Out-of-equilibrium systems are ubiquitous in nature. In contrast to their equilibrium counterparts, which are fairly well-understood using the framework of statistical mechanics, a description of them is challenging and is at the forefront of current-day research. Recently, important experimental and theoretical techniques have been developed, and advances have been made in both classical and quantum systems that are out-of-equilibrium. This conference aims to discuss the current trends in this broad area of research by bringing together experts, early-career researchers, postdocs, and students working in this field. Topics would include (but are not limited to) quantum many-body scars, driven and dissipative systems, disordered, noisy setups, stochastic processes, first-passage problems, and random walks. An important goal is to bring together the community of people working on the classical and quantum fronts and foster collaborations between them. https://www.imsc.res.in/current_trends_classical_and_quantum_out_of_equilibrium_systems_ctcqoes

We introduce a new method to obtain lower bounds for the 2k-th moment of a broad class of L-functions for all real k>0. In particular, our method applies to families of higher degree L-functions which were previously inaccessible. https://www.imsc.res.in/~anupdixit/IMSc-CMI-NT-seminar.html

Out-of-equilibrium systems are ubiquitous in nature. In contrast to their equilibrium counterparts, which are fairly well-understood using the framework of statistical mechanics, a description of them is challenging and is at the forefront of current-day research. Recently, important experimental and theoretical techniques have been developed, and advances have been made in both classical and quantum systems that are out-of-equilibrium. This conference aims to discuss the current trends in this broad area of research by bringing together experts, early-career researchers, postdocs, and students working in this field. Topics would include (but are not limited to) quantum many-body scars, driven and dissipative systems, disordered, noisy setups, stochastic processes, first-passage problems, and random walks. An important goal is to bring together the community of people working on the classical and quantum fronts and foster collaborations between them. https://www.imsc.res.in/current_trends_classical_and_quantum_out_of_equilibrium_systems_ctcqoes

Out-of-equilibrium systems are ubiquitous in nature. In contrast to their equilibrium counterparts, which are fairly well-understood using the framework of statistical mechanics, a description of them is challenging and is at the forefront of current-day research. Recently, important experimental and theoretical techniques have been developed, and advances have been made in both classical and quantum systems that are out-of-equilibrium. This conference aims to discuss the current trends in this broad area of research by bringing together experts, early-career researchers, postdocs, and students working in this field. Topics would include (but are not limited to) quantum many-body scars, driven and dissipative systems, disordered, noisy setups, stochastic processes, first-passage problems, and random walks. An important goal is to bring together the community of people working on the classical and quantum fronts and foster collaborations between them. https://www.imsc.res.in/current_trends_classical_and_quantum_out_of_equilibrium_systems_ctcqoes

The tensor product problem, a classical question in representation theory, concerns decomposing the tensor product (also known as the Kronecker product) of two irreducible representations into a direct sum of irreducibles. In the setting of finite groups, understanding tensor product decompositions is fundamental to the study of their representation theory. The groups of our interest are defined as follows. Let $F$ be a non-Archimedean local field with ring of integers $\mathfrak{o}$, maximal ideal $ \mathfrak{p}$, and finite residue field $\mathbb{F}_q $ with odd characteristic. Let $\ell \geq 1$ be an integer. Define $\mathfrak{o}_\ell \coloneq \mathfrak{o} / \mathfrak{p}^{\ell}$ as the finite principal ideal local ring of length $\ell$ associated to $\mathfrak{o}$. For $\ell=1$, $\mathfrak{o}_1=\mathbb{F}_q$. Let $GL_2(\co_\ell)$ be the group of $2 \times 2$ invertible matrices. It is known that the irreducible representations of $GL_2(\co_\ell)$ are built from certain distinguished regular representations, which are classified into three types: cuspidal, split semisimple, and split non-semisimple. In this talk, we will discuss the decomposition of tensor products of the regular representations of $GL_2(\mathfrak{o}_\ell)$ and give some results about the multiplicities of the constituents. In the second part of the talk, we discuss the real conjugacy classes and real characters for the group $GL_2(\mathfrak{o}_\ell)$, which are closely related to the tensor product problem. An element of a group is called real if it is conjugate to its inverse. We describe all real conjugacy classes and characterize the real characters of $GL_2(\mathfrak{o}_\ell)$ for $\ell>1 $ by identifying irreducible representations whose tensor square contains the trivial representation.