Calendar

We shall first explain the relation between a family of deformations of genus zero p-adic string worldsheet action and Tate’s thesis. We then propose a genus one p-adic string worldsheet action. The key is the definition of a p-adic Laplacian operator on the Tate curve. We show that the genus one p-adic Green’s function exists, is unique under some obvious constraints, is locally constant off diagonal, and has a reflection symmetry. It can also be numerically computed exactly off the diagonal, thanks to some simplifications due to the p-adic setup. Numerics suggest that at least in some special cases, the asymptotic behavior of the Green’s function near the diagonal is a direct p-adic counterpart of the familiar Archimedean case, although the p-adic Laplacian is not a local operator. Joint work in progress with Rebecca Rohrlich.

A striking correspondence between dynamics of automorphism groups of countable first order structures and Ramsey theory of finitary approximation of the structures was established in 2005 by Kechris, Pestov, and Todocevic. Since then, their work has been generalized and applied in many directions. It also struck a fresh wave of interest in finite Ramsey theory. Many classes of finite structures are shown to have the Ramsey property by encoding their problem in a known Ramsey class and translating a solution back. This is often a case-by-case approach and naturally there is a great need for abstracting the process. There has been much success on this front, however, none of the tools captures every situation. We will discuss one such encoding via a model-theoretic notion of semi-retraction introduced by Lynn Scow in 2012. In a joint work, we showed that a semi-retraction transfers the Ramsey property from one class of structures to another under quite general conditions. We compare semi-retractions to a category-theoretical notion of pre-adjunction revived by Mašulović in 2016. If time permits, I will mention a transfer theorem of the Ramsey property from a class of finite structures to their uncountable ultraproducts, which is an AIMSQuaRE project with Džamonja, Patel, and Scow.

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

Password: cmsa

We will investigate if some well known properties of log canonical singularities over the complex numbers still hold true over perfect fields of positive characteristic and over excellent rings with perfect residue fields. We will discuss both pathological behavior in characteristic p as well as some positive results for threefolds. We will see that the pathological behavior of these singularities in positive characteristic is closely linked to the failure of certain vanishing theorems in positive characteristic. Additionally, we will explore how these questions are related to the moduli theory of varieties of general type.
This is based on joint work with F. Bernasconi and Zs. Patakfalvi, as well as joint work with Q. Posva.

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

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

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

Discrete Morse theory, developed by Forman, is an efficient tool to determine the homotopy type of a regular CW complex. The theory has been reformulated by Chari in purely combinatorial terms of acyclic matchings on the face poset. In this talk, I will discuss explicit constructions of such acyclic matchings on Bruhat intervals using reflection orders. As an application, we show the totally nonnegative Springer fibres are contractible, verifying a conjecture of Lusztig. This is based on work in progress with Xuhua He.

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

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.