In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in R^r with angle arccos(alpha) and gave a partial answer in the regime r <= 1/alpha^2 – 2. At the other extreme where r is at least exponential in 1/alpha, recent breakthroughs have led to an almost complete resolution of this problem. In this talk, we introduce a new method for obtaining upper bounds which unifies and improves upon all previous approaches, thereby yielding bounds which bridge the gap between the aforementioned regimes and are best possible either exactly or up to a small multiplicative constant. A crucial new ingredient of our approach is orthogonal projection of matrices with respect to the Frobenius inner product and it also yields the first extension of the Alon-Boppana theorem to dense graphs, with equality for strongly regular graphs corresponding to r(r+1)/2 equiangular lines in R^r. Applications of our method in the complex setting will be discussed as well.

The fractional Dehn twist coefficient (FDTC) is an invariant of a self-map of a surface which is some measure of how the map twists near a boundary component of the surface. It has been studied for compact (or finite-type) surfaces; in this setting the invariant is always a fraction. I will discuss work to extend this invariant to infinite-type surfaces and show that it has surprising properties in this setting. In particular, the invariant no longer needs to be a fraction – any real number amount of twisting can be achieved! I will also discuss a new set of examples of (tame) big mapping classes called wagon wheel maps which exhibit irrational twisting behavior. This is joint work with Diana Hubbard and Peter Feller.

Gravitational instantons are non-compact Calabi-Yau metrics with L^2 bounded curvature and are categorized into six types. We will discuss one such type called ALH* metrics which has a non-compact end modelled by the Calabi ansatz with inhomogeneous collapsing near infinity. Such metrics  appeared recently in the works on SYZ conjecture, as well as the scaling bubble limits for codimension-3 collapsing of K3 surfaces, where the study of its Laplacian played a central role. In this talk I will talk about the Fredholm mapping property and L^2 cohomology of such metrics. This is ongoing work joint with Rafe Mazzeo.

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

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In characteristic zero, Ueno’s conjecture states that if X is a smooth projective variety with Kodaira dimension zero, then its Albanese morphism in an algebraic fiber space and the Kodaira dimension of the general fiber is again zero. This was proven by Cao and Păun in 2016.

Building on the generic vanishing techniques of Hacon and Patakfalvi, we prove a positive characteristic version of this result. We use it to deduce new cases of Iitaka’s subadditivity conjecture in positive characteristics. The goal of this talk is to explain how these techniques work, and how we can use them to prove such results.

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

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

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https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09
Password: cmsa

There is much excitement about the opportunity to harness the power of large language models (LLMs) when building problem-solving assistants. However, the standard methodology of evaluating LLMs based on static pairs of inputs and outputs is insufficient to be able to make an informed decision about which LLMs, and under what assistive settings they can be sensibly utilised. Static assessment fails to take into account the essential interactive element in their deployment, and therefore limits how we understand language model capabilities. In this talk, we present our recent work introducing CheckMate, an adaptable prototype platform for humans to interact with and evaluate LLMs. We discuss our study with CheckMate to evaluate three language models (InstructGPT, ChatGPT, and GPT-4) as assistants in proving undergraduate-level mathematics, with a mixed cohort of participants from undergraduate students to professors of mathematics. We release the resulting interaction and rating dataset, MathConverse. By analysing MathConverse, we derive a taxonomy of human behaviours and uncover that despite a generally positive correlation, there are notable instances of divergence between correctness and perceived helpfulness in LLM generations, amongst other findings. Further, we identify useful scenarios and existing issues of GPT-4 in mathematical reasoning through a series of case studies contributed by expert mathematicians. We conclude with actionable takeaways for ML practitioners and mathematicians: models which communicate uncertainty, respond well to user corrections, and are more interpretable and concise may constitute better assistants; interactive evaluation is a promising way to navigate the capability of these models; humans should be aware of language models’ algebraic fallibility and discern where they are appropriate to use.

https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09
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I will report on joint work in progress with Andrew Booker on the practical implementation of an axiomatic approach to the enumeration of arithmetic L-functions that lie in a certain subset of the Selberg class that is expected to include all L-functions of abelian varieties. As in the work of Farmer, Koutsoliotas, and Lemurell, our approach is based on the approximate functional equation. We obtain additional constraints by considering twists (and more general Rankin-Selberg convolutions) of our unknown L-function that yield a system of linear constraints that can be solved using the simplex method. This allows us to significantly extend the range of our computations for the family of L-functions associated to abelian surfaces over ℚ. We also introduce a method for certifying the completeness of our enumeration.

Abstract TBA

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

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

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Flows in the fluidic interior of living cells can serve biological function or act as signatures of how intracellular forces are exerted. I’ll discuss examples of each. One is understanding the emergence of cell-spanning vortical flows in large developing egg cells, while the other arises in studying the nature of force transduction in single cell embryos moving towards their first cell division. Both involve the cytoskeleton, that set of polymers, cross-linkers, and molecular motors that underlie much of the active mechanics within cells, and has led to the development of new coarse-grained active matter models and novel instabilities.

Lunch served at 12:30.

This seminar will be held in person and on Zoom.

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

Password: cmsa

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

Password: cmsa

Symmetric mass generation (SMG) is a novel interaction-driven mechanism that generates fermion mass without breaking symmetry, unlike the standard Anderson-Higgs mechanism. SMG can occur in the fermion system without quantum anomalies. In this talk, I will focus on the SMG for the systems with finite fermion density, i.e., the Fermi surface. I will discuss the Fermi surface anomaly and Fermi surface SMG. Lastly, I will talk about its application in the newly found nickelate superconductors, where the superconductivity emerges without a nearby spontaneous symmetry-breaking phase.

The physics of M-theory and Type IIA strings on a projective nodal CY 3-folds is determined by the geometry of a small resolution, even if the latter is not Kähler.
We will demonstrate this explicitly in the context of a family of Calabi-Yau double covers of P^3.
Using conifold transitions, we prove that the exceptional curves in any small resolution are torsion while M-theory develops a discrete gauge symmetry.
This leads to a torsion refinement of the ordinary Gopakumar-Vafa invariants, that is associated to the singular Calabi-Yau and captures the enumerative geometry of the non-Kähler resolutions.
We further argue that twisted circle compactifications of the 5d theory are dual to IIA compactifications on the nodal CY 3-fold with a flat but topologically non-trivial B-field.
As a result, the torsion refined invariants are encoded in the topological string partition functions with different choices for the global topology of a flat B-field.

The talk is based on 2108.09311, 2212.08655 (with S. Katz, A. Klemm, and E. Sharpe) and 2307.00047 (with S. Katz).

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

Password: cmsa

Singular homology has a number of well-known defects when used to study spaces such as the Hawaiian earring and solenoids. It may not reflect the “shape” of the space and can give counterintuitive information about its dimension. One remedy of this is to develop a homology theory based on approximating spaces by polyhedra, computing their homologies, and then taking a limit. This is the approach taken by Steenrod-Sitnikov homology and Lisica and Mardesic’s strong homology. Even within the class of locally compact second countable spaces though, the properties of these homology theories — and the higher derived limits which underly them — are dependent on axioms of set theory beyond ZFC. Recently it was shown that it is consistent with (and therefore independent of) ZFC that strong homology and Steenrod Sitnikov homology coincide in the class of locally compact second countable spaces — and therefore each of these homology theories enjoys the desirable properties of the other. These results also point to how we might develop variants of these homology theories which enjoy their desirable properties, but which are less sensitive to set theory. This is joint work with Nathaniel Bannister, Jeff Bergfalk, and Stevo Todorcevic.

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