Harvard University Mathematics Department Cambridge MA

## news

##### Alexander Smith Awarded Clay Research Fellowship

Alexander Smith, 2020 Ph.D. recipient, has been named a 2021 Clay Research Fellow. Clay Research Fellows are selected for their research achievements and their potential...

##### Special Colloquium Series–Spring 2021

Spring 2021 Special Colloquium Series: Talks-Videos-Registration All talks on Zoom 3-4pm • February 23, 2021 Allan Sly -- Princeton University Replica Symmetry Breaking for Random...

See Older News

## upcoming events

• March 8, 2021
3:00 pm
Virtually

COLLOQUIUMS

Speaker: Melissa (Chiu-Chu) Liu - Columbia University   Title: Special Colloquium
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• CMSA EVENT: CMSA Mathematical Physics Seminar: Mathematical supergravity and its applications to differential geometry

##### CMSA EVENTCMSA Mathematical Physics Seminar: Mathematical supergravity and its applications to differential geometry

10:00 AM-11:00 AM
March 1, 2021

I will discuss the recent developments in the mathematical theory of supergravity that lay the mathematical foundations of the universal bosonic sector of four-dimensional ungauged supergravity and its Killing spinor equations in a differential-geometric framework.  I will provide the necessary context and background. explaining the results pedagogically from scratch and highlighting several open mathematical problems which arise in the mathematical theory of supergravity, as well as some of its potential mathematical applications. Work in collaboration with Vicente Cortés and Calin Lazaroiu.

• COLLOQUIUMS

##### COLLOQUIUMSSpecial Colloquium

3:00 PM-4:00 PM
March 1, 2021

Title: Robustness Meets Algorithms

Abstract: Starting from the seminal works of Tukey (1960) and Huber (1964), the field of robust statistics asks: Are there estimators that probably work in the presence of noise? The trouble is that all known provably robust estimators are also hard to compute in high-dimensions.

Here, we study a fundamental problem in robust statistics, posed in various forms in the above works. Given corrupted samples from a high-dimensional Gaussian, are there efficient algorithms to accurately estimate its parameters? We give the first algorithm that is able to tolerate a constant fraction of corruptions that is independent of the dimension. Moreover, we give a general recipe for detecting and correcting corruptions based on tensor-spectral techniques that are applicable to many other problems.

I will also discuss how this work fits into the broader agenda of developing mathematical and algorithmic foundations for modern machine learning.

Registration is required to receive the Zoom information

Register here to attend

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• DIFFERENTIAL GEOMETRY SEMINAR

##### DIFFERENTIAL GEOMETRY SEMINARDisc potential functions of Quadrics

8:00 AM-9:00 AM
March 2, 2021

A disc potential function plays an important role in studying a symplectic manifold and its Lagrangian submanifolds. In this talk, I will explain how to compute the disc potential function of quadrics. The potential function provides the Landau—Ginzburg mirror, which agrees with Przyjalkowski’s mirror and a cluster chart of Pech—Rietsch—Williams’ mirror

• MATHEMATICAL PICTURE LANGUAGE SEMINAR

##### MATHEMATICAL PICTURE LANGUAGE SEMINARIntegrability of Liouville Theory

10:00 AM-11:00 AM
March 2, 2021

Polyakov introduced Liouville Conformal Field theory (LCFT) in 1981 as a way to put a naturalmeasure on the set of Riemannian metrics over a two dimensional manifold. Ever since, the work of Polyakov has echoed in various branches of physics and mathematics, ranging from string theory to probability theory and geometry. In the context of 2D quantum gravity models, LCFT is related through the Knizhnik-Polyakov-Zamolodchikov relationsto the scaling limit of Random Planar Maps and through the Alday-Gaiotto-Tachikava correspondence LCFT is conjecturally related to certain 4D Yang-Mills theories. Through the work of Dorn, Otto, Zamolodchikov and Zamolodchikov and Teschner LCFT is believed to be to a certain extent integrable. I will review a probabilistic construction of LCFT and recent proofs concerning the integrability of LCFT developed together with F. David, C. Guillarmou, R. Rhodes and V. Vargas.

