**CANCELED**

Societal accumulation of knowledge is a complex, and arguably error-prone, process. The correctness of new units of knowledge depends not only on the correctness of the new reasoning, but also on the correctness of old units that the new one builds on. If left unchecked, errors could completely ruin the validity of most of this knowledge so there must some error-correcting going on. What are the error-corrections processes employed in nature and how effective are they? In this talk, we describe our attempts to model such phenomena using probablistic models – we describe models for growth of cumulative knowledge, emergence of errors and methods to check for errors and eliminate them. We then analyze in this compound model, when effects of errors may survive, and when they are totally eliminated.

The central discovery in our work is the following optimistic statement: If we do checking correctly (most of the time) investing just a constant factor (<1) of our effort in checking (and saving the remaining constant factor towards deriving new units of knowledge), then effects of errors can be kept in check. Notably the amount of effort expended on checking does not scale with the volume of total knowledge or the depth of dependencies in the new units of knowledge, either of which would be overwhelming.

Based on the papers:

Is this correct? Let’s check!
Omri Ben-Eliezer, Dan Mikulincer, Elchanan Mossel, Madhu Sudan
arXiv:2211.12301

Errors are Robustly Tamed in Cumulative Knowledge Processes
Anna Brandenberger, Cassandra Marcussen, Elchanan Mossel, Madhu Sudan
arXiv:2309.05638

The eigenvalues of a uniformly distributed unitary matrix have the physical interpretation of a system of particles subject to a logarithmic pair interaction, restricted to lie on the unit circle and at inverse temperature 2. In this talk, I will present a more general model in which the unit circle is replaced by a sufficiently regular Jordan curve, at any positive temperature. I will show how to obtain the asymptotic partition function and Laplace transform of a linear statistic. These can be expressed using either the exterior conformal mapping of the curve or its associated Grunsky operator. Based on joint work with Kurt Johansson.

Abstract TBA

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

Talk at 3 pm in Science Center 507; Tea at 4 pm in the Math Common Room

Please see website for more details: www.math.harvard.edu/~ctm/sem.

The derived category of a variety is an important and difficult invariant. In this talk, I discuss a purely convex-geometric and combinatorial approach to these categories for toric varieties. Along the way, we will run into the curious question of studying the contractibility of set differences of polytopes. I will focus on the toric variety of the permutahedron which has played an important role in many recent developments in matroid theory. I will discuss how this derived category can be described through the theory of matroids.

===============================

For more info, see https://math.mit.edu/combin/

Materials that increase in size offer intriguing possibilities for shape morphing applications. Here, we explore two such systems—swelling polyacrylamide hydrogels and expanding polyurethane foams. The hydrogels swell by absorbing water into crosslinked polymer networks. They can therefore be modeled by coupling solvent migration with the deformations of a hyperelastic solid. In contrast, the foams initially behave as liquids with viscosity and volume increasing in time, responding elastically only when close to solidification. We investigate how these expanding materials are sculpted by complex environments with obstacles and trenches.

This seminar will be held in person and on Zoom.

https://harvard.zoom.us/j/96657833341

Password: cmsa

I will discuss the action induced by Adams operations on V times THH(\ell), where V is a type 2 complex.

Last summer, in this seminar, I introduced a dynamical system called “complex billiards” that extends the classical real billiards map to the setting of curves over algebraically closed fields. The dynamical degree of complex billiards is an invariant that describes the “algebraic entropy”, or chaos, of the system. In my previous talk, I proved an upper bound on this dynamical degree. In this talk, we derive lower bounds on the dynamical degree (a more challenging problem) through a careful study of one very special billiard table, the Fermat hyperbola. Independently of this result, we construct an algebraically stable model for complex billiards in the Fermat hyperbola using a little bit of complex dynamics. It is NOT necessary to have attended (or to remember) the previous talk, as I will begin again with all the key definitions.

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

Current Developments in Mathematics 2024

April 5-6, 2024

Harvard University Science Center

Friday—Lecture Hall C

Saturday—Lecture Hall A

Register Here

• Speakers:
• Daniel Cristofaro-Gardiner – University of Maryland
• Samit Dasgupta – Duke University
• Jiaoyang Huang – University of Pennsylvania
• Daniel Litt – University of Toronto
• Lisa Piccirillo – MIT/University of Texas

Funding application is closed as of March 12.

