# Next Week

Here is a selection of seminars that might be of interest to string theorists in Oxford:

On Monday at 14:00 in Dennis Sciama Lecture Theatre, a seminar in
the Astrophysics Colloquia series:

Prof. Steven Finkelstein (Univ. of Texas)

Reionizing the Universe with Low Galaxy Ionizing Escape Fractions and Implications for First Light with JWST

On Monday at 14:15 in L5, a seminar in
the Geometry and Analysis series:

Brent Doran (Oxford)

Geometry of subrings

*The basic algebra-geometry dictionary for finitely generated k-algebras is one of the triumphs of 19th and early 20th century mathematics. However, classes of related rings, such as their k-subalgebras, lack clean general properties or organizing principles, even when they arise naturally in problems of smooth projective geometry. “Stabilization” in smooth topology and symplectic geometry, achieved by products with Euclidean space, substantially simplifies many problems. We discuss an analog in the more rigid setting of algebraic and arithmetic geometry, which, among other things (e.g., applications to counting rational points), gives some structure to the study of k-subalgebras. We focus on the case of the moduli space of stable rational n-pointed curves to illustrate.*

**Further information:**
On Tuesday at 15:45 in L4, a seminar in
the Algebraic Geometry series:

Dominic Joyce (Oxford)

Lie brackets on the homology of moduli spaces, and wall-crossing formulae

*Let $\mathbb K$ be a field, and $\mathcal M$ be the “projective linear' moduli stack of objects in a suitable $\mathbb K$-linear abelian category $\mathcal A$ (such as the coherent sheaves coh($X$) on a smooth projective $\mathbb K$-scheme $X$) or triangulated category $\mathcal T$ (such as the derived category $D^b$coh($X$)). I will explain how to define a Lie bracket [ , ] on the homology $H_*({\mathcal M})$ (with a nonstandard grading), making $H_*({\mathcal M})$ into a graded Lie algebra. This is a new variation on the idea of Ringel-Hall algebra. There is also a differential-geometric version of this: if $X$ is a compact manifold with a geometric structure giving instanton-type equations (e.g. oriented Riemannian 4-manifold, $G_2$-manifold, Spin(7)-manifold) then we can define Lie brackets both on the homology of the moduli spaces of all $U(n)$ or $SU(n)$ connections on $X$ for all $n$, and on the homology of the moduli spaces of instanton $U(n)$ or $SU(n)$ connections on $X$ for all $n$. All this is (at least conjecturally) related to enumerative invariants, virtual cycles, and wall-crossing formulae under change of stability condition. Several important classes of invariants in algebraic and differential geometry — (higher rank) Donaldson invariants of 4-manifolds (in particular with $b^2_+=1$), Mochizuki invariants counting semistable coherent sheaves on surfaces, Donaldson-Thomas type invariants for Fano 3-folds and CY 4-folds — are defined by forming virtual classes for moduli spaces of “semistable” objects, and integrating some cohomology classes over them. The virtual classes live in the homology of the “projective linear' moduli stack. Yuuji Tanaka and I are working on a way to define virtual classes counting strictly semistables, as well as just stables / stable pairs. I conjecture that in all these theories, the virtual classes transform under change of stability condition by a universal wall-crossing formula (from my previous work on motivic invariants) in the Lie algebra $(H_*({\mathcal M}), [ , ])$.*

**Further information:**
On Thursday at 12:45 in L6, a seminar in
the Strings Junior series:

Mark van Loon

6d (2,0) SCFT

On Thursday at 13:00 in Dalitz Institute, a seminar in
the Dalitz Seminar in Fundamental Physics series:

Edward Hardy (Liverpool)

The QCD axion and other possible new light particles

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**Further information:**
On Thursday at 16:00 in L6, a seminar in
the Number Theory series:

Lucia Mocz (Princeton)

A New Northcott Property for Faltings Height

*The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.*

**Further information:**
On Thursday at 16:15 in Dennis Sciama Lecture Theatre, a seminar in
the Theoretical Particle Physics series:

Peter Richardson (CERN Geneva)

Tools for colliders: HERWIG