• CMSA EVENT: CMSA Computer Science for Mathematicians: Randomized Dimensionality Reduction for Clustering

##### CMSA EVENTCMSA Computer Science for Mathematicians: Randomized Dimensionality Reduction for Clustering

11:30 AM-12:30 PM
March 2, 2021

Random dimensionality reduction is a versatile tool for speeding up algorithms for high-dimensional problems. We study its application to two clustering problems: the facility location problem, and the single-link hierarchical clustering problem, which is equivalent to computing the minimum spanning tree. We show that if we project the input pointset $X$ onto a random $d = O(d_X)$-dimensional subspace (where $d_X$ is the doubling dimension of $X$), then the optimum facility location cost in the projected space approximates the original cost up to a constant factor. We show an analogous statement for minimum spanning tree, but with the dimension $d$ having an extra $\log \log n$ term and the approximation factor being arbitrarily close to $1$. Furthermore, we extend these results to approximating solutions instead of just their costs. Lastly, we provide experimental results to validate the quality of solutions and the speedup due to the dimensionality reduction.

Unlike several previous papers studying this approach in the context of $k$-means and $k$-medians, our dimension bound does not depend on the number of clusters but only on the intrinsic dimensionality of $X$.

Joint work with Shyam Narayanan, Piotr Indyk, Or Zamir.

• HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

##### HARVARD-MIT ALGEBRAIC GEOMETRY SEMINARDecomposition theorem for semisimple local systems

3:00 PM-4:00 PM
March 2, 2021

In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of long papers via harmonic analysis and D-modules. In this talk, I would like to explain a more geometric/topological approach in the case of semisimple local systems adapting de Cataldo-Migliorini. As a byproduct, we can recover a weak form of Saito’s decomposition theorem for variations of Hodge structures. Joint work in progress with Chuanhao Wei.

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• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Symmetry-protected sign problem and magic in quantum phases of matter

##### CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: Symmetry-protected sign problem and magic in quantum phases of matter

10:30 AM-12:00 PM
March 3, 2021

We introduce the concepts of a symmetry-protected sign problem and symmetry-protected magic, defined by the inability of symmetric finite-depth quantum circuits to transform a state into a nonnegative real wave function and a stabilizer state, respectively. We show that certain symmetry protected topological (SPT) phases have these properties, as a result of their anomalous symmetry action at a boundary. For example, one-dimensional Z2 × Z2 SPT states (e.g. cluster state) have a symmetry-protected sign problem, and two-dimensional Z2 SPT states (e.g. Levin-Gu state) have both a symmetry-protected sign problem and magic. We also comment on the relation of a symmetry-protected sign problem to the computational wire property of one-dimensional SPT states and the greater implications of our results for measurement based quantum computing.

• CMSA EVENT: CMSA New Technologies in Mathematics: Neural Theorem Proving in Lean using Proof Artifact Co-training and Language Models

##### CMSA EVENTCMSA New Technologies in Mathematics: Neural Theorem Proving in Lean using Proof Artifact Co-training and Language Models

3:00 PM-4:00 PM
March 3, 2021

Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer language models to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT (Proof Artifact Co-Training), a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer language model and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32\% to 48\%.

• NUMBER THEORY SEMINAR

##### NUMBER THEORY SEMINAREquidistribution and Uniformity in Families of Curves

3:00 PM-4:00 PM
March 3, 2021

In the talk, I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, namely that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in a few select cases by work of David–Philippon and DeMarco–Krieger–Ye. Finally, one can obtain a rather uniform version of the Mordell-Lang conjecture as well by complementing a result of Dimitrov–Gao–Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz–Rabinoff–Zureick-Brown).

