L4

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 $(H_*({\mathcal M}), [ , ])$. 

In a given ambient Riemannian manifold with boundary, geometric objects of particular interest are those properly embedded submanifolds that are critical points of the volume functional, when allowed variations are only those that preserve (but not necessarily fix) the ambient boundary. This variational condition translates into a quite nice geometric condition, namely, minimality and orthogonal intersection with the ambient boundary. Even when the ambient manifold is simply a ball in the Euclidean space, the theory of these objects is very rich and interesting. We would like to discuss several aspects of the theory, including our own contributions to the subject on topics such as: classification results, index estimates and compactness (joint works with different groups of collaborators - I. Nunes, A. Carlotto, B. Sharp, R. Buzano -  will be appropriately mentioned). 

We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold X and define DT4 invariants by integrating the Euler class of a tautological vector bundle against the virtual class. We conjecture a formula for their generating series, which we prove in certain cases when L corresponds to a smooth divisor on X. A parallel equivariant conjecture for toric Calabi-Yau 4-folds is proposed. This conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples. Combining the equivariant conjecture with a vertex calculation, we find explicit positive rational weights, which can be assigned to solid partitions. The weighted generating function of solid partitions is given by exp(M(q) − 1), where M(q) denotes the MacMahon function. This is joint work with Martijn Kool.

Starting from the seminal paper of Caporaso-Harris-Mazur, it has been proved that if Lang's Conjecture holds in arbitrary dimension, then it implies a uniform bound for the number of rational points in a curve of general type and analogue results in higher dimensions. In joint work with Kenny Ascher we prove analogue statements for integral points (or more specifically stably-integral points) on curves of log general type and we extend these to higher dimensions. The techniques rely on very recent developments in the theory of moduli spaces for stable pairs, a higher dimensional analogue of pointed stable curves.
If time permits we will discuss how very interesting problems arise in dimension 2 that are related to the geometry of the log-cotangent bundle.

Following Grothendieck's vision that many cohomological invariants of of an algebraic variety should be captured by a common motive, Voevodsky introduced a triangulated category of mixed motives which partially realises this idea. After describing this category, I will explain how to define the motive of certain algebraic stacks in this context. I will then report on joint work in progress with Victoria Hoskins, in which we study the motive of the moduli stack of vector bundles on a smooth projective curve and show that this motive can be described in terms of the motive of this curve and its symmetric powers.
 

There have been enormous advances in both our ability to represent scattering amplitudes at the integrand-level (for an increasingly wide variety of quantum field theories), and also in our integration technology (and our understanding of the functions that result). In this talk, I review both sides of these recent developments. At the integrand-level, I describe the "prescriptive" refinement of generalized unitarity, and show how closed, integrand-level formulae can be given for all leading-weight contributions to any amplitude in any quantum field theory. Regarding integration, I describe some new results that could be summarized as "dual-conformal sufficiency": that all planar, ultraviolet-finite integrands can be regulated and computed directly in terms of manifestly dual-conformal integrals. I illustrate the power of having such representations, and discuss the role played by a (conjectural) cluster-algebraic structure for kinematic dependence. 

In this talk, I will present the construction of a family of solutions to
vacuum Einstein equations which consist of an arbitrary number of high
frequency waves travelling in different directions. In the high frequency
limit, our family of solutions converges to a solution of Einstein equations
coupled to null dusts. This construction is an illustration of the so called
backreaction, studied by physicists (Isaacson, Burnet, Green, Wald...) : the
small scale inhomogeneities have an effect on the large scale dynamics in
the form of an energy impulsion tensor in the right-hand side of Einstein
equations. This is a joint work with Jonathan Luk (Stanford).

Some of the most studied examples of conformal field theories
include
the Wess-Zumino-Witten models. These are conformal field theories exhibiting
affine Lie algebra symmetry at non-negative integers levels. In this talk I
will
discuss conformal field theories exhibiting affine Lie algebra symmetry at
certain rational (hence fractional) levels whose structure is arguably even
more intricate than the structure of the non-negative integer levels,
provided
one is prepared to look beyond highest weight modules.

Hardy's axiomatic approach to quantum theory revealed that just one axiom
distinguishes quantum theory from classical probability theory: there should
be continuous reversible transformations between any pair of pure states. It
is the single word `continuous' that gives rise to quantum theory. This
raises the question: Does there exist a finite theory of quantum physics
(FTQP) which can replicate the tested predictions of quantum theory to
experimental accuracy? Here we show that an FTQP based on complex Hilbert
vectors with rational squared amplitudes and rational phase angles is
possible providing the metric of state space is based on p-adic rather than
Euclidean distance. A key number-theoretic result that accounts for the
Uncertainty Principle in this FTQP is the general incommensurateness between
rational $\phi$ and rational $\cos \phi$. As such, what is often referred to
as quantum `weirdness' is simply a manifestation of such number-theoretic
incommensurateness. By contrast, we mostly perceive the world as classical
because such incommensurateness plays no role in day-to-day physics, and
hence we can treat $\phi$ (and hence $\cos \phi$) as if it were a continuum
variable. As such, in this FTQP there are two incommensurate Schr\"{o}dinger
equations based on the rational differential calculus: one for rational
$\phi$ and one for rational $\cos \phi$. Each of these individually has a

simple probabilistic interpretation - it is their merger into one equation
on the complex continuum that has led to such problems over the years. Based
on this splitting of the Schr\"{o}dinger equation, the measurement problem
is trivially solved in terms of a nonlinear clustering of states on $I_U$.
Overall these results suggest we should consider the universe as a causal
deterministic system evolving on a finite fractal-like invariant set $I_U$
in state space, and that the laws of physics in space-time derive from the
geometry of $I_U$. It is claimed that such a  deterministic causal FTQP will
be much easier to synthesise with general relativity theory than is quantum
theory.