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Mike Hopkins Gives Lecture at 2022 Abel Prize Celebration

Dr. Michael J. Hopkins, George Putnam Professor of Pure and Applied Mathematics and Harvard Department of Mathematics Chair, gave a lecture titled "The great wild...

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...

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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

• July 30, 2022 - August 1, 2022
Advances in Mathematical Physics: A Conference in Honor of Elliott H. Lieb on his 90th Birthday.

ANNOUNCEMENTS, CONFERENCE

• August 2, 2022 - August 5, 2022
CMSA, 20 Garden St, G10

CMSA EVENT

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• CMSA EVENT: Math Science Lectures in Honor of Raoul Bott

CMSA EVENTMath Science Lectures in Honor of Raoul Bott

11:00 AM-12:15 PM
October 4, 2021-October 5, 2021

On October 4th and October 5th, 2021, Harvard CMSA will host its annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker will be Michael Freedman (Microsoft). The lectures will take place from 11:00am – 12:15pm (ET) on Zoom.

You must register to attend. Register here.

This will be the third annual lecture series held in honor of Raoul Bott.

Lecture 1:

Monday, October 4, 11:00 am – 12:15 pm ET

Title: The Universe from a single Particle

Abstract: I will explore a toy model  for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709

Lecture 2:

Tuesday, October 5, 11 am – 12:15 pm ET

Title: Controlled Mather Thurston Theorems.

Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374

The previous Bott Lectures lecture featured Mina Aganagic (UC Berkeley), who spoke on “Two math lessons from string theory.” Information can be found here.

• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics Seminar: UV/IR and Effective Field Theory

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics Seminar: UV/IR and Effective Field Theory

8:30 PM-10:00 PM
October 4, 2021

**rescheduled from October 4**

https://harvard.zoom.us/j/977347126

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• CMSA EVENT: CMSA Combinatorics, Physics and Probability Seminar: Geodesic Geometry on Graphs

CMSA EVENTCMSA Combinatorics, Physics and Probability Seminar: Geodesic Geometry on Graphs

9:30 AM-10:30 AM
October 5, 2021

In a graph G = (V, E) we consider a system of paths S so that for every two vertices u,v in V there is a unique uv path in S connecting them. The path system is said to be consistent if it is closed under taking subpaths, i.e. if P is a path in S then any subpath of P is also in S. Every positive weight function w: E–>R^+ gives rise to a consistent path system in G by taking the paths in S to be geodesics w.r.t. w. In this case, we say w induces S. We say a graph G is metrizable if every consistent path system in G is induced by some such w.

We’ll discuss the concept of graph metrizability, and, in particular, we’ll see that while metrizability is a rare property, there exists infinitely many 2-connected metrizable graphs.

Joint work with Nati Linial.

• CMSA EVENT: Math Science Lectures in Honor of Raoul Bott

CMSA EVENTMath Science Lectures in Honor of Raoul Bott

11:00 AM-12:15 PM
October 5, 2021-October 5, 2021

On October 4th and October 5th, 2021, Harvard CMSA will host its annual Math Science Lectures in Honor of Raoul Bott. This year’s speaker will be Michael Freedman (Microsoft). The lectures will take place from 11:00am – 12:15pm (ET) on Zoom.

You must register to attend. Register here.

This will be the third annual lecture series held in honor of Raoul Bott.

Lecture 1:

Monday, October 4, 11:00 am – 12:15 pm ET

Title: The Universe from a single Particle

Abstract: I will explore a toy model  for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108.12709

Lecture 2:

Tuesday, October 5, 11 am – 12:15 pm ET

Title: Controlled Mather Thurston Theorems.

Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374

The previous Bott Lectures lecture featured Mina Aganagic (UC Berkeley), who spoke on “Two math lessons from string theory.” Information can be found here.

• HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

HARVARD-MIT ALGEBRAIC GEOMETRY SEMINARBott periodicity, algebro-geometrically

3:00 PM-4:00 PM
October 5, 2021
1 Oxford Street, Cambridge, MA 02138 USA
I will report on joint work with Hannah Larson, and joint work in progress with Jim Bryan, in which we try to make sense of Bott periodicity from a naively algebro-geometric point of view.
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• CMSA EVENT: CMSA Colloquium: Strings, Knots and Quivers

CMSA EVENTCMSA Colloquium: Strings, Knots and Quivers

9:30 AM-10:30 AM
October 6, 2021

I will discuss a recently discovered relation between quivers and knots, as well as – more generally – toric Calabi-Yau manifolds. In the context of knots this relation is referred to as the knots-quivers correspondence, and it states that various invariants of a given knot are captured by characteristics of a certain quiver, which can be associated to this knot. Among others, this correspondence enables to prove integrality of LMOV invariants of a knot by relating them to motivic Donaldson-Thomas invariants of the corresponding quiver, it provides a new insight on knot categorification, etc. This correspondence arises from string theory interpretation and engineering of knots in brane systems in the conifold geometry; replacing the conifold by other toric Calabi-Yau manifolds leads to analogous relations between such manifolds and quivers.

• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics Seminar: Line defects in CFTs: renormalization group flows and semiclassical limits

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics Seminar: Line defects in CFTs: renormalization group flows and semiclassical limits

10:30 AM-12:00 PM
October 6, 2021

I will discuss line defects in d-dimensional Conformal Field Theories (CFTs). In the first part of the talk, I will argue that the ambient CFT places nontrivial constraints on Renormalization Group (RG) flows on such line defects. I will show that the flow on line defects is consequently irreversible and furthermore a canonical decreasing entropy function exists. This construction generalizes the g theorem to line defects in arbitrary dimensions. In the second part of the talk, I will present some applications. In particular, I will discuss impurities with large isospin S for some O(3) symmetric theories in the epsilon expansion. For sufficiently large S diagrammatic perturbation theory breaks down, and these are studied in a semiclassical expansion at fixed epsilon S.

—–
Subscribe to Harvard CMSA seminar videos (more to be uploaded):

• CMSA EVENT: CMSA New Technologies in Mathematics Seminar: New results in Supergravity via ML Technology

CMSA EVENTCMSA New Technologies in Mathematics Seminar: New results in Supergravity via ML Technology

2:00 PM-3:00 PM
October 6, 2021

The infrastructure built to power the Machine Learning revolution has many other uses beyond Deep Learning. Starting from a general architecture-level overview over the lower levels of Google’s TensorFlow machine learning library, we review how this has recently helped us to find all the stable vacua of SO(8) Supergravity in 3+1 dimensions, has allowed major progress on other related questions about M theory, and briefly discuss other applications in field theory and beyond.

https://harvard.zoom.us/j/99651364593?pwd=Q1R0RTMrZ2NZQjg1U1ZOaUYzSE02QT09

• NUMBER THEORY SEMINAR

NUMBER THEORY SEMINARHigher-dimensional modular equations and point counting on abelian surfaces

3:00 PM-4:00 PM
October 6, 2021
1 Oxford Street, Cambridge, MA 02138 USA

Given a prime number l, the elliptic modular polynomial of level l is an explicit equation for the locus of elliptic curves related by an l-isogeny. These polynomials have a large number of algorithmic applications: in particular, they are an essential ingredient in the celebrated SEA algorithm for counting points on elliptic curves over finite fields of large characteristic.

More generally, modular equations describe the locus of isogenous abelian varieties in certain moduli spaces called PEL Shimura varieties. We will present upper bounds on the size of modular equations in terms of their level, and outline a general strategy to compute an isogeny A->A’ given a point (A,A’) where the modular equations are satisfied. This generalizes well-known properties of elliptic modular polynomials to higher dimensions.

The isogeny algorithm is made fully explicit for Jacobians of genus 2 curves. In combination with efficient evaluation methods for modular equations in genus 2 via complex approximations, this gives rise to point counting algorithms for (Jacobians of) genus 2 curves whose
asymptotic costs, under standard heuristics, improve on previous results.

• OPEN NEIGHBORHOOD SEMINAR

OPEN NEIGHBORHOOD SEMINARAlternative twin prime problems

4:30 PM-5:30 PM
October 6, 2021
1 Oxford Street, Cambridge, MA 02138 USA

It is conjectured that there are infinitely many pairs of primes that differ by 2. After presenting some motivation, results toward the conjecture, and obstructions to further progress, we will consider an analogous problem where the primes are replaced by irreducible polynomials with coefficients in the integers modulo 3 (or modulo 5). Alternative problems of this type often boil down to counting solutions to algebraic equations. Work of Grothendieck and Deligne reduces the latter to (topological) questions about the shape of the geometric figures cut out by our equations. I will report on joint work with Will Sawin obtaining some control on the number of higher-dimensional holes inside the figures in question. This allows us to make progress on some alternative twin prime problems, similar to the ones mentioned above.

• DIFFERENTIAL GEOMETRY SEMINAR

DIFFERENTIAL GEOMETRY SEMINARJoint Harvard-CUHK-YMSC Differential Geometry Seminar

9:30 PM-10:30 PM
October 6, 2021

will speak on:

“Angular momentum in general relativity”

Tuesday, October 5, 2021

9:30 – 10:30 AM  *Hong Kong time*

9:30 – 10:30 PM *Eastern Time*

Abstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960′, due to the discovery of “supertranslation ambiguity”: the angular momentums recorded by two distant observers of the same system may not be the same. In this talk, I shall show how the mathematical theory of optimal isometric embedding and quasilocal angular momentum identifies a correction term, and leads to a new definition of angular momentum that is free of any supertranslation ambiguity. This is based on joint work with Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tung Yau.

Meeting ID: 931 1453 7124, Passcode: 20211006

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• CMSA EVENT: CMSA Interdisciplinary Science Seminar: SiRNA Targeting TCRb: A Proposed Therapy for the Treatment of Autoimmunity

CMSA EVENTCMSA Interdisciplinary Science Seminar: SiRNA Targeting TCRb: A Proposed Therapy for the Treatment of Autoimmunity

9:00 AM-10:00 AM
October 7, 2021

As of 2018, the United States National Institutes of Health estimate that over half a billion people worldwide are affected by autoimmune disorders. Though these conditions are prevalent, treatment options remain relatively poor, relying primarily on various forms of immunosuppression which carry potentially severe side effects and often lose effectiveness overtime. Given this, new forms of therapy are needed. To this end, we have developed methods for the creation of small-interfering RNA (siRNA) for hypervariable regions of the T-cell receptor β-chain gene (TCRb) as a highly targeted, novel means of therapy for the treatment of autoimmune disorders.