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

• OPEN NEIGHBORHOOD SEMINAR

##### OPEN NEIGHBORHOOD SEMINARSave the Pilgrim!

4:30 PM-5:30 PM
March 3, 2021

An evil mathematician has kidnapped the Harvard Pilgrim! To win his freedom, a group of undergrads must each find their name in a row of boxes. The odds look dire—but we’ll use some probability theory and combinatorics to find a strategy that dramatically improves our chances. Can you help save our hapless mascot?

Please go to the College Calendar to register.

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• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Generalized ‘t Hooft anomalies in vector-like theories

##### CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: Generalized ‘t Hooft anomalies in vector-like theories

10:30 AM-12:00 PM
March 4, 2021

‘t Hooft anomalies provide a unique handle to study the nonperturbative infrared dynamics of strongly-coupled theories.  Recently, it has been realized that higher-form global symmetries can also become anomalous, leading to further constraints on the infrared dynamics.  In this talk, I show how one can turn on ‘t Hooft twists in the color, flavor, and baryon number directions in vector-like asymptotically-free gauge theories, which can be used to find new generalized ‘t Hooft anomalies. I give examples of the constraints the generalized anomalies impose on strongly-coupled gauge theories. Then, I argue that the anomaly inflow can explain a non-trivial intertwining that takes place between the light and heavy degrees of freedom on axion domain walls, which leads to the deconfinement of quarks on the walls.  This phenomenon can be explicitly seen in a weakly-coupled model of QCD compactified on a small circle.

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• ALGEBRAIC DYNAMICS SEMINAR

##### ALGEBRAIC DYNAMICS SEMINARIrreducibility of periodic curves of cubic polynomials

10:00 AM-12:00 PM
March 5, 2021

In the moduli space of one variable complex cubic polynomials with a marked critical point, given any $p \ge 1$, we prove that the loci formed by polynomials with the marked critical point periodic of period $p$ is an irreducible curve.  Our methods rely on techniques used to study one-complex-dimensional parameter spaces.  This is joint work with Matthieu Arfeux.

Go to http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for Zoom information.

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• CMSA EVENT: CMSA Mathematical Physics Seminar: Virasoro constraints for stable pairs

##### CMSA EVENTCMSA Mathematical Physics Seminar: Virasoro constraints for stable pairs

10:00 AM-11:00 AM
March 8, 2021

The theory of stable pairs (PT) with descendents, defined on a 3-fold X, is a sheaf theoretical curve counting theory. Conjecturally, it is equivalent to the Gromov-Witten (GW) theory of X via a universal (but intricate) transformation, so we can expect that the Virasoro conjecture on the GW side should have a parallel in the PT world. In joint work with A. Oblomkov, A. Okounkov, and R. Pandharipande, we formulated such a conjecture and proved it for toric 3-folds in the stationary case. The Hilbert scheme of points on a surface S might be regarded as a component of the moduli space of stable pairs on S x P1, and the Virasoro conjecture predicts a new set of relations satisfied by tautological classes on S[n] which can be proven by reduction to the toric case.

• COLLOQUIUMS

##### COLLOQUIUMSSpecial Colloquium

3:00 PM-4:00 PM
March 8, 2021

Title: Topological Recursion and Enumerative Geometry

Abstract: Given a holomorphic curve in the complex 2-plane together with a suitably normalized symmetric meromorphic bilinear differential, the Chekhov-Eynard-Orantin Topological Recursion defines an infinite sequence of symmetric meromorphic multilinear differentials W_{g,n} on the curve. In many examples, the invariants W_{g,n} provide answers to enumerative problems. I will describe Topological Recursion and present several examples in which the answers are Hodge integrals (which are intersection numbers on moduli of curves) or Gromov-Witten invariants (which are virtual counts of holomorphic maps from Riemann surfaces to a Kahler manifold).

