## news

##### Undergraduate Prizes and Awards 2021-2022

Congratulations to this year’s prize and award recipients! Thomas Temple Hoopes Prize From the estate of Thomas T. Hoopes, Class of 1919, Harvard received a...

##### Mark Kisin Elected to American Academy of Arts and Sciences

We are thrilled to announce that Perkins Professor of Mathematics and Director of Graduate Studies Mark Kisin is among sixteen Harvard faculty elected to the...

##### Demystifying Math 55

By Anastasia Yefremova Few undergraduate level classes have the distinction of nation-wide recognition that Harvard University’s Math 55 has. Officially comprised of Mathematics 55A “Studies...

##### Math Question Center helps undergraduates with PSets

If you're a Harvard University student taking an entry-level math course and you haven't heard of the Math Question Center (MQC) yet, you could be...

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## announcements

##### **New Dates: June 15-17, 2022**: THE CIRCLE AT INFINITY: AN INTERNATIONAL COLLOQUIUM IN HONOR OF CURTIS T. MCMULLEN
June 15, 2022 - June 17, 2022
New Dates: June 15-17, 2022 Conference Poster Confirmed speakers: Serge Cantat, Université de Rennes I Danny Calegari, University of Chicago Laura DeMarco, Harvard...
##### Advances in Mathematical Physics: A Conference in Honor of Elliott H. Lieb on his 90th Birthday.
July 30, 2022 - August 1, 2022
Advances in Mathematical Physics A Conference in Honor of Elliott H. Lieb on his 90th Birthday Dates: July 30-August 1, 2022 Harvard University July 30...
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## upcoming events

• June 6, 2022 - June 8, 2022
2022 Symposium on Foundations of Responsible Computing (FORC)

CMSA EVENT

• June 15, 2022 - June 17, 2022
Science Center Hall C

ANNOUNCEMENTS, CONFERENCE

• June 21, 2022 - June 24, 2022
CMSA, 20 Garden St, G10

CMSA EVENT

< 2020 >
December
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• MATHEMATICAL PICTURE LANGUAGE SEMINAR

##### MATHEMATICAL PICTURE LANGUAGE SEMINARTopological order, tensor networks and subfactors

10:00 AM-11:00 AM
December 1, 2020

We present recent progress on studies of 2-dimensional topological order in terms of tensor networks and its connections to subfactor theory. We explain how Drinfel’d centers and higher relative commutants naturally appear in this context and use of picture language in this study.

• CMSA EVENT: CMSA Computer Science for Mathematicians: Some extensions on argumentation frameworks via hypergraphs

##### CMSA EVENTCMSA Computer Science for Mathematicians: Some extensions on argumentation frameworks via hypergraphs

11:30 AM-12:30 PM
December 1, 2020
The Dung Abstract Argumentation Framework (AAF) is an effective formalism for modelling disputes between two or more agents. Generally, the Dung AF is extended to include some unique interactions between agents. This has further been explained with the Bipolar Argumentation Framework (BAF). In the academic space, the use of AAF is highly signified. We can use the AF as a means to resolve disagreements that allows for the determination of a winning argument. In general, there can be imperfect ontologies that affect how reasoning is defined. Typical logic-based AFs apply to the incoherent/uncertain ontologies. However, Dung demonstrated a stable extension of AF to support an “acceptable standard of behavior”. This talk will align with present endeavors on extending the Dung AAF to consider the notion of conflict-freeness in relation to persistence over a hypergraph. With a generic type of argumentation, there are some methods that can exploit certain complex decision procedures. Argument and attack relations within the Dung AAF, thus are further defined to obtain a graphical formula of Kripke groundedness. The incorporating of multiple levels of knowledge aligns with a computational linguistics aspect for the defining of a classification criteria for AAF. In the construction, I will provide some treatment of ‘good’ model-theoretic properties that bridge AAF with Zarankiewicz’s problem to introduce how arguments are consistent with bipartite hypergraphs. The Zarankiewicz problem appears with the communication complexity on AF graphs.
• HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

##### HARVARD-MIT ALGEBRAIC GEOMETRY SEMINARPositroid varieties and q,t-Catalan numbers

3:00 PM-4:00 PM
December 1, 2020

Positroid varieties are subvarieties of the Grassmannian obtained by intersecting cyclic rotations of Schubert varieties.  We show that the “top open positroid variety” has mixed Hodge polynomial given by the q,t-rational Catalan numbers (up to a simple factor).  Unlike the Grassmannian, the cohomology of open positroid varieties is not pure.

