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Geometric realizations of Brauer classes on K3 surfaces
Sarah Frei, Dartmouth College
Geometric realizations (as etale projective bundles) of Brauer classes on surfaces have many applications, such as the arithmetic of surfaces over non-closed fields and rationality of fourfolds (even over the complex numbers). In this talk, I will discuss joint work with Jack Petok and Anthony Varilly-Alvarado, in which we consider constructions of these etale projective bundles for Brauer classes on K3 surfaces. This builds on recent results of van Geemen and Kaputska, who show that some 2-torsion Brauer classes on K3 surfaces have realizations as the exceptional locus of a divisorial contraction on a hyperkahler fourfold.
On Lack of Existence of Local Formulas for $Z_2$ invariants
Ahmad Reza Haj Saeedi Sadegh, Dartmouth College
In the 5th volume of their celebrated articles (The Index of Elliptic Operators: V), Atiyah and Singer discuss elliptic differential operators over real numbers instead of complex numbers. These real elliptic operators usually produce more refined indices than the usual index of complex elliptic operators. Such indices seem not to be accessible via local expressions in general. However, there are still topological formulas for these indices in K-theoretical terms. In certain settings, we give topological formulas in terms of the integral of characteristic forms.
Wednesday, February 5
Thursday, February 6
13:20–14:20 Combinatorics Seminar, Zoom
A New Symmetric Function Identity With an Application to Symmetric Group Character Values
Karlee Westrem, Michigan Tech
Symmetric functions show up in several areas of mathematics including enumerative combinatorics and representation theory. Tewodros Amdeberhan conjectures equalities of $\Sigma_n$ characters sums over a new set called $Ev(\lambda)$. When investigating the alternating sum of characters for $Ev(\lambda)$ written in terms of the inner product of Schur functions and power sum symmetric functions, we found an equality between the alternating sum of power sum symmetric polynomials and a product of monomial symmetric polynomials. As a consequence, a special case of an alternating sum of $\Sigma_n$ characters over the set $Ev(\lambda)$ equals $0$.
Path induction and the indiscernibility of identicals
Emily Riehl, Johns Hopkins University
Mathematics students learn a powerful technique for proving theorems about an arbitrary natural number: the principle of mathematical induction. This talk introduces a closely related proof technique called "path induction," which can be thought of as an expression of Leibniz's "indiscernibility of identicals": if x and y are identified, then they must have the same properties, and conversely. What makes this interesting is that the notion of identification referenced here --- given by Per Martin-Löf's intensional identity types --- encodes a more flexible notion of sameness than traditional equality because an identification can carry data, for instance of an explicit isomorphism or equivalence. The nickname "path induction" for the elimination rule for identity types derives from a new homotopical interpretation of type theory, in which the terms of a type define the points of a space and identifications correspond to paths. In this homotopical context, indiscernibility of identicals is a consequence of the path lifting property of fibrations and path induction is justified by the fact that based path spaces are contractible.
Queer theory challenges essentialist or normative conceptions of identity, often using personal stories to show the limits of conventional categories. In this talk, I will share my journey as a queer mathematician, tracing the formation of a pair of identities that were more often developed in parallel rather than fully integrated. I will also describe a recent "queering" of the concept of "identity" in mathematics, which enables a more expansive notion of mathematical equality than appears in traditional mathematical foundations.