This talk will review the general mechanism by which autoimmune diseases occur and discuss the pros and cons of conventional pharmaceutical therapies as they pertain to autoimmune disease treatment. I will then examine the rational and design methodology for the proposed siRNA therapy and how it contrasts with contemporary methods for the treatment of these conditions. Additionally, the talk will compare the efficacy of multiple design strategies for such molecules by comparison over several metrics and discuss how this will be guiding future research.

Zoom ID: 950 2372 5230 (Password: cmsa)

• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics Seminar: A tour of categorical symmetry

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics Seminar: A tour of categorical symmetry

10:30 AM-12:00 PM
October 7, 2021

I will discuss some perspectives on symmetry coming from the study of topological defects in quantum field theory. I will argue that we should take topological defects themselves to define the symmetries of QFT. This gives us a view of the “category of QFTs”. I will describe some examples of these “categorical symmetries”, their applications, and some open problems.

—–
Subscribe to Harvard CMSA seminar videos (more to be uploaded):

• COLLOQUIUMS

COLLOQUIUMSFinding distinct-variable solutions to linear equations in \mathbb{F}_p^n

4:30 PM-5:30 PM
October 7, 2021
1 Oxford Street, Cambridge, MA 02138

For a fixed prime p and large n, what is the largest size of a subset of \mathbb{F}_p^n which does not contain a non-trivial solution to some given linear equation or system of linear equations? This is a fundamental question in additive combinatorics with a long history. While there are various different notions of “non-trivial solutions”, in this talk we say that a solution is non-trivial if all of its variables are distinct.

A particularly famous and important instance of the question above is the problem of bounding the largest size of a subset of \mathbb{F}_p^n which does not contain a three-term arithmetic progression. In 2016, Ellenberg and Gijswijt made a breakthrough on this problem and the approach in their proof was later generalized by Tao yielding what is now called the slice rank polynomial method. Unfortunately, for other instances of the question above, the slice rank polynomial method cannot handle the condition for the variables in a non-trivial solution to be distinct.

In this talk, we will first give a brief survey on the slice rank polynomial method and some of its applications, and we will then discuss two different results concerning the question above. These results combine the slice rank polynomial method with additional combinatorial ideas in order to handle the distinctness condition. We also discuss an application of one of these results to the Erdös-Ginzburg-Ziv problem in discrete geometry.

• CMSA EVENT: CMSA Algebraic Geometry in String Theory Seminar: A mirror theorem for GLSMs

CMSA EVENTCMSA Algebraic Geometry in String Theory Seminar: A mirror theorem for GLSMs

10:00 PM-11:00 PM
October 7, 2021

A gauged linear sigma model (GLSM) consists roughly of a complex vector space V, a group G acting on V, a character \theta of G, and a G-invariant function w on V.  This data defines a GIT quotient Y = [V //_\theta G] and a function on that quotient.  GLSMs arise naturally in a number of contexts, for instance as the mirrors to Fano manifolds and as examples of noncommutative crepant resolutions. GLSMs provide a broad setting in which it is possible to define an enumerative curve counting theory, simultaneously generalizing FJRW theory and the Gromov-Witten theory of hypersurfaces. Despite a significant effort to rigorously define the enumerative invariants of a GLSM, very few computations of these invariants have been carried out.  In this talk I will describe a new method for computing generating functions of GLSM invariants. I will explain how these generating functions arise as derivatives of generating functions of Gromov-Witten invariants of Y.

https://harvard.zoom.us/j/98781914555?pwd=bmVzZGdlRThyUDREMExab20ybmg1Zz09

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• CMSA EVENT: CMSA Combinatorics, Physics and Probability Seminar: On counting algebraically defined graphs

CMSA EVENTCMSA Combinatorics, Physics and Probability Seminar: On counting algebraically defined graphs

9:30 AM-10:30 AM
October 12, 2021

For many classes of graphs that arise naturally in discrete geometry (for example intersection graphs of segments or disks in the plane), the edges of these graphs can be defined algebraically using the signs of a finite list of fixed polynomials. We investigate the number of n-vertex graphs in such an algebraically defined class of graphs. Warren’s theorem (a variant of a theorem of Milnor and Thom) implies upper bounds for the number of n-vertex graphs in such graph classes, but all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving a lower bound for this number (under some reasonable assumptions on the fixed list of polynomials), and this lower bound essentially matches the upper bound from Warren’s theorem.