Download PDF for a detailed schedule of lectures and events.

Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard).

Sponsored by the National Science Foundation, Harvard University Mathematics, and the Massachusetts Institute of Technology.

Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.

In this talk, I’ll discuss the remarkable connections between the modular bootstrap and sphere packing or ground state problems discovered by Hartman, Mazáč, and Rastelli in 2019, with a focus on opportunities for further progress.

We investigate a minimal model of a nematopolar system. We analytically uncover a phase diagram consisting of a locked phase where the polar order and nematic order are locked, and unlocked phases which could be ordered or disordered. In particular, we develop two complementary perspectives on the locked phase: (i) the nematic order induces polar order, (ii) in the locked phase, all 1/2 integral nematic topological charges are confined. In particular, a polar +1 defect fattens from a point along a string with constant tension and confines a pair of nematic +1/2 defects at its ends.

Friday, Apr. 5th at 12pm, with lunch, lounge at CMSA (20 Garden Street).

Also by Zoom: https://harvard.zoom.us/j/92410768363

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

Abstract TBA

Hierarchical percolation is a toy model for percolation on Z^d that, much like percolation on Euclidean lattices, is expected to exhibit mean-field behavior in high dimensions, non-mean-field behavior in low dimensions, and logarithmic corrections to mean-field behavior at the upper-critical dimension. The hierarchical lattice allows for a renormalization group—style analysis which is currently inaccessible for percolation on Euclidean lattices. Building on Hutchcroft’s work on cluster volumes in all dimensions, we examine the distribution of the chemical distance, extremal distance (also known as the effective resistance), and pivotal distance in high dimensions and the upper-critical dimension. Joint work with Tom Hutchcroft.

https://harvard.zoom.us/j/94035561793?pwd=VUZ3aml1eVovb2tRc1h6OS9sdlh6UT09

Password: 849612

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

Let pi be a cuspidal automorphic representation of $\mathrm{Sp}_{2n}$ over $\mathbb{Q}$ which is holomorphic discrete series at infinity, and $\chi$ a Dirichlet character. Then one can attach to $\pi$ an orthogonal $p$-adic Galois representation $\rho$ of dimension $2n+1$. Assume $\rho$ is irreducible, that pi is ordinary at $p$, and that $p$ does not divide the conductor of $\chi$. I will describe work in progress which aims to prove that the Bloch–Kato Selmer group attached to the twist of $\rho$ by $\chi$ vanishes, under some mild ramification assumptions on $\pi$; this is what is predicted by the Bloch–Kato conjectures.

The proof uses “ramified Eisenstein congruences” by constructing $p$-adic families of Siegel cusp forms degenerating to Klingen Eisenstein series of nonclassical weight, and using these families to construct ramified Galois cohomology classes for the Tate dual of the twist of $\rho$ by $\chi$.

For more info, see https://ashvin-swaminathan.github.io/home/NTSeminar.html

For a poset $P$, Galashin introduced a simple polytope $\mathscr A(P)$ called the $P$-associahedron. We will discuss a simple realization of poset associahedra and show that the $f$-vector of $\mathscr A(P)$ depends only on the comparability graph of $P$. Furthermore, we will show that when $P$ is a rooted tree, the 1-skeleton of $\mathscr A(P)$ orients to a lattice, answering a question of Laplante-Anfossi. These lattices naturally generalize both the weak order on permutations and the Tamari lattice. This is joint work with Colin Defant and Son Ngyuen.

===============================

For more info, see https://math.mit.edu/combin/

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

Last time we saw how to identify THH(ell^hZ)times V with THH(ell^BZ)times V at the level of spectra with Frobenius, for V a type 2 complex. I will explain how this spectrum level statement can be upgraded to the level of cyclotomic spectra, after replacing V with an arbitrary type 3 complex, and the Z-action by p^kZ for large k. I will then explain the Bockstein argument that shows that for a large enough k, the coassembly map for the T(2)-homology of TC(ell^hp^kZ) behaves as if the action were trivial. This finishes the proof that the T(2)-local TC of ell^{hp^kZ} is not K(2)-local.

