Mathematics and Mathematical Physics

Ahmed Ittihad Hasib:

Topic 1 : Representation Theory of Quivers/Hecke Algebras

Topic 2 : Symmetric Groups and Young Diagram Combinatorics

Prerequisites : Linear Algebra, Representation Theory of Groups, Basic Category Theory

Onirban Islam:

Topic 1 : Hawking-Penrose singularity theorems

Synopsis: Singularity theorems are one of the classic predictions of general relativity, originally due to Penrose and propounded by Hawking in collaboration with Penrose in the 1970s. Since then, a number of refinements have been reported. The aim of this project is to achieve a mathematically rigorous understanding of singularity theorems, beginning with the classic Hawking-Penrose singularity theorem. If time permits and the student is interested, I plan to touch on some of the modern singularity theorems.

Reference: Ettore Minguzzi. Lorentzian causality theory. Living Rev Relativ 22, 3 (2019).

Topic 2: Cauchy problems for Einstein equations

Synopsis: The local well-posedness of the Einstein equations as an the initial value problem was proven by Choquet-Bruhat in her PhD thesis. Subsequently, a global version of this result was reported by her and

Geroch under some assumptions. The generic situation is still an open problem, which is related to some other global aspects of general

relativity, for instance, the cosmic censorship. The aim of this project is to get a sound understanding of these results from a modern viewpoint.

Reference: Hans Ringström, The Cauchy Problem in General Relativity, EMS

Press (2009).

Prerequisites : A solid foundation in general relativity is essential.

The student must be interested in analytic computation and mathematical

analysis. Backgrounds in differential (Lorentzian) geometry (Topic 1)

and partial differential equations (Topic 2) are useful but can be built

as a part of these projects.

Mentor Profile: https://inspirehep.net/authors/1298353

Siddiqur Rahman Milon:

Topic 1 :Hodge Theory on Compact Manifold

Prerequisites 1: Linear Algebra, Smooth Manifold Theory, basic co(homology) theory (recommended), and Real Analysis (at least undergraduate level).

Topic 2: Symplectic Geometry (finite-dimensional versions)

Textbook: Lectures on Symplectic Geometry by Ana Cannas da Silva

Goal: To complete this book within two months so that the students can start studying the infinite-dimensional version (Marsden has a good book).

Prerequisites 2 : Linear Algebra and Smooth Manifold Theory. Students should be comfortable with the materials from the following books:

a)Linear Algebra Done Right Textbook by Sheldon Axler

b) An Introduction to Manifolds by Tu

Astronomy and Astrophysics

Asif ud-Dowla:

Topic 1 : Supernova Types and their distribution across the Universe

Topic 2: Possibility of Life around M Dwarf planets and their moons

Prerequisites : Basic astronomy, calculus, computer language such as Python or Fortran/C/C++

Mentor Profile: https://orcid.org/0000-0001-7721-6713

Istiak Akib:

Topic 1 : Dynamics of the Local Group Galaxies

Prerequisites : Classical Mechanics, Scientific Computing in Python, Linear Algebra, Probability & Statistics

Mentor Profile: https://scholar.google.com/citations?user=PrHJE_EAAAAJ&hl=en

Syeda Lammim Ahad:

Topic 1 : Galaxy clusters from large sky surveys

Topic 2: Galaxies in different cosmic environments

Prerequisites : Undergrad physics courses, any background in astronomy will be great, familiarity with python programming is necessary

Mentor Profile: https://scholar.google.com/citations?hl=en&user=glCxJTAAAAAJ

Lamiya Mowla:

Topic 1 : Extragalactic astronomy using the James Webb Space Telescope

Prerequisites : Introductory physics courses (Mechanics, EM, Waves, Optics, Thermo, Stat Mech, Diff Eq).

Students interested in working with our group are required to complete an online onboarding tutorial: https://www.mowlaastrogroup.com/onboarding.

This tutorial includes readings from a first-year university-level astronomy textbook, along with introductory Python exercises. These materials are designed to help students assess their interest in learning more about extragalactic astronomy.

After completing the onboarding tutorial, students will advance to the Launchpad tutorials. https://www.mowlaastrogroup.com/launchpad

These include first-year graduate-level readings from review articles, as well as selected papers from our group that provide a detailed overview of our research. While students are not expected to fully grasp every concept at this stage, these readings are intended to lay the groundwork for engaging with our research projects.

