Calendar

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/

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/