My thesis discusses a bridge between equivariant topology and combinatorics. The kind of problem I look at is an inherently discrete problem which can be solved by translating the problem into showing the nonexistence of a certain map of topological spaces. We will see examples stemming from graph theory, such as the Lovász Conjecture discrete geometry, such as the Randakumar and Rao Conjecture, and general combinatorics.

When is a real smooth manifold secretly a complex manifold? For this, it is necessary, but not sufficient, for the manifold’s tangent bundle to be a complex vector bundle, a condition called being “almost complex”. In this talk, I will give several examples of complex, almost complex, and (orientable, even-dimensional) not-even-almost complex manifolds. I will then discuss how the Atiyah-Singer Index Theorem can be used to show that certain smooth manifolds are not almost complex, focusing on the case of the twisted Dirac operator on spinor bundles on spheres.

I will speak about my recent work (joint with Sam Panitch) constructing the 3d quantum trace map, a homomorphism from the Kauffman bracket skein module of an ideally triangulated 3-manifold to its (square root) quantum gluing module, thereby giving a precise relationship between the two quantizations of the character variety of ideally triangulated 3-manifolds. Our construction is based on the study of stated skein modules and their behavior under splitting, especially into face suspensions.

Friday, Apr. 12th at 12pm, with lunch, lounge at CMSA (20 Garden Street).

Also by Zoom: https://harvard.zoom.us/j/92410768363

It is important to understand if the `solutions’ of non-linear evolutionary PDEs persist for all time or become extinct in finite time through the blow-up of invariant entities. Now the question of this global existence or finite time blow up in the PDE settings is well defined if the regularity of the solution is specified. Most physically interesting scenarios demand control of the point-wise behavior of the solution. Unfortunately, most times this level of regularity is notoriously difficult to obtain for non-linear equations. In this talk, I will discuss very low regularity solutions namely distributional (or weak) solutions of vacuum Einsten’s equations in 3+1 dimensions. I prove that on a globally hyperbolic spacetime foliated by closed connected oriented negative Yamabe slices, weak solutions of the Einstein equations exist for all time. The monotonicity of a Coercive Entity called reduced Hamiltonian that controls the minimum regularity required for the weak solution is employed. This is in the same spirit as Leray’s global weak solutions of Navier-Stokes in 3+1 dimensions and the first result in the context of Einstein equations.

Friday, Apr. 12th at 12pm, with lunch, lounge at CMSA (20 Garden Street).

Also by Zoom: https://harvard.zoom.us/j/92410768363

Transcendental numbers form a mysterious and large class of complex numbers: they are defined as complex numbers that are not the solution of a polynomial equation, and include the numbers pi and e, for example. Within this class, we find the periods that were first studied by Newton and Kepler in the context of celestial mechanics, and which present many curious properties that are the subject of very active research. In this talk, I will give a glimpse of almost 500 years of history of periods, right up to the most recent developments.

Erdos-Kleitman-Rothschild proved that the number of triangle-free graphs on n vertices is asymptotic to the number of bipartite graphs; or in other words, a typical triangle-free graph is a random subgraph of a nearly balanced complete bipartite graph. Osthus-Promel-Taraz extended this result to much lower densities: when m >(\sqrt{3}/4 +eps) n^{3/2} \sqrt{\log n}, a typical triangle-free graph with m edges is a random subgraph of size m from a nearly balanced complete bipartite graph (and this no longer holds below this threshold). What do typical triangle-free graphs at sparser densities look like and how many of them are there? We consider what we call the “ordered” regime, in which typical triangle-free graphs are not bipartite but do align closely with a nearly balanced bipartition. In this regime we prove asymptotic formulas for the number of triangle-free graphs and give a precise probabilistic description of their structure. Joint work with Matthew Jenssen and Aditya Potukuchi.

===============================

For more info, see https://math.mit.edu/combin/

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

The notion of an abstract group encapsulates and illuminates concrete manifestations of symmetry. Recently in quantum field theory there have been discussions of “higher symmetry” and “noninvertiblesymmetry” and their applications.  In joint work with Greg Moore and Constantin Teleman, we propose a conceptual framework for symmetry in quantum field theory, built on the ongoing developments in topological field theory.  It incorporates these newer forms of symmetry, at least with sufficient finiteness conditions.