The q,t-rational Catalan numbers satisfy remarkable symmetry and unimodality properties, and these arise from the Koszul duality phenomenon in the derived category of the flag variety, and from the curious Lefschetz phenomenon for cluster varieties.  Our work is also related to knot homology and to the cohomology of compactified Jacobians.

This talk is based on joint work with Pavel Galashin.

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• CMSA EVENT: CMSA Math Science Literature Lecture Series

##### CMSA EVENTCMSA Math Science Literature Lecture Series

8:00 AM-9:30 AM
December 2, 2020

TITLE: Is relativity compatible with quantum theory?

ABSTRACT: We review the background, mathematical progress, and open questions in the effort to determine whether one can combine quantum mechanics, special relativity, and interaction together into one mathematical theory. This field of mathematics is known as “constructive quantum field theory.” Physicists believe that such a theory describes experimental measurements made over a 70 year period and now refined to 13-decimal-point precision—the most accurate experiments ever performed.

Talk chair: Zhengwei Liu

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

##### Register here to attend.
• CMSA EVENT: CMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal

##### CMSA EVENTCMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal

12:00 PM-1:30 PM
December 2, 2020

I discuss the interplay between non-Fermi liquid behaviour and pairing near a quantum-critical point (QCP) in a metal. These tendencies are intertwined in the sense that both originate from the same interaction mediated by gapless fluctuations of a critical order parameter. The two tendencies compete because fermionic incoherence destroys the Cooper logarithm, while the pairing eliminates scattering at low energies and restores fermionic coherence. I discuss this physics for a class of models with an effective dynamical interaction V (Ω) ~1/|Ω|^γ (the γ-model). This model describes, in particular, the pairing at a 2D Ising-nematic critical point in (γ=1/3), a 2D antiferromagnetic critical point (γ=1/2) and the pairing by an Einstein phonon with vanishing dressed Debye frequency (γ=2). I argue the pairing wins, unless the pairing component of the interaction is artificially reduced, but because of fermionic incoherence in the normal state, the system develops a pseudogap, preformed pairs behaviour in the temperature range between the onset of the pairing at Tp and the onset of phase coherence at the actual superconducting Tc. The ratio Tc/Tp decreases with γ and vanishes at γ =2. I present two complementary arguments of why this happens. One is the softening of longitudinal gap fluctuations, which become gapless at γ =2. Another is the emergence of a 1D array of dynamical vortices, whose number diverges at γ =2. I argue that once the number of vortices becomes infinite, quasiparticle energies effectively get quantized and do not get re-arranged in the presence of a small phase variation. I show that a new non-superconducting ground state emerges at γ >2.

• RANDOM MATRIX SEMINAR

##### RANDOM MATRIX SEMINARJoint Dept. of Mathematics and CMSA Random Matrix & Probability Theory Seminar: Thermodynamics of a hierarchical mixture of cubes

2:00 PM-3:00 PM
December 2, 2020

The talk discusses a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes of side-lengths 2^j, j\in \N_0. Cubes belong to an admissible set such that if two cubes overlap, then one cube is contained in the other, a picture reminiscent of Mandelbrot’s fractal percolation model. I will present exact formulas for the entropy and pressure, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and briefly sketch some broader questions on renormalization and cluster expansions that motivate the model. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309-340).

• NUMBER THEORY SEMINAR

##### NUMBER THEORY SEMINARThe 2-torsion subgroups of the class groups in families of cubic fields

3:00 PM-4:00 PM
December 2, 2020

The Cohen–Lenstra–Martinet conjectures have been verified in
only two cases. Davenport–Heilbronn compute the average size of the
3-torsion subgroups in the class group of quadratic fields and Bhargava
computes the average size of the 2-torsion subgroups in the class groups of cubic fields. The values computed in the above two results are remarkably stable. In particular, work of Bhargava–Varma shows that they do not change if one instead averages over the family of quadratic or cubic fields satisfying any finite set of splitting conditions.

However for certain “thin” families of cubic fields, namely, families of
monogenic and n-monogenic cubic fields, the story is very different. In
this talk, we will determine the average size of the 2-torsion subgroups of
the class groups of fields in these thin families. Surprisingly, these
values differ from the Cohen–Lenstra–Martinet predictions! We will also
provide an explanation for this difference in terms of the Tamagawa numbers of naturally arising reductive groups. This is joint work with Manjul Bhargava and Jon Hanke.