Mentor Profile: https://scholar.google.com/citations?user=bVtATjgAAAAJ&hl=en&oi=ao

High Energy Physics

Ahmed Rakin Kamal:

Topic 1 : Vacuum Decay in Quantum Field Theory

Prerequisite 1: Advanced Quantum Mechanics, Special Relativity, Basics of Quantum Field Theory (Optional)

Topic 2: Einstein-Maxwell Theories in Diverse Dimensions

Prerequisite 2 : General Relativity, Electrodynamics, Basics of Quantum Field Theory (Optional)

Mentor Profile: https://inspirehep.net/authors/2763317?ui-citation-summary=true

Nafiz Ishtiaque:

Topic 1 : Quantum to classical map for (Z_N or more general) toric codes.

Prerequisites : Good grasp of linear algebra is essential. Familiarity with quantum mechanics and statistical mechanics is necessary. Encounters with Ising models and/or toric codes would be great, but not essential.

Mentor Profile: https://inspirehep.net/authors/1321472

Rafid Mahbub:

Topic 1 : Oscillon formation in the post-inflationary universe

Topic 2: Analytical and numerical investigation of stochastic inflation with consequences for PBH formation; deep learning techniques in inflationary model building

Prerequisites : Basic knowledge on inflationary cosmology; comfortable with programming in Python/Julia

Mentor Profile:https://inspirehep.net/authors/1622353?ui-citation-summary=true

Ratul Mahanta:

Topic 1 : Properties of scattering amplitudes: Feynman integrals (which serve as building blocks in perturbation theory) could be introduced to advanced undergraduate or Masters students, and analytic properties of a generic such integral could be examined. This may potentially lead to a research project.

Prerequisites 1 : Mathematical Methods in Physics, Complex analysis: functions of a complex variable. Analytic continuations.

Topic 2: Reading project on D branes and gauge fields following Barton Zweibach’s string theory book.

Prerequisites 2 : Electrodynamics (preferably in tensor notations), Classical Mechanics: Lagrangian and Hamiltonian dynamics (​​prior exposure to the classical theory of fields is a plus), Quantum Mechanics: Harmonic oscillators, raising and lowering operators.

Topic 3 : String thermodynamics and Boltzmann equations.

Prerequisites 3 : Mathematical Methods in Physics, Quantum Mechanics: Harmonic oscillators, raising and lowering operators, Statistical mechanics and physics: canonical and microcanonical ensembles, Numerical methods (required if interested in numerical simulations in solving Boltzmann equations in string thermodynamics).

Mentor Profile: https://inspirehep.net/authors/1744386

Shadman Salam:

Topic 1 : Path Integral approach to QFT for simple systems (e.g. scalar field)

Prerequisites 1: Expertise in Lagrangian formulation of classical mechanics. Good understanding of QM. Familiarity with path integral approach to QM is good but not necessary.

Topic 2 : Free Bosonic String Theory - A first step to understanding string theory.

Prerequisites 2: Expertise in Lagrangian formulation of classical mechanics. Basics of canonical quantization of QFT.

Mentor Profile: https://inspirehep.net/authors/1975235?ui-citation-summary=true

Samanta Saha:

Topic 1 : Scattering Amplitudes, Symmetry and Soft Theorems

Topic 2 : Causality Constraints on Effective Field Theory and Gravity

Topic 3 : Conformal Field Theory

Prerequisites: Quantum Mechanics, Basics of QFT and GR.

Mentor Profile: https://inspirehep.net/authors/2648795

Wasif Ahmed

Topic 1 : Matrix Models and JT Gravity.

It is well known that Jackiw-Teitelboim (JT) gravity is dual to an ensemble of random matrix models. Using the powerful techniques of random matrix models, we have got a lot of insights on quantum gravity in general. We can start by learning about the basics of random matrix models. Our goal should be to numerically solve the string equations (some nonlinear differential equations) for different ensembles. If it works out we can apply these in the context of modified JT gravity theories.

Prerequisites : Quantum mechanics and basic quantum field theory (Path integrals), General Relativity (Einstein-Hilbert Action), Numerically solving nonlinear differential equations.

Mentor Profile: https://inspirehep.net/authors/1748423?ui-citation-summary=true

N.B: Internship Period for this topic (Matrix Models and JT gravity) is from September to November.