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

Zoom: https://harvard.zoom.us/j/7855806609

We consider a complex Ginibre ensemble of random matrices with a deformation $H=H_0+A$, where $H_0$ is a Gaussian complex Ginibre matrix and $A$ is a rather general deformation matrix. The analysis of such ensemble is motivated by many problems of random matrix theory and its applications. We use the Grassmann integration methods to obtain integral representation of spectral correlation functions of the first and the second order and discuss the analysis of these representations with a saddle point method.

Abstract TBA

The Cohen-Lenstra heuristic studies the distribution of the p-part of the class group of quadratic number fields for odd prime $p$. Gerth’s conjecture regards the distribution of the $2$-part of the class group of quadratic fields. The main difference between these conjectures is that while the (odd) $p$-part of the class group behaves completely “randomly”, the $2$-part of the class group does not since the $2$-torsion of the class group is controlled by the genus field. In this talk, we will discuss a new conjecture generalizing Cohen-Lenstra and Gerth’s conjectures. The techniques involve Galois cohomology and the embedding problem of global fields.

For more info, see https://ashvin-swaminathan.github.io/home/NTSeminar.html

The derived category of moduli spaces of vector bundles on a curve is expected to be decomposed into the derived categories of symmetric products of the base curve. I will briefly explain the expectation and known results, and some consequences. This is joint work in progress with Kyoung-Seog Lee.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool not just within representation theory. It also has applications to number theory and other areas, and in particular enables progress on various very different forms of the Langlands program.

In this talk, I will introduce p-adic groups and explain how the category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. I will then provide an overview of what we know about the structure of these Bernstein blocks. In particular, I will sketch how to use a joint project in progress with Adler, Mishra and Ohara to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or are at least easier to achieve.

Talk at 3 pm in Science Center 507; Tea at 4 pm in the Math Common Room

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

Suppressing errors is one of the central challenges for useful quantum computing, requiring quantum error correction for large-scale processing. However, the overhead in the realization of error-corrected “logical” qubits, where information is encoded across many physical qubits for redundancy, poses significant challenges to large-scale logical quantum computing. In this talk we will discuss recent advances in quantum information processing using dynamically reconfigurable arrays of neutral atoms. With this platform we have realized programmable quantum processing with encoded logical qubits, combining the use of 280 physical qubits, high two-qubit gate fidelities, arbitrary connectivity, and mid-circuit readout. Using this logical processor with various types of error-correcting codes, we demonstrate that we can improve logical two-qubit gates by increasing code size, outperform physical qubit fidelities, create logical GHZ states, and perform computationally complex scrambling circuits using 48 logical qubits and hundreds of logical gates. We find that this logical encoding substantially improves algorithmic performance with error detection, outperforming physical qubits at both benchmarking and quantum simulations. These results herald the advent of early errorcorrected quantum computation, enabling new applications and inspiring a shift in both the challenges and opportunities that lay ahead.

*In-person and on Zoom*

QR Code & Link:
https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09
Passcode: 657361
https://mathpicture.fas.harvard.edu/seminar

The advent of error-corrected quantum computers is anticipated to usher in a new era in computing, with Shor’s algorithm poised to demonstrate practical quantum advantages in prime number factorization. However, cryptography problems are typically not categorized as scientific computing problems. This raises the question: which scientific computing challenges are likely to benefit from quantum computers? I will first discuss some essential criteria and considerations towards realizing quantum advantages in these problems. I will then introduce some recent advancements in quantum algorithms, especially for simulating non-unitary quantum dynamics and open quantum system dynamics. The first half of the presentation is intended to be accessible to a broad audience, including both theoretical and experimental researchers.

The Cohen-Lenstra heuristic studies the distribution of the p-part of the class group of quadratic number fields for odd prime $p$. Gerth’s conjecture regards the distribution of the $2$-part of the class group of quadratic fields. The main difference between these conjectures is that while the (odd) $p$-part of the class group behaves completely “randomly”, the $2$-part of the class group does not since the $2$-torsion of the class group is controlled by the genus field. In this talk, we will discuss a new conjecture generalizing Cohen-Lenstra and Gerth’s conjectures. The techniques involve Galois cohomology and the embedding problem of global fields.

For more info, see https://ashvin-swaminathan.github.io/home/NTSeminar.html

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

Please see website for more details: www.math.harvard.edu/~ctm/sem.