Registration is required to receive the Zoom information

Register here to attend

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• MATHEMATICAL PICTURE LANGUAGE SEMINAR

##### MATHEMATICAL PICTURE LANGUAGE SEMINARSome Analysis Aspects in Subfactor Theory

10:00 AM-11:00 AM
March 9, 2021

One of the most fascinating aspects about non-commutative spaces (aka von Neumann algebras), is the way their building data, which is often geometric in nature, impacts on the properties of their quantized symmetries. This is particularly the case for II1 factors, where symmetries are encoded by their subfactors of finite Jones index. I will discuss some results and open problems that illustrate the unique interplay between analysis and algebra/combinatorics entailed by this interdependence, that’s specific to subfactor theory.

• CMSA EVENT: CMSA Computer Science for Mathematicians: Optimal Mean Estimation without a Variance

##### CMSA EVENTCMSA Computer Science for Mathematicians: Optimal Mean Estimation without a Variance

11:30 AM-12:30 PM
March 9, 2021

Estimating the mean of a distribution from i.i.d samples is a fundamental statistical task. In this talk, we will focus on the high-dimensional setting where we will design estimators achieving optimal recovery guarantees in terms of all relevant parameters. While optimal one-dimensional estimators have been known since the 80s (Nemirovskii and Yudin ’83), optimal estimators in high dimensions have only been discovered recently beginning with the seminal work of Lugosi and Mendelson in 2017 and subsequent work has led to computationally efficient variants of these estimators (Hopkins 2018). We will discuss statistical and computational extensions of these results by developing optimal estimators for settings where the data distribution only obeys a finite fractional moment condition as opposed to the existence of a second moment as assumed previously.

Joint work with Peter Bartlett, Nicolas Flammarion, Michael I. Jordan and Nilesh Tripuraneni.

The talk will be based on the following papers: https://arxiv.org/abs/2011.12433https://arxiv.org/abs/1902.01998.

• HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

##### HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAROn extension of pluricanonical forms for Kaehler families

3:00 PM-4:00 PM
March 9, 2021

We will report on a recent joint work with Junyan Cao, cf. arXiv:2012.05063. The main topics we will discuss are revolving around the extension of pluricanonical forms defined on the central fiber of a family of Kaehler manifolds. For our results to hold we need the divisor of zeros of the said forms to be sufficiently “nice”, in a sense that will become clear during the talk.

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• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Supersymmetric quantum field theory with exotic symmetry in 3+1 dimensions and fermionic fracton phases

##### CMSA EVENTCMSA Quantum Matter in Mathematics and Physics: Supersymmetric quantum field theory with exotic symmetry in 3+1 dimensions and fermionic fracton phases

10:30 AM-12:00 PM
March 10, 2021

Fracton phases show exotic properties, such as sub-extensive entropy, local particle-like excitation with restricted mobility, and so on. In order to find natural fermionic fracton phases, we explore supersymmetric quantum field theory with exotic symmetry. We use superfield formalism and write down the action of a supersymmetric version of one of the simplest models with exotic symmetry, the φ theory in 3+1 dimensions. It contains a large number of ground states due to the fermionic higher pole subsystem symmetry. Its residual entropy is proportional to the area instead of the volume. This theory has a self-duality similar to that of the φ theory. We also write down the action of a supersymmetric version of a tensor gauge theory, and discuss BPS fractons.

• CMSA EVENT: CMSA New Technologies in Mathematics: The Ramanujan Machine: Using Algorithms for the Discovery of Conjectures on Mathematical Constants

##### CMSA EVENTCMSA New Technologies in Mathematics: The Ramanujan Machine: Using Algorithms for the Discovery of Conjectures on Mathematical Constants

