List of Talks

Alhun Aydın - Size-invariant Shape Transformation

Koç University, Department of Physics & Harvard University, Department of Physics

Abstract: "Can One Hear the Shape of a Drum?" This interesting question was asked by a famous Polish mathematician Mark Kac in 1966 [1]. Shape of a drumhead determines the sound the drum makes. We all experienced this fact in one rock concert or another. But can we figure out the shape of a drum by just listening to it? If we would like to add a bit more mathematical rigor, we can translate the question into this: "Are there different shapes with identical spectra?" It has been found that there, in fact, are isospectral domains [2]. So one cannot always hear the shape of a drum. On the other hand, these kinds of domains are very rare and specially constructed. Almost all shapes that we encounter in real systems are "hearable" in principle [3]. Physical properties of particles confined in nanostructures are directly determined by the spectrum of the confining domain. Confining particles in quantum wells of different sizes has been a common practice in nanotechnology with many applications ranging from enhancing the desired properties of materials to use them in energy harvesting. Here we ask another related question: "Can one change the shape of a domain without changing its size?" and its immediate follow-up: "If yes, then how it sounds?" We developed a technique called size-invariant shape transformation, which leads to a new phenomenon called quantum shape effect {4,5]. In this talk, we will "hear" the shapes of size-invariant quantum wells and investigate the statistical thermodynamics of particles confined in such domains.

Keywords: Spectral geometry, Dirichlet problem, quantum confinement, size effects, shape effects

References:

[1] Kac, M., Can one hear the shape of a drum?. Am. Math. Monthly. 73 pp. 1 (1966).

[2] Giraud, O. & Thas, K. Hearing shapes of drums: Mathematical and physical aspects of isospectrality. Rev. Mod. Phys.. 82, 2213-2255 (2010).

[3] Baltes, H. & Hilf, E., Spectra of finite systems. Bibliographisches Institut, (1976).

[4] Aydin, A. & Sisman, A., Quantum shape effects and novel thermodynamic behaviors at nanoscale. Phys. Lett. A. 383 pp. 655 (2019).

[5] Aydin, A., Quantum Shape Effects. (arXiv:2102.04332 [cond-mat.mes-hall] (P.hD. Thesis). Energy Institute, Istanbul Technical University, Istanbul,2020).

Oğul Esen - On the Legendre Transformation

Gebze Technical University, Department of Mathematics

Abstract: There are two main approaches to Classical Mechanics. One is Lagrangian Mechanics and the other is Hamiltonian Mechanics. In this talk, we consider the former on the tangent bundle (velocity phase space) and the latter on the cotangent bundle (momentum phase space) of configuration space. The passage between these approaches is achieved by means of the Legendre transformation. For degenerate (singular) cases, the Legendre transformation is not immediate. I shall review the Tulczyjew triplet which is a geometric structure enabling the Legendre transformation even for degenerate theories.

Keywords: Lagrangian mechanics, Hamiltonian mechanics, Legendre transformation.

References:

[1] Abraham, R., & Marsden, J. E. Foundations of mechanics, , 2nd ed. Benjamin/Cummings, Reading, MA, 1978.

[2] Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd ed. Springer, New York, 1989.

[3] Tulczyjew W. M., The legendre transformation. In Annales de l’IHP Physique théorique, (27) No. 1, pp. 101-114) 1977.

Victor Py - T\bar{T} Deformations in Curved Space from 4D Chern-Simons Theory

University of Edinburgh, School of Mathematics

Abstract: T\bar{T} deformations of conformal field theories have been widely studied in the past years, in particular in light of their holographic relation to AdS_3 at finite cutoff. After having defined these deformations and motivated their importance, I will introduce a new way of thinking about them to first order as coming from 4D Chern-Simons theories. Namely, I will show that starting with a 4D Chern-Simons theory coupled with 2D chiral and anti-chiral CFTs, one can recover a 2D (infinitesimally) T\bar{T}-deformed theory by integrating out the Chern-Simons gauge field. Different parameters of the 4D Chern-Simons theory will yield deformed CFTs on different geometries.

Keywords: T\bar{T} deformations, Chern-Simons theory, effective field theory, conformal field theory.

References:

[1] Yunfeng Jiang, A pedagogical review on solvable irrelevant deformations of 2D quantum field theory, Commun. Theor. Phys. 73 (2021), Arxiv: 1904.13376.

[2] Kevin Costello, Edward Witten and Masahito Yamazaki, Gauge Theory and Integrability, I, ICCM Not. 06 (2018), Arxiv: 1709.09993.

[3] F. A. Smirnov and A. B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017), Arxiv: 1608.05499.

[4] Andrea Cavaglia, Stefano Negro, Istvan M. Szecsenyi, and Roberto Tateo, T \bar{T}-deformed 2D Quantum Field Theories, JHEP 10 (2016), Arxiv: 1608.05534.

[5] Victor Py, T\bar{T} deformations in curved space from 4D Chern-Simons theory, JHEP 08 (2022), Arxiv: 2202.08841.

Atınç Çağan Şengül - Probing Dark Matter with Strong Gravitational Lensing

Harvard University, Department of Physics

Abstract: Galaxy-galaxy strong lensing involves a foreground galaxy producing multiple distorted and highly magnified images of a background galaxy. Smaller perturbers, which can be either subhalos orbiting the foreground galaxy or halos along the line-of-sight, cause slight deviations in these bright arcs. For this reason, strong gravitational lenses have become a popular probe of structures at sub-galactic scales, which are crucial for discerning the properties of dark matter. For massive perturbers, we can distinguish between line-of-sight halos and subhalos by measuring the curl of the angular deflection field. Moreover, for lensing systems with bright source galaxies, we can measure the average power-law slope of the density profile of the perturbers, which can allow us to distinguish between different dark matter models. In the coming years, we expect to have tens of thousands of new strong lensing systems obtained by experiments such as the Vera C. Rubin Observatory, the Dark Energy Survey, and others. We will be able to measure the abundance and density profile of sub-galactic structures with strong gravitational lensing once these observations are followed up with higher resolution observations with telescopes such as JWST and ELT, probing dark matter at scales where it has yet to be tested.

Keywords: Gravitational lensing: strong, cosmology: dark matter

This is a collaboration with Arthur Tsang, Bryan Ostdiek, and Cora Dvorkin

References:

[1] A. Ç. Şengül, C. Dvorkin, B. Ostdiek and A. Tsang, ”Substructure Detection Reanalyzed: Dark Perturber shown to be a Line-of-Sight Halo”, Monthly Notices of the Royal Astronomical Society 515 (2022), Arxiv number:2112.00749.

[2] A. Ç. Şengül and C. Dvorkin, ”Probing Dark Matter with Strong Gravitational Lensing through an Effective Density Slope”, Monthly Notices of the Royal Astronomical Society 516 (2022), Arxiv number:2206.10635.