• CMSA EVENT: CMSA Algebraic Geometry in String Theory Seminar: Derived projectivizations of two-term complexes

CMSA EVENTCMSA Algebraic Geometry in String Theory Seminar: Derived projectivizations of two-term complexes

10:30 AM-11:30 AM
October 12, 2021

For a given two-term complex of vector bundles on a derived scheme (or stack), there are three natural ways to define its “derived projectivizations”: (i) as the derived base-change of the classical projectivization of Grothendieck; (ii) as the derived moduli parametrizing one-dimensional locally free quotients; (iii) as the GIT quotient of the total space by $mathbb{G}_m$-action. In this talk, we first show that these three definitions are equivalent. Second, we prove a structural theorem about the derived categories of derived projectivizations and study the corresponding mutation theory. Third, we apply these results to various moduli situations, including the moduli of certain stable pairs on curves and the Hecke correspondences of one-point modification of moduli of stable sheaves on surfaces. If time allowed, we could also discuss the generalizations of these results to the derived Quot schemes of locally free quotients.

https://harvard.zoom.us/j/98781914555?pwd=bmVzZGdlRThyUDREMExab20ybmg1Zz09

• CMSA EVENT: CMSA Algebraic Geometry in String Theory Seminar: Derived projectivizations of two-term complexes

CMSA EVENTCMSA Algebraic Geometry in String Theory Seminar: Derived projectivizations of two-term complexes

10:30 AM-11:30 AM
October 12, 2021

For a given two-term complex of vector bundles on a derived scheme (or stack), there are three natural ways to define its “derived projectivizations”: (i) as the derived base-change of the classical projectivization of Grothendieck; (ii) as the derived moduli parametrizing one-dimensional locally free quotients; (iii) as the GIT quotient of the total space by $\mathbb{G}_m$-action. In this talk, we first show that these three definitions are equivalent. Second, we prove a structural theorem about the derived categories of derived projectivizations and study the corresponding mutation theory. Third, we apply these results to various moduli situations, including the moduli of certain stable pairs on curves and the Hecke correspondences of one-point modification of moduli of stable sheaves on surfaces. If time allowed, we could also discuss the generalizations of these results to the derived Quot schemes of locally free quotients.

https://harvard.zoom.us/j/98781914555?pwd=bmVzZGdlRThyUDREMExab20ybmg1Zz09

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• CMSA EVENT: CMSA Colloquium: Knot homology and sheaves on the Hilbert scheme of points on the plane

CMSA EVENTCMSA Colloquium: Knot homology and sheaves on the Hilbert scheme of points on the plane

9:30 AM-10:30 AM
October 13, 2021

The knot homology (defined by Khovavov,Rozansky) provide us with a refinements of the knot polynomial knot invariant defined by Jones. However, the knot homology are much harder to compute compare to the polynomial invariant of Jones. In my talk I present recent developments that allow us to use tools of algebraic geometry to compute the homology of torus knots and prove long-standing conjecture on the Poncare duality the knot homology. In more details, using physics ideas of Kapustin-Rozansky-Saulina, in the joint work with Rozansky, we provide a mathematical construction that associates to a braid on n strands a complex of sheaves on the Hilbert scheme of n points on the plane. The knot homology of the closure of the braid is a space of sections of this sheaf. The sheaf is also invariant with respect to the natural symmetry of the plane, the symmetry is the geometric counter-part of the mentioned Poincare duality.

• CMSA EVENT: CMSA New Technologies in Mathematics Seminar: Computer-Aided Mathematics and Satisfiability

CMSA EVENTCMSA New Technologies in Mathematics Seminar: Computer-Aided Mathematics and Satisfiability

2:00 PM-3:00 PM
October 13, 2021

Progress in satisfiability (SAT) solving has made it possible to
determine the correctness of complex systems and answer long-standing
open questions in mathematics. The SAT solving approach is completely
automatic and can produce clever though potentially gigantic proofs.
We can have confidence in the correctness of the answers because
highly trustworthy systems can validate the underlying proofs
regardless of their size.

We demonstrate the effectiveness of the SAT approach by presenting
some recent successes, including the solution of the Boolean
Pythagorean Triples problem, computing the fifth Schur number, and
resolving the remaining case of Keller’s conjecture. Moreover, we
constructed and validated a proof for each of these results. The
second part of the talk focuses on notorious math challenges for which
automated reasoning may well be suitable. In particular, we discuss
our progress on applying SAT solving techniques to the chromatic
number of the plane (Hadwiger-Nelson problem), optimal schemes for
matrix multiplication, and the Collatz conjecture.

https://harvard.zoom.us/j/99651364593?pwd=Q1R0RTMrZ2NZQjg1U1ZOaUYzSE02QT09

• NUMBER THEORY SEMINAR

NUMBER THEORY SEMINARThe plectic conjecture over local fields

3:00 PM-4:00 PM
October 13, 2021
1 Oxford Street, Cambridge, MA 02138 USA

The étale cohomology of varieties over Q enjoys a Galois action. In the case of Hilbert modular varieties, Nekovář-Scholl observed that this Galois action on the level of cohomology extends to a much larger profinite group: the plectic group. They conjectured that this extension holds even on the level of complexes, as well as for more general Shimura varieties.

We present a proof of the analogue of this conjecture for local Shimura varieties. This includes (the generic fibers of) Lubin–Tate spaces, Drinfeld upper half spaces, and more generally Rapoport–Zink spaces. The proof crucially uses Scholze’s theory of diamonds.

