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.

It has long been realized by microlocal analysts that an alternative, ´Euclidean-signature´ semi-classical ansatz for solutions to Schrödinger´s equation is much more natural, from a mathematical point of view, than the classical, physics textbook, W.K.B. ansatz, even though the latter has the conceptual advantage of emphasizing the classical correspondence principle. For technical reasons though this microlocal approach has seemed to be limited to quantum mechanical problems and not to be applicable to quantum field theories. In this seminar I will review this method and discuss an expansive, ongoing project with A. Marini (Yeshiva, L´Aquila) and R. Maitra (Wentworth I.T.) to extend these influential ideas to bosonic quantum field theories including phip type scalar and Yang-Mills fields.

 

« Return of the IHÉS Postdoc Seminar »

 

Abstract: The mod-p Local Langlands Program aims to find a relationship between the mod-p representation theory of p-adic reductive groups and Galois groups, in the spirit of the classical Langlands correspondence on complex vector spaces.  I'll describe several aspects of this theory, and outline how techniques from the classical case fail in the mod-p setting.  Finally, I'll describe how modules over certain Hecke algebras can be used as an avatar for this correspondence in certain situations.

« Return of the IHÉS Postdoc Seminar »

 

Abstract: The Andre-Oort conjecture is an important problem in arithmetic geometry concerning subvarieties of Shimura varieties. It attempts to characterise those subvarieties V for which the special points lying on V constitute a Zariski dense subset. When the ambient Shimura variety is the moduli space of principally polarised abelian varieties of dimension g (in which case, a special point is a point corresponding to the isomorphism class of a CM abelian variety), the conjecture has been obtained by Pila and Tsimerman via the so-called Pila-Zannier strategy, reliant on the Pila-Wilkie counting theorem on o-minimal structures. In this talk, we will outline the Pila-Zannier strategy, providing some introduction to Shimura varieties and Andre-Oort, and explain the state of the art for the full conjecture. In particular, we will mention certain height bounds obtained jointly with Orr.

 

 

« 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.

« 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.

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).)

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.

We consider measurements performed on quantum systems. When the measurement outcomes are lost or ignored, decoherence in the system is unavoidable. When it is taken into account, the system purifies until the information flow is blocked by "darkness".

In the spirit of 4/1 I will confront old and recent speculations in quantum biology and cognitive philosophy with experiments, mostly from neurobiology.