Password: The order of the permutation group on 9 elements.

• MATH TABLE

##### MATH TABLEMath Table/Open Neighborhood Seminar: Why are p-adic numbers useful for fast linear algebra algorithms?

4:30 PM-5:30 PM
December 2, 2020

p-Adic numbers have always been primarily associated with pure Mathematics, and have become especially relevant in algebra and modern number theory. But why did Computer Scientists become interested in them? In this talk we will introduce p-adic numbers and survey their main properties. We will then introduce Dixon’s algorithm, which is the first algorithm that used p-adic numbers to compute the exact rational solution to an integer linear system of equations. We will also explore the latest runtime improvements in p-adic linear algebra algorithms, and discuss whether we can solve linear equation systems faster than matrix multiplication.

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• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Domain Wall Fermions and Chiral Gauge theories: Topological Insulators in Particle Physic

##### CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: Domain Wall Fermions and Chiral Gauge theories: Topological Insulators in Particle Physic

10:30 AM-12:00 PM
December 3, 2020

Ideas from the early 1990s for regulating chiral fermions in lattice gauge theory led to a number of developments which paralleled roughly concurrent and independent discoveries in condensed matter physics.  I show how the Integer Quantum Hall Effect, Chern Insulators, Topological Insulators, and Majorana edge states all play a role in lattice gauge theories, and how one can also find relativistic versions of the Fractional Quantum Hall Effect, the Quantum Spin Hall Effect and related exotic forms of matter.  How to construct a nonperturbative regulator for chiral gauge theories (like the Standard Model!)  remains an open challenge, however, one that may require new insights from condensed matter physics into exotic states of matter.

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• CMSA EVENT: CMSA Math Science Literature Lecture Series

##### CMSA EVENTCMSA Math Science Literature Lecture Series

8:00 AM-9:30 AM
December 4, 2020

TITLE: Michael Atiyah: Geometry and Physics

ABSTRACT: In mid career, as an internationally renowned mathematician, Michael Atiyah discovered that some problems in physics responded to current work in algebraic geometry and this set him on a path to develop an active interface between mathematics and physics which was formative in the links which are so active today. The talk will focus, in a fairly basic fashion, on some examples of this interaction, which involved both applying physical ideas to solve mathematical problems and introducing mathematical ideas to physicists.

Talk chair: Peter Kronheimer

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

##### Register here to attend.
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• CMSA EVENT: CMSA Mathematical Physics Seminar: Moduli Space Holography and the Finiteness of Flux Vacua

##### CMSA EVENTCMSA Mathematical Physics Seminar: Moduli Space Holography and the Finiteness of Flux Vacua

10:30 AM-11:30 AM
December 7, 2020

In this talk I describe a holographic perspective to study field spaces that arise in string compactifications. The constructions are motivated by a  general description of the asymptotic, near-boundary regions in complex structure moduli spaces of Calabi-Yau manifolds using asymptotic Hodge theory. For real two-dimensional field spaces, I introduce an auxiliary bulk theory and describe aspects of an associated sl(2) boundary theory. The bulk reconstruction from the boundary data is provided by the sl(2)-orbit theorem of Schmid and Cattani, Kaplan, Schmid, which is a famous and general result in Hodge theory. I then apply this correspondence to the flux landscape of Calabi-Yau fourfold compactifications and discuss how this allows us, in work with C. Schnell, to prove that the number of self-dual flux vacua is finite.

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• MATHEMATICAL PICTURE LANGUAGE SEMINAR

##### MATHEMATICAL PICTURE LANGUAGE SEMINARSymmetry as a shadow of topological order

10:00 AM-11:00 AM
December 8, 2020

In this talk, I will present a new way to look at symmetry. We show that symmetry can be viewed as a non-invertible gravitational anomaly, and a non-invertible gravitational anomaly is classified by topological order in one higher dimension. This leads to a holographic view of symmetry:  symmetry is a shadow of topological order in one higher dimension. This point of view allows us to see the duality (i.e. the equivalence) between symmetries that look very different. It also gives rise to a more general symmetry – algebraic higher symmetry, which is beyond higher group description.

• HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

##### HARVARD-MIT ALGEBRAIC GEOMETRY SEMINARThe cohomology of moduli of curves at infinite level

3:00 PM-4:00 PM
December 8, 2020

Moduli spaces of curves admit finite covers by moduli spaces which parametrize curves together with so-called level structures. In my talk, I will discuss how the cohomology of these spaces at infinite level is related to a profinite property of the mapping class group. I will then explain why tools from p-adic geometry yield vanishing statements for these cohomologies in high degree.