• DIFFERENTIAL GEOMETRY SEMINAR

DIFFERENTIAL GEOMETRY SEMINARJoint Harvard-CUHK-YMSC Differential Geometry Seminar

4:00 PM-5:00 PM
October 13, 2021

will speak on:

“Some remarks on contact Calabi-Yau 7-manifolds”

Wednesday, October 13, 2021

4:00 – 5:00 PM  *Hong Kong time*

4:00 – 5:00 AM *Eastern Time*

Abstract: In geometry and physics it has proved useful to relate G2 and Calabi-Yau geometry via circle bundles. Contact Calabi-Yau 7-manifolds are, in the simplest cases, such circle bundles over Calabi-Yau 3-orbifolds. These 7-manifolds provide testing grounds for the study of geometric flows which seek to find torsion-free G2-structures (and thus Ricci flat metrics with exceptional holonomy). They also give useful backgrounds to examine the heterotic G2 system (also known as the G2-Hull-Strominger system), which is a coupled set of PDEs arising from physics that involves the G2-structure and gauge theory on the 7-manifold. I will report on recent progress on both of these directions in the study of contact Calabi-Yau 7-manifolds, which is joint work with H. Sá Earp and J. Saavedra.

Meeting ID: 919 3602 5861
Passcode: 20211013

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• CMSA EVENT: CMSA Interdisciplinary Science Seminar: D3C: Reducing the Price of Anarchy in Multi-Agent Learning

CMSA EVENTCMSA Interdisciplinary Science Seminar: D3C: Reducing the Price of Anarchy in Multi-Agent Learning

9:00 AM-10:00 AM
October 14, 2021

In multi-agent systems the complex interaction of fixed incentives can lead agents to outcomes that are poor (inefficient) not only for the group but also for each individual agent. Price of anarchy is a technical game theoretic definition introduced to quantify the inefficiency arising in these scenarios– it compares the welfare that can be achieved through perfect coordination against that achieved by self-interested agents at a Nash equilibrium. We derive a differentiable upper bound on a price of anarchy that agents can cheaply estimate during learning. Equipped with this estimator agents can adjust their incentives in a way that improves the efficiency incurred at a Nash equilibrium. Agents adjust their incentives by learning to mix their reward (equiv. negative loss) with that of other agents by following the gradient of our derived upper bound. We refer to this approach as D3C. In the case where agent incentives are differentiable D3C resembles the celebrated Win-Stay Lose-Shift strategy from behavioral game theory thereby establishing a connection between the global goal of maximum welfare and an established agent-centric learning rule. In the non-differentiable setting as is common in multiagent reinforcement learning we show the upper bound can be reduced via evolutionary strategies until a compromise is reached in a distributed fashion. We demonstrate that D3C improves outcomes for each agent and the group as a whole on several social dilemmas including a traffic network exhibiting Braess’s paradox a prisoner’s dilemma and several reinforcement learning domains.

Zoom ID: 950 2372 5230 (Password: cmsa)

• CMSA EVENT: CMSA Active Matter Seminar: Stochastic PDE as scaling limits of interacting particle systems

CMSA EVENTCMSA Active Matter Seminar: Stochastic PDE as scaling limits of interacting particle systems

1:00 PM-2:00 PM
October 14, 2021

Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models. In this talk, I will illustrate how this challenge can be overcome by elucidating the probabilistic connections between models of different levels of detail. These connections explain how stochastic partial differential equations (SPDE) arise naturally from particle models.

I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of the underlying population dynamics.

• ALGEBRAIC DYNAMICS SEMINAR

ALGEBRAIC DYNAMICS SEMINARA dynamical approach to generalized Weil’s Riemann hypothesis

4:00 PM-6:00 PM
October 14, 2021

Inspired by a result of Esnault and Srinivas on automorphisms of surfaces and recent advances in complex dynamics, Truong raised a question on the comparison of two dynamical degrees, which are defined using pullback actions of dynamical correspondences on numerical cycle class groups and cohomology groups, respectively. An affirmative answer to his question would surprisingly imply Weil’s Riemann hypothesis. In this talk, I first discuss a special case of Abelian varieties. Then I will introduce the so-called dynamical correspondence and its application to certain surfaces. This is based on joint work with Tuyen Truong.

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• CMSA EVENT: CMSA Combinatorics, Physics and Probability Seminar: Ising model, total positivity, and criticality

CMSA EVENTCMSA Combinatorics, Physics and Probability Seminar: Ising model, total positivity, and criticality

9:30 AM-10:30 AM
October 19, 2021

The Ising model, introduced in 1920, is one of the most well-studied models in statistical mechanics. It is known to undergo a phase transition at critical temperature, and has attracted considerable interest over the last two decades due to special properties of its scaling limit at criticality.

The totally nonnegative Grassmannian is a subset of the real Grassmannian introduced by Postnikov in 2006. It arises naturally in Lusztig’s theory of total positivity and canonical bases, and is closely related to cluster algebras and scattering amplitudes.

I will give some background on the above objects and then explain a precise relationship between the planar Ising model and the totally nonnegative Grassmannian, obtained in our recent work with P. Pylyavskyy. Building on this connection, I will give a new boundary correlation formula for the critical Ising model.

https://harvard.zoom.us/j/94191911494?pwd=RnN3ZnIwcFYwd0QyT0MwZWVISmR5Zz09

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• CMSA EVENT: CMSA Colloquium: Categorification and applications

CMSA EVENTCMSA Colloquium: Categorification and applications

9:30 AM-10:30 AM
October 20, 2021

I will give a survey of the program of categorification for quantum groups, some of its recent development and applications to representation theory.

