## A proof of a general slice-Bennequin inequality

In this blog post, I’ll provide a slick proof of a form of the slice-Bennequin inequality (as outlined by Kronheimer in a mathoverflow answer.) The main ingredient is the adjunction inequality for surfaces embedded in closed 4-manifolds. To obtain the slice-Bennequin inequality (which is a statement about surfaces embedded in 4-manifolds with boundary) we use […]

## What Topological Spaces are π₀?

This was a fun question I thought about once.  My answer is at the end, in case you’d like to try solving the problem yourself.  The question is likely more interesting than my solution. A well known theorem says that every group occurs as for some topological space .  It’s not a hard construction; is […]

## Isometries of a product of Riemannian manifolds

Theorem. Let and be two compact Riemannian manifolds with irreducible holonomy groups. Let . Then This result seems to be a folklore, probably well known to the specialists, although it is hard to find it in the literature. The only discussion which I managed to find on Mathoverflow contains an answer by Igor Rivin (due […]

## Demystification of the Willmore integrand

The Willmore energy for a surface in Euclidean 3-space is defined as , where is the mean curvature of and its area form. It’s known to be invariant under the conformal transformations (whereas the mean curvature itself is not). White, and later Bryant noticed that the 2-form , where stands for the Gaussian curvature, is […]

## Simple Proof of Tokuyama’s Formula

Tokuyama’s Formula is a combinatorial result that expresses the product of the deformed Weyl denominator and the Schur polynomial as a sum over strict Gelfand-Tsetlin patterns. This result implies Gelfand’s parametrization of the Schur polynomial, Weyl’s Character Formula, and Stanley’s formula on Hall-Littlewood polynomials — all for ; also, the formula is related to alternating […]

## The Picard number of a Kummer K3 surface

Let be a separably closed field of characteristic not , and an abelian surface. Then it is a basic fact (e.g. see Example 1.3 (iii) of Huybrechts’ “K3 Surfaces”) that one can make a K3 surface out of . The construction is as follows. Consider the involution given by The fixed locus of this involution […]

## The torsion component of the Picard scheme

This post is a continuation of Sean Cotner’s most recent post [see An example of a non-reduced Picard scheme]. Since writing that post, Bogdan Zavyalov shared some notes of his proving the following strengthened version of the results described there. Main Theorem. Let be a noetherian local ring and let be a finite flat commutative […]

## The étale cohomology of curves over finite fields

When I was a graduate student, Zev Rosengarten (a former student of Brian Conrad) and I used to eat dinner at Stanford’s Arrillaga dining hall a lot. We’d talk about math for hours, but one thing that will forever be ingrained in my mind is how Zev was able to do all these complicated spectral […]

## An explicit construction of indecomposable vector bundles over an elliptic curve

In the celebrated paper “Vector bundles over an elliptic curve,” M. Atiyah classifies indecomposable vector bundles, namely he provides a bijection between indecomposable bundles of arbitrary rank and degree (denoted by ) and (where ). The latter is described explicitly: there is a distinguished element such that for any other one has with a line […]

## A shortcut in Kapovich’s proof of Haupt’s theorem

The Teichmüller space of genus curves carries the Hodge bundle , the total space of which maps into the first cohomology space via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image in terms of the integral structure on and the intersection pairing . It […]