 2020

Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory and topological quantum field theory) with analytic estimates. We give an overview of quantum Fourier analysis in this talk. We highlight an application in recent work joint with Sebastien Palcoux and Jinsong Wu: we find new, surprisingly efficient, analytic obstructions of unitary categorification of fusion rings by applying quantum Fourier analysis to the Drinfeld center of unitary fusion categories.

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• CMSA EVENT: CMSA Quantum Matter/Quantum Field Theory Seminar: Holomorphic anomaly in Vafa-Witten theory

##### CMSA EVENTCMSA Quantum Matter/Quantum Field Theory Seminar: Holomorphic anomaly in Vafa-Witten theory

9:30 AM-11:00 AM
June 24, 2020
Vafa-Witten theory is a topologically twisted version of 4d N=4 super

Yang-Mills theory. In my talk I will tell how to derive a holomorphic
anomaly equation for its partition function on CP^2 by two different,
but somewhat analogous, methods. First is the derivation from the path
integral of the effective theory on the Coulomb branch. The second is
the derivation from the path-integral in the effective 2d theory
obtained by compactification of the corresponding 6d (2,0) theory on the
ternsor branch.

examine the attendees the zoombombing repeating)

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• CMSA EVENT: CMSA Condensed Matter/Math Seminar: Insights into type-II fractons via topological order

##### CMSA EVENTCMSA Condensed Matter/Math Seminar: Insights into type-II fractons via topological order

9:30 AM-11:00 AM
June 25, 2020

“Type-II” fracton phases are exotic, gapped phases of matter in which all point-like excitations are deconfined but completely immobile. We present two approaches to constructing and analyzing models for type-II fractons using tools from conventional topological order. First, we use networks of defects in topological quantum field theories to describe Haah’s B code, which is a type-II fracton model. We use this fact as evidence for a conjecture that all fracton models can be described by a suitable defect network. Second, we show how gauging global symmetries of Abelian type-II fracton models produces new, exactly solvable models with non-Abelian type-II fractons, and we analyze the unusual excitations.

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• MATHEMATICAL PICTURE LANGUAGE SEMINAR

##### MATHEMATICAL PICTURE LANGUAGE SEMINARA hidden variable model for universal quantum computation with magic states on qubits

10:00 AM-11:00 AM
June 30, 2020

We show that every quantum computation can be described by Bayesian update of a probability distribution on a finite state space. Negativity in a quasiprobability function is not required, neither in states nor the operations. Our result is consistent with Gleason’s Theorem and the PBR theorem. This is joint work with Michael Zurel and Cihan Okay; see arXiv:2004.01992.

• CMSA EVENT: CMSA Geometry and Physics Seminar: 3-manifolds, q-series, and topological strings

##### CMSA EVENTCMSA Geometry and Physics Seminar: 3-manifolds, q-series, and topological strings

9:30 PM-10:30 PM
June 30, 2020

\hat{Z} is an invariant of 3-manifolds valued in q-series (i.e. power series in q with integer coefficients), which has interesting modular properties. While originally from physics, this invariant has been mathematically constructed for a big class of 3-manifolds, and conjecturally it can be extended to all 3-manifolds. In this talk, I will give a gentle introduction to \hat{Z} and what is known about it, as well as highlighting some recent developments, including the use of R-matrix, generalization to higher rank, large N-limit and interpretation as open topological string partition functions.

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