21 JAN 2021
11:00 - 13:00

Shortly after Eckmann and Schopf's definition of the concept of an injective modules, Matlis showed that the injective hull of a simple module over a commutative Noetherian ring R is Artinian. Vamos showed that the latter condition is actually equivalent to the commutative ring R being locally Noetherian. However, injective hulls of simple modules over non-commutative Noetherian rings might not be Artinian. Slightly different, but related, one might consider Noetherian algebras A over a field F and ask whether any finitely generated submodule of the injective hull of a finite dimensional simple A-module need to be finite dimensional. In other words, is the category of locally finite dimensional representations of A closed under taking injective hulls. This is certainly the case for commutative algebras, but not for non-commutative algebras. Feldvoss proved, using results of Donkin and Dahlberg, that the category of locally finite dimensional representations over an enveloping algebra $U(g)$ of a finite dimensional Lie algebra $g$ is closed under taking injective hulls if and only if $g$ is a solvable Lie algebra. Furthermore, the group ring $F[G]$ of a polycyclic-by-finite group also has this property. Having these motivating examples in mind we intend to find necessary and sufficient conditions for the category of locally finite dimensional representations over a Noetherian algebra A over a field F to be closed under taking injective hulls.

In the first talk we will characterize Noetherian algebras A whose category of locally finite dimensional representations is essentially closed through their finite dual coalgebra
$$A^circ = {f in Hom(A,F): mathrm{Ker}(f) mbox{ contains an ideal of finite codimension}}$$ and provide sufficient conditions that involve the maximal ideals of A of finite codimension. We then consider Ore extensions of an algebra A, which are non-commutative polynomial rings in a variable x such that the coefficients in A obey to $$xa = sigma(a)x+delta(a) , qquad forall a in A$$ for a given automorphism $sigma$ and a $sigma$-derivation $delta$ of $A$. As an important tool we will use the Rees ring of an ideal in the Ore extension and find condition under which it is Noetherian.

In the second talk we will provide some background on the theory of Hopf algebras, having in mind that enveloping algebras of Lie algebras and group algebras are fundamental example. We will show that the question of the category of locally finite dimensional representations of a Noetherian Hopf algebra to be essentially closed can be reduced to the study of finitely generated extensions of the trivial representation. Our main results will show that Ore extension of commutative Noetherian Hopf algebras, certain Hopf crossed product algebra $A#_sigma H$ and all affine Hopf algebra domains of Gelfand-Kirillov dimension $leq 2$ with $Ext_H^1(F,F)
eq 0$ are Noetherian algebras whose category of locally finite representations is essentially closed.

Throughout the talks we will assume that the base field is algebraically closed and has characteristic zero.

The talks will be based on the preprint ``Locally Finite Representations Over Noetherian Hopf algebras" jointly written with my coauthor Can Hatipoglu. A PDF of this preprint can be found on ArXiv: href{https://arxiv.org/abs/2010.14192}{ exttt{https://arxiv.org/abs/2010.14192}}





Access link:


https://meet.google.com/ndk-abim-gxy

21 JAN 2021
11:00 - 12:00

Journal Club
http://meet.google.com/iys-ixfq-gwm

20 JAN 2021
16:30 - 18:00

The ancient Greek word "geometry" is, with its etymological meaning of "measurement of the earth", one of the oldest of mathematics. The discovery of non-euclidean geometries in the 19th century dramatically increased the scope of geometry. This scope was further extended in the 20th century by dividing geometry into branches differing in their objects of study and their methods: topology, differential geometry, Riemannian geometry, symplectic geometry, complex geometry, algebraic geometry, ... Grothendieck is known primarily for having re-founded algebraic geometry on entirely new bases. But he himself considered himself to be a general mathematician, not a specialist. So one may wonder whether the word "geometry", which he used very often without ever defining it, has for him a precise meaning that goes beyond algebraic geometry, and whether certain notions he introduced or to which he gave a central role are likely to apply to everything that one might imagine to be called geometric. The aim of the presentation will be to propose elements of an answer to this question.