Arpit Das:

Topic 1 : Topic: Anomalies and Generalized symmetries in QFT and 2D CFT

In this project we shall explore the novel concepts of higher-form symmetries, non-invertible symmetries, categorical symmetries in various QFTs and 2D CFTs. Along the way, we shall also investigate anomalies in them. If time permits, we shall explore some applications of these concepts in holography and phase structure of QED. Refs: “An introduction to higher-form symmetries” by P. Gomes: https://inspirehep.net/literature/2638369; “Jena lectures on generalized global symmetries: principles and applications” by N. Iqbal: https://inspirehep.net/literature/2812087; “Lectures on generalized symmetries” by L. Bhardwaj: https://inspirehep.net/literature/2677667

Topic 2: Classifying rational conformal field theories using number theoretic techniques Two-dimensional Conformal Field Theories (2D CFTs) play a central role in both Physics and Mathematics. In physics, they describe critical phenomena in condensed matter systems and govern phase transitions. The worldsheet theory in String Theory is also a 2D CFT. A special class known as Rational Conformal Field Theories (RCFTs) is of particular interest due to their finite spectrum of operators. In the mathematical literature, these are referred to as Vertex Operator Algebras (VOAs) and find applications in Representation Theory, Affine Lie Algebras, Modular Tensor Categories (MTCs), and the Theory of Lattices. A classification of RCFTs would, therefore, be of significant value to both communities. This project focuses on classifying RCFTs of low rank using number-theoretic tools such as automorphic forms, Rademacher series, and the theory of Modular Linear Differential Equations (MLDEs). The method, known as the Mathur-Mukhi-Sen (MMS) approach, begins by postulating a general MLDE of a fixed order. Solutions whose series expansions have non-negative integer coefficients are selected as admissible characters. These are then used to construct modular-invariant partition functions. One then employs techniques from affine Lie algebras and MTCs to determine whether these correspond to genuine RCFTs. The MMS approach has successfully identified novel RCFTs and revealed new structures within them. Through this project, we shall also explore the categorical structure of RCFTs. Refs: “On the classification of rational CFTs”: https://inspirehep.net/literature/262800; “Classification of RCFT from Holomorphic Modular Bootstrap: A Status Report ”https://inspirehep.net/literature/1758023; “Modern Approach to 2D Conformal Field Theory”: https://inspirehep.net/literature/2862678 (roughly upto chapter 4)

Topic 3: Study of Laws of Black Hole mechanics and Hawking Radiation In this project, using techniques from general relativity, thermal field theory and bogoliubov transformations (useful for studying scattering in curved spacetimes), we shall derive the four laws of black hole mechanics, Unruh radiation and Hawking radiation. If time permits, we shall explore thermodynamics of more exotic solutions like black holes in higher/lower dimensions, e.g., Myers-Perry black holes in 5D, CGHS black holes in 2D dilaton gravity, etc. and Emparan-Reall black rings solutions. Refs: “An into to BH evaporation” by P. H. Lambert: https://inspirehep.net/literature/1262788; Lectures on Quantum Gravity and BHs by T. Hartman: http://www.hartmanhep.net/topics2015/ (upto chapter 6); Lectures on Adv. GR by M. Guica: https://www.youtube.com/playlist?list=PLbcx3dKCUhgYn3_RsWo8LbRe_fLzjgccC (last 3 lectures).

Pre-requisites for generalized symmetries: Basic knowledge of QFT is essential. Knowledge of CFT is useful but can be learnt as part of the project.

Pre-requisites for rational CFTs: Basic knowledge of Finite Groups and their Representations, ODEs and Complex Analysis is essential. Knowledge of Lie groups, Lie algebras and Conformal Field Theory is useful but can be learnt as part of the project.

Pre-requisites for Black Hole thermodynamics: Basic knowledge of general relativity, quantum field theory and statistical mechanics is essential.

Mentor Profile: https://inspirehep.net/authors/1806329?ui-citation-summary=true

M. Shahnoor Rahman:

Topic 1 : Computation of band structures of solids using quantum espresso

Prerequisites : Classical Mechanics (Lagrangian and Hamiltonian to find equations of motion, normal coordinates and eigenvalue) and Quantum Mechanics (Schrodinger equation, eigenstates and diagonalization, Fourier transformation between coordinate space and momentum space), Basic understanding of command line interface(CLI) and process of compiling a C/C++/Fortran program.

Mentor Profile: https://scholar.google.com/citations?hl=en&view_op=list_works&authuser=1&gmla=ANZ5fUOELhvJmVlyXMftbY8NnKmWpZm82R7MbCEajk62BgbOWdCZ8rIt0AudPZpZgbspTTMwFVwq8hQJFxAVCjT0xnAE&user=ymySJEIAAAAJ

Tanvir Ahmed Masum:

Topic 1 : Design of Superconducting Quantum Circuits

Prerequisites : Basic understanding of classical mechanics, EM theory, Python coding. This is a good project for Junior/senior undergrad student interested in hardware side of quantum information

Mentor Profile: https://scholar.google.com/citations?user=8jSfU0QAAAAJ&hl=en