• NUMBER THEORY SEMINAR

NUMBER THEORY SEMINAREffective height bounds for odd-degree totally real points on some curves

3:00 PM-4:00 PM
October 20, 2021
1 Oxford Street, Cambridge, MA 02138 USA

I will give a finite-time algorithm that, on input (g,K,S) with g > 0, K a totally real number field of odd degree, and S a finite set of places of K, outputs the finitely many g-dimensional abelian varieties A/K which are of GL_2-type over K and have good reduction outside S.

The point of this is to effectively compute the S-integral K-points on a Hilbert modular variety, and the point of that is to be able to compute all K-rational points on complete curves inside such varieties.

This gives effective height bounds for rational points on infinitely many curves and (for each curve) over infinitely many number fields. For example one gets effective height points for odd-degree totally real points on x^6 + 4y^3 = 1, by using the hypergeometric family associated to the arithmetic triangle group \Delta(3,6,6).

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• CMSA EVENT: CMSA Interdisciplinary Science Seminar: Mathematical resolution of the Liouville conformal field theory

CMSA EVENTCMSA Interdisciplinary Science Seminar: Mathematical resolution of the Liouville conformal field theory

9:00 AM-10:00 AM
October 21, 2021

The Liouville conformal field theory is a well-known beautiful quantum field theory in physics describing random surfaces. Only recently a mathematical approach based on a well-defined path integral to this theory has been proposed using probability by David, Kupiainen, Rhodes, Vargas.
Many works since the 80’s in theoretical physics (starting with Belavin-Polyakov-Zamolodchikov) tell us that conformal field theories in dimension 2 are in general « Integrable », the correlations functions are solutions of PDEs and can in principle be computed explicitely by using algebraic tools (vertex operator algebras, representations of Virasoro algebras, the theory of conformal blocks). However, for Liouville Theory this was not done at the mathematical level by algebraic methods.

I’ll explain how to combine probabilistic, analytic and geometric tools to give explicit (although complicated) expressions for all the correlation functions on all Riemann surfaces in terms of certain holomorphic functions of the moduli parameters called conformal blocks, and of the structure constant (3-point function on the sphere). This gives a concrete mathematical proof of the so-called conformal bootstrap and of Segal’s gluing axioms for this CFT. The idea is to break the path integral on a closed surface into path integrals on pairs of pants and reduce all correlation functions to the 3-point correlation function on the Riemann sphere $S^2$. This amounts in particular to prove a spectral resolution of a certain operator acting on $L^2(H^{-s}(S^1))$ where $H^{-s}(S^1)$ is the Sobolev space of order -s<0 equipped with a Gaussian measure, which is viewed as the space of fields, and to construct a certain representation of the Virasoro algebra into unbounded operators acting on this Hilbert space.

This is joint work with A. Kupiainen, R. Rhodes and V. Vargas.

Zoom ID: 950 2372 5230 (Password: cmsa)

• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics Seminar: Electric-magnetic duality and the Geometric Langlands duality

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics Seminar: Electric-magnetic duality and the Geometric Langlands duality

1:30 PM-3:00 PM
October 21, 2021

I will give a pedagogical review of the connection between electric-magnetic duality and the Geometric Langlands duality.
*Note special time*

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• ALGEBRAIC DYNAMICS SEMINAR

ALGEBRAIC DYNAMICS SEMINARArithmetic intersection and measures of maximal entropy

4:00 PM-6:00 PM
October 21, 2021

About 10 years ago, Xinyi Yuan and Shouwu Zhang proved that if two holomorphic maps f and g on P^N have the same sets of preperiodic points (or if the intersection of Preper(f) and Preper(g) is Zariski dense in P^N), then they must have the same measure of maximal entropy.  This was new even in dimension N=1.  I will describe some ingredients in their proof, while emphasizing the dynamical history behind this result.  I will also sketch the proof of a theorem of Levin and Przytycki from the 1990s, in dimension N=1, that two (non-exceptional) maps have the same measure of maximal entropy if and only if they “essentially” share an iterate.

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• MATHEMATICAL PICTURE LANGUAGE SEMINAR

MATHEMATICAL PICTURE LANGUAGE SEMINARA Mathematical Introduction to Machine Learning

9:30 AM-10:30 AM
October 26, 2021

The heart of modern machine learning (ML) is the approximation of high-dimensional functions. Traditional approaches, such as approximation by piecewise polynomials, wavelets, or other linear combinations of fixed basis functions, suffer from the curse of dimensionality (CoD). This does not seem to be the case for the neural network-based ML models. To quantify this, we need to develop the corresponding mathematical framework. At the same time, we might be able to use ML to solve problems in computational science that we could not solve before due to CoD. In this talk, I will report the progress made so far at the theoretical front, and highlight the main remaining challenges. I will also discuss some examples along the lines of “AI for Science”.

https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

• CMSA EVENT: CMSA Combinatorics, Physics and Probability Seminar: The n-queens problem

CMSA EVENTCMSA Combinatorics, Physics and Probability Seminar: The n-queens problem

9:30 AM-10:30 AM
October 26, 2021

The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3. Much more recently Luria showed that T(n) is at most ((1+o(1))ne^{-3})^n and conjectured equality when n is not divisible by 2 or 3. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^{-3})^n for all natural numbers n which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both Q(n) and T(n).

