« Return of the IHÉS Postdoc Seminar »

 

Abstract: In this talk, I will talk about the motivic McKay correspondence studied by Batyrev and Denef-Loeser and its generalization to positive and mixed characteristics by myself partly in collaboration with M.M. Wood. The latter relates stringy invariants of singularities to weighted counts of Galois extensions of a local field.

For a finite dimensional linear representation of a finite group, the motivic McKay correspondence says that the motivic stringy invariant of the associated quotient variety is equal to a finite sum of classes of affine spaces in a certain modification of the Grothendieck ring of varieties.

In order to understand wild quotient singularities, which are known to be typical “bad” singularities in positive/mixed characteristics, I started an attempt to generalize the motivic McKay correspondence to positive and mixed characteristics and formulated a conjectural generalization. Here, the finite sum of classes of affine varieties is replaced with a motivic integral over the (conjectural) moduli space of G-covers of formal discs. When the base field (or the residue field if working over a complete DVR) is finite, then the point counting realization of this motivic integral is a weighted count of Galois extensions of the power series field (or the fraction field of the DVR). Such counts were previously studied by number-theorists including Krasner, Serre, Bhargava, Kedlaya and Wood. The point counting version of the McKay correspondence was recently proved by myself. A part of my motivation of this work is a search for a counterexample of resolution of singularities, which has not been successful till now.

If time allows, I will also explain that from this result and a heuristic argument, one can relate Malle’s conjecture on distribution of Galois extensions of number fields and Manin’s conjecture on distribution of rational points on Fano varieties.

Small compact objects orbiting supermassive back holes are an important potential source of gravitational radiation. Detection of such waves and the parameter estimation of their sources will require accurate waveform templates. To this eventual end, I present work on bound eccentric motion around a static black hole. In two separate approaches, I examine solutions to the first order (in mass-ratio) field equations. First, I consider solving the field equations entirely analytically in a double post-Newtonian/small-eccentricity expansion. Then I show numeric work wherein we use the MST formalism to solve the field equations to 200 digits. We use this extreme accuracy to fit for previously unknown PN energy flux parameters, extending the previous state of the art from 3PN to 7PN.

« Return of the IHÉS Postdoc Seminar »

 

Abstract: In this talk, we will recall the theory of generalized HKR isomorphisms, due to Arinkin and Cāldāraru. Then we will explain how deformation theory allows us to describe arbitrary first order derived intersections.

« Return of the IHÉS Postdoc Seminar »

 

Abstract: Given a complex reductive group G, there is expected to be a generalization of Donaldson-Thomas theory whose goal is to count, in an appropriate sense, stable principal G-bundles over a Calabi-Yau threefold. The standard Donaldson-Thomas theory arises when G is a general linear group. I will present some recent results on such a generalization when G is a classical group using the framework of quiver representations. The key new tool is a representation of Kontsevich and Soibelman's cohomological Hall algebra which is constructed from the cohomology of moduli stacks of quiver theoretic analogues of G-bundles.

We establish connections between two objects, naturally arising in the theory of the Kadomtsev-Petviashvily equation: totally positive Grassmannians and rational degenerations of the M-curves (Riemann surfaces with an anitiholomorphic involutuin and the maximal possible number of real ovals) with a collection of marked points. More precisely, we show that, at least all points from the principal cell of the Grassmannian can be obtained from degenerate M-curves. (Joint work with Simonetta Abenda (Universita degli Studi di Bologna).)

Many of the main conjectures in Iwasawa theory can be phrased as saying that the first Chern class of an Iwasawa module is generated by a p-adic L-series. In this talk I will describe how higher Chern classes pertain to the higher codimension behavior of Iwasawa modules. I'll then describe a template for conjectures which would link such higher Chern classes to elements in the K-theory of Iwasawa algebras which are constructed from tuples of Katz p-adic L-series. I will finally describe an instance in which a result of this kind, for the second Chern class of an unramified Iwasawa module, can be proved over an imaginary quadratic field. This is joint work with F. Bleher, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi and M. J. Taylor.

I will discuss the use of semi-classics and instanton calculus and argue that, contrary to common wisdom, complex solutions of the equations of motion are a necessary ingredient of semi-classical expansion. In particular, I will show that without the complex solutions semi-classical expansion of supersymmetric theories cannot be reconciled with the constraints of supersymmetry. This has a natural interpretation in the Picard-Lefschetz theory.

In this blackboard presentation, we will outline a new Palatini formalism for Galileon scalar field theories. After a pedagogical review of the Palatini formalism for GR, we will outline how this carries over to the spin-0 case. This will uniquely single out the Galileon symmetry and provides a first-order formalism for such theories. Finally, we present possible extensions including the coupling to gravity.

The standard perturbation theory leads to the asymptotic series because of the illegal interchange of the summation and integration. However, changing the initial approximation of the perturbation theory, one can generate the convergent series. We study the lattice phi4-model and compare observables calculated using the convergent series and Monte Carlo simulations. Then, we discuss the generalization of the same ideas for the continuum phi4-model and QCD.

After overviewing the representation of biological molecules, I introduce the representation of interactions between modules in the proteins with the SO(3) rotation of the peptide unit introduced by Penner et al.. Using protein data in PDB, I will show our representation with the SO(3) rotation is useful in characterizing the structure and its change due to the mutation or the native dynamical change with time. I also discuss some possible potential extensions of our approach to the chromatin structure.

It is well known that ideal projective measurements are paradigmatic non-deterministic and irreversible processes in Quantum Mechanics. Nevertheless it is also known that the associated probabilities satisfy a time-symmetry property: the conditional probabilities for prediction and retrodiction take the same form. I shall argue that this feature of Quantum Mechanics may be used to discuss it in a more natural way and to present it as a less mysterious theory than is usually done. This will be shown for the Algebraic Formulation and the Quantum Logic Formulation. If time permits, I shall ask some naïve questions about the Quantum Information Formulations and Quantum Gravity.