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• CMSA EVENT: CMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: Signatures of anomalous symmetry breaking in the cuprates

##### CMSA EVENTCMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: Signatures of anomalous symmetry breaking in the cuprates

10:30 AM-12:00 PM
December 9, 2020

The temperature versus doping phase diagram of the cuprate high-Tsuperconductors features an enigmatic pseudogap region whose microscopic origin remains a subject of intensive study. Experimentally resolving its symmetry properties is imperative for narrowing down the list of possible explanations. In this talk I will give an overview of how optical second harmonic generation (SHG) can be used as a sensitive probe of symmetry breaking, and recap the ways it has been used to solve outstanding problems in condensed matter physics. I will then describe how we have been applying SHG polarimetry and spectroscopy to interrogate the cuprate pseudogap. In particular, I will discuss our data on YBa2Cu3Oy [1], which show an order parameter-like increase in SHG intensity below the pseudogap temperature T* across a broad range of doping levels. I will then focus on our more recent results on a model parent cuprate Sr2CuO2Cl2 [2], where evidence of anomalous broken symmetries surprisingly also exists. Possible connections between these observations will be speculated upon.

[1] L. Zhao, C. A. Belvin, R. Liang, D. A. Bonn, W. N. Hardy, N. P. Armitage and D. Hsieh, “A global inversion-symmetry-broken phase inside the pseudogap region of YBa2Cu3Oy,” Nature Phys. 13, 250 (2017).

[2] A. de la Torre, K. L. Seyler, L. Zhao, S. Di Matteo, M. S. Scheurer, Y. Li, B. Yu, M. Greven, S. Sachdev, M. R. Norman and D. Hsieh. “Anomalous mirror symmetry breaking in a model insulating cuprate Sr2CuO2Cl2,” Preprint at https://arxiv.org/abs/2008.06516.

• NUMBER THEORY SEMINAR

##### NUMBER THEORY SEMINARHermite interpolation and counting number fields

3:00 PM-4:00 PM
December 9, 2020

There are several ways to specify a number field. One can provide the minimal polynomial of a primitive element, the multiplication table of a $\bf Q$-basis, the traces of a large enough family of elements, etc. From any way of specifying a number field one can hope to deduce a bound on the number $N_n(H)$ of number fields of given degree $n$ and discriminant bounded by $H$. Experimental data suggest that the number of isomorphism classes of number fields of degree $n$ and discriminant bounded by $H$ is equivalent to $c(n)H$ when $n\geqslant 2$ is fixed and $H$ tends to infinity. Such an estimate has been proved for $n=3$ by Davenport and Heilbronn and for $n=4$, $5$ by Bhargava. For an arbitrary $n$ Schmidt proved a bound of the form $c(n)H^{(n+2)/4}$ using Minkowski’s theorem. Ellenberg et Venkatesh have proved that the exponent of $H$ in $N_n(H)$ is less than sub-exponential in $\log (n)$. I will explain how Hermite interpolation (a theorem of Alexander and Hirschowitz) and geometry of numbers combine to produce short models for number fields and sharper bounds for $N_n(H)$.

Password: The order of the permutation group on 9 elements.

• CMSA EVENT: CMSA New Technologies in Mathematics: Machine learning and SU(3) structures on six manifolds

##### CMSA EVENTCMSA New Technologies in Mathematics: Machine learning and SU(3) structures on six manifolds

3:00 PM-4:00 PM
December 9, 2020

In this talk we will discuss the application of Machine Learning techniques to obtain numerical approximations to various metrics of SU(3) structure on six manifolds. More precisely, we will be interested in SU(3) structures whose torsion classes make them suitable backgrounds for various string compactifications. A variety of aspects of this topic will be covered. These will include learning moduli dependent Ricci-Flat metrics on Calabi-Yau threefolds and obtaining numerical approximations to torsional SU(3) structures.

