The cohomology annihilator of Noetherian algebras was defined by Iyengar and Takahashi in their work on strong generation in the module category. For a commutative Noetherian local ring, it can be observed that the cohomology annihilator ideal is the entire ring if and only if the ring is regular. Motivated by this, I will consider the question: When is the cohomology annihilator ideal of a local ring equal to the maximal ideal? I will discuss various ring-theoretic and category-theoretic conditions towards understanding this question and describe applications for understanding when the test ideal of the module closure operation on cyclic surface quotient singularities is the maximal ideal.

Assume we have a sequence of polynomials whose asymptotic zero distribution is known. What can be said about the zeros of their derivatives? Especially if we differentiate each polynomial several times, proportional to its degree? This simple-to-formulate problem has recently attracted the attention of researchers. Both the problem and the methods of its solution have exciting connections with free probability, random matrices, and approximation theory on the complex plane. In this talk, I will explain some known results in this direction and our approach to the problem, which uses only some elementary complex analysis. This is a joint work with E. Rakhmanov from the University of South Florida.

Simulation of quantum dynamics, emerging as the original motivation for quantum computers, is widely viewed as one of the most important applications of a quantum computer. Quantum algorithms for Hamiltonian simulation with unbounded operators Recent years have witnessed tremendous progress in developing and analyzing quantum algorithms for Hamiltonian simulation of bounded operators. However, many scientific and engineering problems require the efficient treatment of unbounded operators, which may frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure, quantum differential equations solver and quantum optimization. We will introduce some recent progresses in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter type splitting and Magnus expansion based algorithms in the interaction picture. (The talk does not assume a priori knowledge on quantum computing.)

N. Zabusky coined the word "soliton" in 1965 to describe a curious feature he and M. Kruskal observed in their numerical simulations of the initial-value problem for a simple nonlinear PDE. The first part of the talk will be a broad introduction to the theory of solitons/solitary waves and integrable PDEs (the KdV and modified KdV equation in particular), describing classical results in the field. The second (and main) part of the talk will focus on some new developments and growing interest into a special case of solutions defined as "soliton gas".

I will describe a collection of works done in collaborations with K. McLaughlin (Tulane U.), T. Grava (SISSA/Bristol), R. Jenkins (UCF) and A. Minakov (U. Karlova). We analyze the case of a regular, dense KdV soliton gas and its large time behaviour with the presence of a single trial soliton travelling through it. We are able to derive a series of physical quantities that precisely describe the dynamics, such as the local phase shift of the gas after the passage of the soliton, and the velocity of the soliton peak, which is highly oscillatory and it satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El (at leading order).

I will finally present some ongoing work where we establish that the soliton gas is the universal limit for a large class of N-solutions with random initial data.

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