In this talk we’ll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a 4-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost’ configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption.

This is joint work with Peter Keevash.

• HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

HARVARD-MIT ALGEBRAIC GEOMETRY SEMINARCompactified Jacobians and the Double Ramification Cycle

3:00 PM-4:00 PM
October 26, 2021

The double ramification cycle — roughly speaking, the cycle of curves admitting a rational function with prescribed ramification profile — is an algebraic cycle in the moduli space of curves, intimately connected to Gromov-Witten theory and classical Abel-Jacobi theory. The DR cycle has been extensively studied in recent years; one of the outcomes of this study is a remarkable formula in terms of simple classes in the tautological ring of \bar{M}_{g,n}. However, for certain more delicate questions involving the DR, such as computing its higher dimensional analogues or its behavior under intersection, one must study certain refinements of the DR, for which the existing methods do not give analogous formulas. In this talk I will discuss joint work with Holmes, Pandharipande, Pixton and Schmitt on how one can obtain such formulas by studying the DR via compactified Jacobians.

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• CMSA EVENT: CMSA Colloquium: Anisotropy, biased pairing theory and applications

CMSA EVENTCMSA Colloquium: Anisotropy, biased pairing theory and applications

9:30 AM-10:30 AM
October 27, 2021

Not so long ago, the relations between algebraic geometry and combinatorics were strictly governed by the former party, with results like log-concavity of the coefficients of the characteristic polynomial of matroids shackled by intuitions and techniques from projective algebraic geometry, specifically Hodge Theory. And so, while we proved analogues for these results, combinatorics felt subjugated to inspirations from outside of it.

In recent years, a new powerful technique has emerged: Instead of following the geometric statements of Hodge theory about signature, we use intuitions from the Hall marriage theorem, translated to algebra: once there, they are statements about self-pairings, the non-degeneracy of pairings on subspaces to understand the global geometry of the pairing. This was used to establish Lefschetz type theorems far beyond the scope of algebraic geometry, which in turn established solutions to long-standing conjectures in combinatorics.

I will survey this theory, called biased pairing theory, and new developments within it, as well as new applications to combinatorial problems. Reporting on joint work with Stavros Papadaki, Vasiliki Petrotou and Johanna Steinmeyer.

• CMSA EVENT: CMSA New Technologies in Mathematics Seminar: Why explain mathematics to computers?

CMSA EVENTCMSA New Technologies in Mathematics Seminar: Why explain mathematics to computers?

2:00 PM-3:00 PM
October 27, 2021

A growing number of mathematicians are having fun explaining mathematics to computers using proof assistant softwares. This process is called formalization. In this talk I’ll describe what formalization looks like, what kind of things it teaches us, and how it could even turn out to be useful (in our usual sense of “useful”). This will not be a talk about foundations of mathematics, and I won’t assume any prior knowledge about formalization.

https://harvard.zoom.us/j/99651364593?pwd=Q1R0RTMrZ2NZQjg1U1ZOaUYzSE02QT09

• NUMBER THEORY SEMINAR

NUMBER THEORY SEMINARp-adic Heights of the arithmetic diagonal cycles

3:00 PM-4:00 PM
October 27, 2021
1 Oxford Street, Cambridge, MA 02138 USA

This is a work in progress joint with Daniel Disegni. We formulate a p-adic analogue of the Arithmetic Gan–Gross–Prasad conjecture for unitary groups, relating the p-adic height pairing of the arithmetic diagonal cycles to the first central derivative (along the cyclotomic direction) of a p-adic Rankin—Selberg L-function associated to cuspidal automorphic representations. In the good ordinary case we are able to prove the conjecture, at least when the ramifications are mild at inert primes. We deduce some application to the p-adic version of the Bloch-Kato conjecture.

• NUMBER THEORY SEMINAR

NUMBER THEORY SEMINARp-adic Heights of the arithmetic diagonal cycles

3:00 PM-4:00 PM
October 27, 2021
1 Oxford Street, Cambridge, MA 02138 USA

This is a work in progress joint with Daniel Disegni. We formulate a p-adic analogue of the Arithmetic Gan–Gross–Prasad conjecture for unitary groups, relating the p-adic height pairing of the arithmetic diagonal cycles to the first central derivative (along the cyclotomic direction) of a p-adic Rankin—Selberg L-function associated to cuspidal automorphic representations. In the good ordinary case we are able to prove the conjecture, at least when the ramifications are mild at inert primes. We deduce some application to the p-adic version of the Bloch-Kato conjecture.