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• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: An analytic bootstrap approach for CFTs on RP^d and CFTs with boundaries

##### CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: An analytic bootstrap approach for CFTs on RP^d and CFTs with boundaries

10:30 AM-12:00 PM
December 10, 2020

In this talk, I will introduce an analytic bootstrap approach for two-point correlation functions in CFTs on real projective space, and CFTs with a conformal boundary. We will use holography as a kinematical tool to derive universal results. By examining the conformal block decomposition properties of exchange diagrams in AdS space, we identify a useful new basis for decomposing correlators. The dual basis gives rise to a basis of functionals, whose actions we can compute explicitly via holography. Applying these functionals to the crossing equations, we can systematically extract constraints on the CFT data in the form of sum rules. I will demonstrate this analytic method in the canonical example of \phi^4 theory in d=4-\epsilon, fixing the CFT data to \epsilon^2.

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• MATHEMATICAL PICTURE LANGUAGE SEMINAR

##### MATHEMATICAL PICTURE LANGUAGE SEMINARTopology in shallow-water waves: A violation of bulk-edge correspondence

10:00 AM-11:00 AM
December 15, 2020

A two-dimensional rotating shallow-water model describes a layer of water, in guise of oceans covering the Earth. It is formally analogue to a Schrödinger equation where the tools from topological insulators are relevant. Once regularized at small scale by an odd-viscous term, such a model has a well-defined bulk topological index. However, in presence of a sharp boundary, the number of edge modes depends on the boundary condition, showing an explicit violation of the bulk-edge correspondence. We study a continuous family of boundary conditions with a rich phase diagram, and explain the origin of this mismatch. Our approach relies on scattering theory and Levinson’s theorem. The latter does not apply at infinite momentum because of the analytic structure of the scattering amplitude there, which is ultimately the reason for the violation. (Joint work with H. Jud and C. Tauber.)

• CMSA EVENT: CMSA Computer Science for Mathematicians: Lightning Network Economics: Cost-minimal channels and their implications for network structure

##### CMSA EVENTCMSA Computer Science for Mathematicians: Lightning Network Economics: Cost-minimal channels and their implications for network structure

11:30 AM-12:30 PM
December 15, 2020

The Lightning Network is a second-layer solution built above Bitcoin, aimed to solve Bitcoin’s scalability and immediacy problems. A channel in the Lightning Network allows two parties to secure bitcoin payments and escrow holdings between them. Designed to increase transaction immediacy and reduce blockchain congestion, this has the potential to solve many issues associated with Bitcoin.

In this talk, we study the economics of the Lightning Network. We present conditions under which two parties optimally establish a channel and give explicit formulas for channels’ costs. Using these, we derive implications for the network’s structure under cooperation assumptions among small sets of users. We show both local implications, such as the wastefulness of certain structures, and global implications, such as a (low) upper bound on the Lightning Network’s average degree.

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• CMSA EVENT: CMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: Organizing Principle of Mottness and Complex Phenomenon in High Temperature Superconductors

##### CMSA EVENTCMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: Organizing Principle of Mottness and Complex Phenomenon in High Temperature Superconductors

10:30 AM-12:00 PM
December 16, 2020

The complex phenomenon in the high-Tc cuprate calls for a microscopic understanding based on general principles. In this Lecture, an exact organizing principle for a typical doped Mott insulator will be presented, in which the fermion sign structure is drastically reduced to a mutual statistics. Its nature as a long-range spin-charge entanglement of many-body quantum mechanics will be exemplified by exact numerical calculations. The phase diagram of the cuprate may be unified in a “bottom-up” fashion by a “parent” ground state ansatz with hidden orders constructed based on the organizing principle. Here the pairing mechanism will go beyond the “RVB” picture and the superconducting state is of non-BCS nature with modified London equation and novel elementary excitations. In particular, the Bogoliubov/Landau quasiparticle excitation are emerging with a two-gap structure in the superconducting state and the Fermi arc in a pseudogap regime. A mathematic framework of fractionalization and duality transformation guided by the organizing principle will be introduced to describe the above emergent phenomenon.

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• MATHEMATICAL PICTURE LANGUAGE SEMINAR

##### MATHEMATICAL PICTURE LANGUAGE SEMINARBrain-storming in Math and Physics–dedicated to Arthur Jaffe

10:00 AM-11:00 AM
December 22, 2020

Imagine that we are in Arthur’s office, and I will give an informal lecture. This lecture has no main theme; instead I will talk about some insightful examples and fresh ideas inspired by math and physics—as well as some potential connections between different areas.

• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Gapped Chiral Fermions

##### CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: Gapped Chiral Fermions

10:30 AM-12:00 PM
December 22, 2020

I’ll describe some quantum field theories that gap fermions without breaking chiral symmetries.

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