Tropical intersection theory aims to create analogues of intersection-theoretic tools from algebraic geometry in a piecewise-linear setting. In this talk, I’ll describe a few aspects of tropical intersection theory and discuss how these ideas can be used to build bridges between algebraic geometry, combinatorics, and convex geometry.

===============================

For more info, see https://math.mit.edu/combin/

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

**Special Time and Location**

The degeneracy of a graph is a central measure of sparseness in extremal graph theory. In 1966, Erdős conjectured that $d$-degenerate bipartite graphs have Turán number $O(n^{2-1/d})$. Though this is still far from solved, the bound $O(n^{2-1/4d})$ was proved by Alon, Krivelevich, and Sudakov in 2003. In a similar vein, the Burr–Erdős conjecture states that graphs of bounded degeneracy have Ramsey number linear in their number of vertices. (This is in contrast to general graphs whose Ramsey number can be as large as exponential in the number of vertices.) This conjecture was proved in a breakthrough work of Lee in 2017.In this talk, we investigate the hypergraph analogues of these two questions. Though the typical notion of hypergraph degeneracy does not give any information about either the Ramsey or Turán numbers of hypergraphs, we instead define a notion that we call skeletal degeneracy. We prove the hypergraph analogue of the Burr–Erdős conjecture: hypergraphs of bounded skeletal degeneracy have Ramsey number linear in their number of vertices. Furthermore, we give good bounds on the Turán number of partite hypergraphs in terms of their skeletal degeneracy. Both of these results use the technique of dependent random choice.

===============================

For more info, see https://math.mit.edu/combin/

Time:  Friday April 19 10:00 am – 11:30 am ET  

Location: Harvard CMSA G10

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

Password: cmsa

Markov chains provide a natural approach to sample from various distributions on the independent sets of a graph. For the uniform distribution on independent sets of a given size $k$ in a graph, perhaps the most natural Markov chain is the so-called “down-up walk”. The down-up walk, which essentially goes back to the foundational work of Metropolis, Rosenbluth, Rosenbluth, Teller and Teller on the Markov Chain Monte Carlo method, starts at an arbitrary independent set of size $k$, and in every step, removes an element uniformly at random and adds a uniformly random legal choice.

Davies and Perkins showed that there is a critical $k = \alpha(\Delta)n$ such that it is hard to (approximately) sample from the uniform distribution on independent sets for the class of graphs $G$ with $n$ vertices and maximum degree at most $\Delta$. They conjectured that for $k$ below this critical value, the down-up walk mixes in polynomial time. I will discuss a resolution of this conjecture, which additionally shows that the down-up walk mixes in (optimal) time $O_{\Delta}(n\log{n})$.

Based on joint work with Marcus Michelen, Huy Tuan Pham, and Thuy-Duong Vuong.

===============================

For more info, see https://math.mit.edu/combin/

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

We describe an approach to Bialynicki-Birula theory for holomorphic C^∗ actions on complex analytic spaces and Morse-Bott theory for Hamiltonian functions for the induced circle actions. A key principle is that positivity of a suitably defined “virtual Morse-Bott index” at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum even when it is a singular point in the moduli space. Inspired by Hitchin’s 1987 study of the moduli space of Higgs monopoles over Riemann surfaces, we apply our method in the context of the moduli space of non-Abelian monopoles or, equivalently, stable holomorphic pairs over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and show that these indices are positive in a setting motivated by a conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type (including symplectic four-manifolds) obey the Bogomolov-Miyaoka-Yau inequality.

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

===============================

For more info, see https://math.mit.edu/combin/

Recent progress in gravitational wave astronomy has spurred the development of efficient tools to describe gravitational binary dynamics. One such tool is classical worldline effective field theory (EFT). In the first part of my talk, I will show how to use this EFT for systematic studies of tidal heating and deformations (Love numbers) of compact objects. I will present a gauge-invariant definition of Love numbers and show how to extract them in a coordinate-independent way from scattering amplitudes of the gravitational Raman process. I will show that the worldline EFT exhibits strong fine-tuning when applied to black holes. This gives rise to a naturalness paradox associated with the vanishing of black hole static Love numbers. In the second part of my talk, I will present a new symmetry of black holes (Love symmetry) that elegantly resolves this paradox. The Love symmetry is tightly connected to isometries of extremal black holes that appear in many holographic constructions. It also provides a curious example of IR/UV mixing, which may give insights for other hierarchy problems.