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• CMSA EVENT: CMSA Interdisciplinary Science Seminar: ARCH: Know What Your Machine Doesn’t Know

CMSA EVENTCMSA Interdisciplinary Science Seminar: ARCH: Know What Your Machine Doesn’t Know

9:00 AM-10:00 AM
October 28, 2021

Despite their impressive performance, machine learning systems remain prohibitively unreliable in safety-, trust-, and ethically sensitive domains. Recent discussions in different sub-fields of AI have reached the consensus of knowledge need in machine learning; few discussions have touched upon the diagnosis of what knowledge is needed. In this talk, I will present our ongoing work on ARCH, a knowledge-driven, human-centered, and reasoning-based tool, for diagnosing the unknowns of a machine learning system. ARCH leverages human intelligence to create domain knowledge required for a given task and to describe the internal behavior of a machine learning system; it infers the missing or incorrect knowledge of the system with the built-in probabilistic, abductive reasoning engine. ARCH is a generic tool that can be applied to machine learning in different contexts. In the talk, I will present several applications in which ARCH is currently being developed and tested, including health, finance, and smart buildings.

Zoom ID: 950 2372 5230 (Password: cmsa)

• CMSA EVENT: CMSA Active Matter Seminar: Drivers of Morphological Complexity

CMSA EVENTCMSA Active Matter Seminar: Drivers of Morphological Complexity

1:00 PM-2:00 PM
October 28, 2021

During development, organisms interact with their natural habitats while undergoing morphological changes, yet we know little about how the interplay between developing systems and their environments impacts animal morphogenesis. Cnidaria, a basal animal lineage that includes sea anemones, corals, hydras, and jellyfish, offers unique insight into the development and evolution of morphological complexity.  In my talk, I will introduce our research on “ethology of morphogenesis,” a novel concept that links the behavior of organisms to the development of their size and shape at both cellular and biophysical levels, opening new perspectives about the design principle of soft-bodied animals. In addition, I will discuss a fascinating feature of cnidarian biology. For humans, our genetic code determines that we will grow two arms and two legs. The same fate is true for all mammals. Similarly, the number of fins of a fish or legs and wings of an insect is embedded in their genetic code. I will describe how sea anemones defy this rule.
References
Anniek Stokkermans, Aditi Chakrabarti, Ling Wang, Prachiti Moghe, Kaushikaram Subramanian, Petrus Steenbergen, Gregor Mönke, Takashi Hiiragi, Robert Prevedel, L. Mahadevan, and Aissam Ikmi. Ethology of morphogenesis reveals the design principles of cnidarian size and shape development. bioRxiv 2021.08.19.456976
Ikmi A, Steenbergen P, Anzo M, McMullen M, Stokkermans M, Ellington L, and Gibson M (2020). Feeding-dependent tentacle development in the sea anemone Nematostella vectensisNature communications, Sept 02; 11:4399
He S, Del Viso F, Chen C, Ikmi A, Kroesen A, Gibson MC (2018). An axial Hox code controls tissue segmentation and body patterning in Nematostella vectensisScience, Vol. 361, Issue 6409, pp. 1377-1380.
Ikmi A, McKinney SA, Delventhal KM, Gibson MC (2014). TALEN and CRISPR/Cas9 mediated genome editing in the early-branching metazoan Nematostella vectensisNature communications. Nov 24; 5:5486.

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• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics Seminar: Anomaly resolution via decomposition

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics Seminar: Anomaly resolution via decomposition

2:15 PM-3:45 PM
October 29, 2021

*Note special day and time*

In this talk we will discuss a method of anomaly resolution due to Wang-Wen-Witten in the special case of (1+1) dimensional theories. Briefly, for our purposes, Wang-Wen-Witten argued that an ill-defined anomalous orbifold [X/G] could be resolved by extending G to a larger group and adding suitable phases. We analyze this process from the perspective of decomposition, a property of (1+1)-dimensional theories with “one-form symmetries” first described in 2006. Examples of such theories include orbifolds with trivially-acting subgroups, of which the extensions of [X/G] are examples. After a review of decomposition, we will see that decomposition implies that in (1+1) dimensions, the Wang-Wen-Witten procedure results in orbifolds that are equivalent to disjoint unions of orbifolds of X by explicitly nonanomalous subgroups of G.
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• CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics Seminar: Integrability and chaos of 1+1d chiral edge states

CMSA EVENTCMSA Quantum Matter in Mathematics and Physics Seminar: Integrability and chaos of 1+1d chiral edge states

4:00 PM-5:30 PM
October 29, 2021

*Note special day and time*

I will talk about the integrability and chaos of 1+1d interacting chiral edge states, which may arise on the edge of 2+1d topological phases. We show that integrable chiral Luttinger liquid is not always a good low energy description of the edge states, and marginal interactions can significantly affect their spectrum and integrability. We first study N identical chiral Majorana fermion modes with random 4-fermion interactions, where we show that the system undergoes a transition from integrable to quantum chaotic as N increases. The large N limit defines a chiral SYK model where the Lyapunov exponent in the out-of-time-ordered correlation can be solved analytically. I will also present a chiral SY model consisting of N interacting SU(M)_1 WZW models, which host anyons and exhibits similar quantum chaos for Abelian anyons. Lastly, I will talk about the analytical and numerical study of the 4/3 FQH edge theory, which shows unusual behavior in its integrability.
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