3:00 PM-4:00 PM
March 10, 2021

In the past, new conjectures about fundamental constants were discovered sporadically by famous mathematicians such as Newton, Euler, Gauss, and Ramanujan. The talk will present a different approach – a systematic algorithmic approach that discovers new mathematical conjectures on fundamental constants. We call this approach “the Ramanujan Machine”. The algorithms found dozens of well-known formulas as well as previously unknown ones, such as continued fraction representations of π, e, Catalan’s constant, and values of the Riemann zeta function. Some of the conjectures are in retrospect simple to prove, whereas others remain so far unproven. We will discuss these puzzles and wider open questions that arose from this algorithmic investigation – specifically, a newly-discovered algebraic structure that seems to generalize all the known formulas and connect between fundamental constants. We will also discuss two algorithms that proved useful in finding conjectures: a variant of the meet-in-the-middle algorithm and a gradient descent algorithm tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values; consequently, they conjecture formulas without providing proofs or requiring prior knowledge of the underlying mathematical structure. This way, our approach reverses the conventional usage of sequential logic in formal proofs; instead, using numerical data to unveil mathematical structures and provide leads to further mathematical research.

• NUMBER THEORY SEMINAR

##### NUMBER THEORY SEMINARTBD

3:00 PM-4:00 PM
March 10, 2021

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

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• COLLOQUIUMS

##### COLLOQUIUMSSpecial Colloquium

3:00 PM-4:00 PM
March 16, 2021

Title: Recent progress on random field Ising model

Abstract: Random field Ising model is a canonical example to study the effect of disorder on long range order. In 70’s, Imry-Ma predicted that in the presence of weak disorder, the long-range order persists at low temperatures in three dimensions and above but disappears in two dimensions. In this talk, I will review mathematical development surrounding this prediction, and I will focus on recent progress on exponential decay and on correlation length in two dimensions. The talk is based on a joint work with Jiaming Xia and a joint work with Mateo Wirth.

Registration is required to receive the Zoom information

Register here to attend

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• NUMBER THEORY SEMINAR

##### NUMBER THEORY SEMINARTBD

3:00 PM-4:00 PM
March 17, 2021

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

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• ALGEBRAIC DYNAMICS SEMINAR

##### ALGEBRAIC DYNAMICS SEMINARBounded contraction, hyperbolicity, and J-stability in non-archimedean dynamics

10:00 AM-12:00 PM
March 19, 2021

Let K be a complete and algebraically closed field, such as C or the p-adic field C_p, and let f\in K(z) be a rational function of degree d\geq 2. The map f is said to be hyperbolic if there is some metric on its Julia set with respect to which it is expanding. A celebrated 1983 theorem of Mane, Sad, and Sullivan shows that for K=C, hyperbolic maps are J-stable, meaning that nearby maps in moduli space have topologically conjugate dynamics on their Julia sets. In this talk, we show that if K is non-archimedean, an a priori weaker bounded-contraction condition also yields J-stability. This project is joint work with Junghun Lee.

Go to http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for Zoom information.

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• COLLOQUIUMS

##### COLLOQUIUMSSpecial Colloquium

3:00 PM-4:00 PM
March 22, 2021

Title: TBD

Abstract: TBD

Registration is required to receive the Zoom information

Register here to attend

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• NUMBER THEORY SEMINAR

##### NUMBER THEORY SEMINARTBD

3:00 PM-4:00 PM
March 24, 2021

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

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• CMSA EVENT: CMSA New Technologies in Mathematics: Doing Mathematics with Simple Types: Infinitary Combinatorics in Isabelle/HOL

##### CMSA EVENTCMSA New Technologies in Mathematics: Doing Mathematics with Simple Types: Infinitary Combinatorics in Isabelle/HOL

3:00 PM-4:00 PM
March 31, 2021

Are proof assistants relevant to mathematics? One approach to this question is to explore the breadth of mathematical topics that can be formalised. The partition calculus was introduced by Erdös and R. Rado in 1956 as the study of “analogues and extensions of Ramsey’s theorem”. Highly technical results were obtained by Erdös-Milner, Specker and Larson (among many others) for the particular case of ordinal partition relations, which is concerned with countable ordinals and order types. Much of this material was formalised last year (with the assistance of Džamonja and Koutsoukou-Argyraki). Some highlights of this work will be presented along with general observations about the formalisation of mathematics, including ZFC, in simple type theory.

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