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

Password: cmsa

In collider physics, perturbative quantum field theory is the workhorse framework for computing theoretical predictions to compare to experimental measurements. An observable is called “safe” if its cross section can be predicted order-by-order in perturbation theory with controlled non-perturbative corrections. In this talk, I show that naive definitions of “safety” are inadequate to determine which observable are perturbatively calculable. I then argue for a more refined definition of safety based on principles from optimal transport theory.

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

Password: cmsa

Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials).

In this work, we present a unified approach to constructing stationary measures for several colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (mASEP); (2) the q-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the q-Boson particle system; (3) the q-deformed Pushing Totally Asymmetric Simple Exclusion Process (q-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang–Baxter equation. We express the stationary measures as partition functions of new “queue vertex models” on the cylinder. The stationarity property is a direct consequence of the Yang–Baxter equation. This is joint work with A. Aggarwal and L. Petrov.

===============================

For more info, see https://math.mit.edu/combin/

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemeredi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.

In this talk, we shall discuss various quantitative refinements of this theorem and explain the various ingredients that recently led to the best quantitative bounds for this theorem. This is joint work with Ashwin Sah and Mehtaab Sawhney.

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For more info, see https://math.mit.edu/combin/

Knot Floer homology is a powerful invariant of knots and links, developed by Ozsvath and Szabo in the early 2000s. Among other properties, it detects the genus, detects fiberedness, and gives a lower bound to the 4-ball genus. The original definition involves counting homomorphic curves in a high-dimensional manifold, and as a result the invariant can be hard to compute. In 2007, Manolecu, Ozsvath, and Sarkar came up with a purely combinatorial description of knot Floer homology for knots in the 3-sphere, called grid homology. Soon after, Ozsvath, Szabo, and Thurston defined invariants of Legendrian knots using grid homology. We show that the filtered version of these GRID invariants, and consequently their associated invariants in a certain spectral sequence for grid homology, obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure, strengthening a result of Baldwin, Lidman, and Wong. This is joint work with Jubeir, Schwartz, Winkeler, and Wong.

The CMSA will be hosting a Workshop on Global Categorical Symmetries from April 29–May 3, 2024.

Organizers:

Dan Freed (Harvard CMSA & Math)
Constantin Teleman  (UC Berkeley)

Participation in the workshop is by invitation.

At the Large Hadron Collider, the copious scattering of quarks and gluons in quantum chromodynamics (QCD) produces Higgs bosons and many backgrounds to searches for new physics. At short distances, scattering in QCD can be evaluated in perturbation theory and leads to highly intricate, multivariate mathematical functions such as generalized polylogarithms. To gain further insight, one can study a cousin of QCD called planar N=4 super-Yang-Mills theory. Some processes in this theory can be computed to eighth order in perturbation theory, versus second or third order in QCD. The computation and analysis of these results rely on a Hopf algebra coaction on polylogarithms. Its maximal iteration is called the `symbol’, which serves as a `genetic code’ for amplitudes. The symbol is a linear combination of words, sequences of letters analogous to sequences of DNA base pairs. Understanding the alphabet, and then reading the code, exposes the physics and mathematics of quantum scattering, including bizarre new symmetries. For example, the two scattering amplitudes that are known to the highest orders in perturbation theory (8 loops) are related to each other by an `antipodal duality’, which involves reading the code backwards as well as forwards. A third scattering amplitude, which contains the other two as limits, has an antipodal self-duality which `explains’ the other duality. However, we still don’t know `who ordered’ this property, or what it really means.

The CMSA will be hosting a Workshop on Global Categorical Symmetries from April 29–May 3, 2024.

Organizers:

Dan Freed (Harvard CMSA & Math)
Constantin Teleman  (UC Berkeley)

Participation in the workshop is by invitation.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

Dr. Monique Harrison will be sharing her work with the Harvard Math Department, the conclusion of a survey and interview study in the Fall 2023 semester. Math courses at Harvard serve the majority of undergraduates who matriculate and often can have large impacts on subsequent student decision-making. She will discuss the findings of this study, which targeted first-year course selection and course experiences. Implications for department and university policies will be discussed.