I review the construction of maximal supergravities in two dimensions. These are parametrized by an embedding tensor transforming in the basic representation of the affine algebra E9. Among the examples is the SO(9) theory describing the low-energy effective action upon reduction of the type IIA theory on the D0-brane near-horizon geometry.

Recently, Weinstein finds some affinoids in the Lubin-Tate perfectoid space and computes their reduction in equal characteristic case. The cohomology of the reduction realizes the local Langlands correspondence for some representations of GLh, which are unramified in some sense. In this talk, we introduce other affinoids in the Lubin-Tate perfectoid space in equal characteristic case, whose reduction realizes "ramified" representations of conductor exponent h+1. We call them ramified affinoids. We study the cohomology of the reduction and its relation with the local Langlands correspondence. This is a joint work with Takahiro Tsushima.

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Morphogens are secreted molecules that influence the development of cells in embryonic tissue. Patterns of gene expression or signalling immediately downstream of many morphogens such as the bone morphogenetic protein (BMP) and decapentaplegic (Dpp) are highly reproducible and robust to perturbations. This is perhaps surprising in view of experimentally determined low concentration (approximately picomolar) range of Dpp, tight receptor binding and very slow kinetic rates, all of which should favor large fluctuations in signalling.

To shed light on this issue, we developed a stochastic model of Dpp signalling in Drosophila melanogaster and used the model to quantify the extent to which stochastic fluctuations would lead to errors in spatial patterning. This was done both with and without a surface-associated BMP-binding protein (SBP), which is an auxiliary protein that plays no direct role in signalling. By analyzing this model over a wide range of parameter values, we find that such SBPs are likely to play an important role in the reliability of morphogen patterning.

In his fundamental paper on the projective line minus three points, Deligne constructed certain extensions of mixed Tate motives arising from the fundamental group of the projective line minus three points. Since then, motivic structures on homotopy groups have been studied by many authors. In this talk, we will construct a motivic structure on the (nilpotent completion of) n-th homotopy group of Pn minus n+2 hyperplanes in general position.

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Coleman and Vologodsky integration theories give canonical parallel transports for unipotent differential equations – Coleman on overconvergent spaces with good reduction and Vologodsky on algebraic varieties, both over p-adic fields. In Coleman’s theory the transport is via a path invariant under the action of Frobenius while in Vologodsky’s theory one adds a condition involving the monodromy operator. While both theories can be formulated in fairly similar terms, the precise relationship between them is a bit unclear.

In this talk, based on joint work in progress with Sarah Zerbes, I will describe some background on the two integration theories and I will describe the simplest non-trivial case – a holomorphic form on a curve with semi-stable reduction, where we can say what the relation should be. Time permitting I’ll discuss possible applications to syntomic regulators.

I will present a new approach for a formula of Deligne and Kato that computes the dimension of the nearby cycle complex of an $ell$-adic sheaf on a smooth relative curve over a strictly henselian trait of characteristic $p ne ell$. Deligne considered the case where the sheaf has no vertical ramification and Kato extended the formula to the general case. My approach is based on ramification theory of Abbes and Saito. It computes the nearby cycle complex in terms of the refined Swan conductor. In fact, I compare Abbes-Saito’s refined Swan conductor with Kato’s Swan conductor with differential values, which is the key ingredient in Kato’s formula; the case of rank one sheaves is due to Abbes and Saito. My approach provides also a new independent proof of Delinge-Kato’s formula.

Thomas’ conjecture about the necessity of a positive circuit for the multistationarity has been proved both in the discrete and continuous setting. Nevertheless it has mostly been used in the discrete case since, as we will demonstrate, it is almost always satisfied by biochemical reaction networks. In order to work around this shortcoming, we will go back to the decomposition of the Jacobian matrix determinant and notice that, for dynamical systems corresponding to biochemical reactions, certain patterns allow the statement of a stronger necessary condition. We will illustrate how that new condition rules out the cases that trivially satisfied the classical condition of Thomas on a simple example of enzymatic reaction.

The infrared structure of quantum gravity is explored by solving a lattice version of the Wheeler-DeWitt equations. For this talk, first, the case of 2+1 dimensions is presented. The wavefunction solutions only depend on the geometric quantities indicating preservation of diffeomorphism. Properties of the lattice vacuum are consistent with the existence of an ultraviolet fixed point in G located at the origin, thus precluding the existence of a weak coupling perturbative phase. The correlation length exponent is determined exactly and found to be nu=6/11. The results obtained lend support to the claim that the Lorentzian and Euclidean formulations belong to the same field-theoretic universality class. I then discuss some results in 3+1 dimensions. Investigations of the vacuum wave functional further indicate that for weak enough coupling, G < Gc, a pathological ground state with no continuum limit appears, where configurations with small curvature have vanishingly small probability.