To get more information about the colloquium, join to the following google group:
https://groups.google.com/g/ipm-math-colloquium

20 JAN 2021
11:00 - 12:00

Abstract: We propose a mass-dependent MOM scheme to renormalize UV divergence of unpolarized PDFs at one-loop order. This approach which is based on a once subtracted dispersion relation does not need any regulator. The overall counterterms are obtained from the imaginary part of large transverse momentum region in loop integrals.
The mass-dependent characteristic of the scheme yields to mass-dependent splitting functions for the DGLAP evolution equations. While the flavor number is fixed at any renormalization scale, the decoupling theorem is automatically imposed by the mass-dependent splitting functions. The required symmetries are also automatically respected by our prescription.


https://meet.google.com/dmk-zowd-cmo


Indico: https://indico.particles.ipm.ir/indico/event/398

20 JAN 2021
13:30 - 15:00

The equilibrium state of fields in the causal wedge of a dS observer is thermal, though realistic observers have only partial access to the state. To them, out-of-equilibrium states of a light scalar field appear to thermalize in a Markovian fashion. We show this by formulating a systematic expansion for tracing out the environment. As an example, we calculate the O(\lambda) correction to the result of Starobinsky and Yokoyama for the relaxation exponents of $\lambda\phi^4$ theory.

20 JAN 2021
14:00 - 15:00

Abstract:

Because of the many-body interactions, the standard concepts of nonlinear response to electromagnetic fields break down in two dimensional Dirac systems, like graphene, in the quantum regime close to the Dirac point. I this talk, I will first discuss the dramatic impact of long-range Coulomb interaction on the nonlinear optical responses owing to the plasmonic and screening effects [1]. Then, I will present how using a conserving Baym-Kadanoff approach, we can provide a fully compelling theory of nonlinear optical and transport response of a Dirac system to electric fields in the presence of disorder scattering. We show that the nonlinear terms are strikingly ruled by the appearance of a dominant two-photon vertex which is absent at the bare level and finite even in the weak-coupling limit. Such two-photon vertex self-generation [2] highlights the crucial role of the frequency and field dependence of the scattering rates in the nonlinear regime. Moreover, we surprisingly predict four- and five-photon incoherent features in the third order optical response of the system [3]. Our study reveals a novel many-body mechanism in the nonlinear response of Dirac materials whose effects are predicted to be observable in nonlinear dc transport, terahertz third-harmonic generation, and the second-order difference-frequency response of graphene.

[1] H. Rostami, M.I. Katsnelson, and M. Polini, PRB 95, 035416 (2017).
[2] H. Rostami and E. Cappelluti, arXiv:2007.08282 (2020).
[3] H. Rostami and E. Cappelluti, arXiv:2011.03824 (2020).


Date: Wednesday, January 20, 2021

Time: 14:00-15:00

To join the webinar, please follow this link:
https://www.skyroom.online/ch/schoolofphysics/condensed-matter-and-statistical-physics
and log in as guest.

Organized by: Condensed Matter and Statistical Physics Group, School of Physics (IPM)

Queries about this event should be addressed to Dr. A. Faridi at azadeh.faridi@ipm.ir

19 JAN 2021
14:00 - 15:00

Abstract:

We construct a top-down holographic model of Weyl semimetal states from the AdS/CFT correspondence. In the presence of a constant external axial gauge field in a spatial direction, we find the defining characteristic of a Weyl semi-metal: a quantum phase transition from a topological state with non-zero anomalous Hall conductivity to a trivial insulator. The transition is first order. We also compute entropy, heat capacity, and speed of sound, and find a peak. Our model allows for calculation of the DC conductivity in a constant electric field, with interesting results. Based on arXiv: 2012.11434.


Date: Tuesday, 19 January

2 p.m. - 3 p.m. (Tehran time)


Zoom. (click)

Meeting ID: 895 9440 6684

Passcode: 347401