Date/Time : 13 Nov 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Dilip Raghavan, National University of Singapore

I will present some recent consistency results on nowhere dense ultrafilters and a weakening of selectivity.

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Date/Time : 12 Nov 2024 16:00
Venue : S16 #03-06
Speaker : Cao Yu, Shanghai Jiao Tong University

Quantum dynamical systems are essential models for understanding the time evolution of our nature. The mathematically simplest models in this context are quantum semigroup dynamics. When a system is closed (as special instances), its dynamics are described by Schrödinger’s equation or equivalently, the von Neumann equation. There is growing interest in simulating the time-dependent Schrödinger’s equation using quantum devices due to its applications in quantum adiabatic computing, quantum control theory, and solving partial differential equations (PDEs). We will explore Sambe-Howland’s clock as a unifying framework for addressing time-dependent problems through time-independent techniques. This approach is compatible with both analog and digital quantum computing and can address some unresolved questions in recent studies. On the other hand, in open systems, semigroup dynamics are described by the Lindblad equation, which models the interaction between the system and its environment. In this talk, we will also discuss a family of structure-preserving algorithms for simulating Lindblad equations, which have close connections to designing stochastic and quantum algorithms.

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Date/Time : 07 Nov 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Euiyong Park, University of Seoul

Let $C$ be a generalized Cartan matrix and $U_q(\mathfrak{g})$ the quantum group associated with $C$. In this talk, I am going to talk about the PBW theory for the bosonic extension $\widehat{A}_{\mathfrak{g}}$ of the negative half $U_q^-(\mathfrak{g})$. When $U_q(\mathfrak{g})$ is of finite simply-laced type, the algebra $\widehat{A}_{\mathfrak{g}}$ is isomorphic to the quantum Grothendieck ring of the Hernandez-Leclerc category over a quantum affine algebra. The PBW theory allows us to define a new family of subalgebras, denoted by $\widehat{A}_{\mathfrak{g}}(b)$, for any element $b$ in the (generalized) Braid group associated with $C$. I will explain why this algebra $\widehat{A}_{\mathfrak{g}}(b)$ can be viewed as a natural extension of the quantum unipotent coordinate ring $A_q(n(w))$ for a Weyl group element $w$. This talk is based on arXiv:2401.04878 (with Se-jin Oh) and arXiv:2408.07312 (with Masaki Kashiwara, Myungho Kim and Se-jin Oh).

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Date/Time : 06 Nov 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Michael Takaaki Leong,

A weakening of a Suslin tree, known as a Suslin lattice, was introduced by Dilworth, Odell, and Sari in 2007, and subsequently investigated by Raghavan and Yorioka in 2012. In this talk, we will show that the compatibility of a Suslin lattice with Martin’s Axiom and its variants mirrors that of a Suslin tree by showing that a fragment of Martin’s Axiom suffices to imply the non-existence of a Suslin lattice. We will also discuss the possible consistency of a Suslin lattice with the P-ideal Dichotomy.

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Date/Time : 06 Nov 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Lei Yang, National University of Singapore

In this talk, I will give a new (and shorter) proof of Strombergsson’s Theorem on effective equidistribution of unipotent orbits in ASL(2, R)/ASL(2,Z). The proof is based on a general framework developed in recent works by Lindenstrauss-Mohammadi-Wang and myself, which relates the problem to incidence geometry. It relies on the study of a Kakeya-type model. This is joint with Elon Lindenstrauss.

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Date/Time : 01 Nov 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Nathan Ross, University of Melbourne

Neural networks are random parametric functions, where the parameters are chosen according to stochastic gradient descent, based on the sample of training data. It has long been known that at initialisation, a neural network with “large” width and “fixed” depth can be well-approximated by a limiting Gaussian random field. But what quantities are well-approximated, and how wide does a neural network need to be for the approximation to be good? There has been a flurry of recent work trying to answer these and related questions, by deriving rates of convergence between the distribution of the neural network and the limiting Gaussian random field in various probability metrics, including some at the process level.

The aim of these lectures is to explain a general approach that can be applied to derive such rates at the process level, which uses Stein’s method, functional smoothing inequalities, and classical Gaussian process concepts such as Karhunen-Loeve expansion and Cameron-Martin/Reproducing Kernel Hilbert Space. The target audience is PhD students planning to do research in probability or statistics. In particular, the definitions and relevant properties of the concepts above will be covered in detail. Open questions and future directions will also be discussed.

The approach is based on joint work with (A.D. Barbour, K. Balasubramanian, L. Goldstein, A. Salim, and G. Zheng).

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Date/Time : 30 Oct 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Nathan Ross, University of Melbourne

Neural networks are random parametric functions, where the parameters are chosen according to stochastic gradient descent, based on the sample of training data. It has long been known that at initialisation, a neural network with “large” width and “fixed” depth can be well-approximated by a limiting Gaussian random field. But what quantities are well-approximated, and how wide does a neural network need to be for the approximation to be good? There has been a flurry of recent work trying to answer these and related questions, by deriving rates of convergence between the distribution of the neural network and the limiting Gaussian random field in various probability metrics, including some at the process level.

The aim of these lectures is to explain a general approach that can be applied to derive such rates at the process level, which uses Stein’s method, functional smoothing inequalities, and classical Gaussian process concepts such as Karhunen-Loeve expansion and Cameron-Martin/Reproducing Kernel Hilbert Space. The target audience is PhD students planning to do research in probability or statistics. In particular, the definitions and relevant properties of the concepts above will be covered in detail. Open questions and future directions will also be discussed.

The approach is based on joint work with (A.D. Barbour, K. Balasubramanian, L. Goldstein, A. Salim, and G. Zheng).

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Date/Time : 30 Oct 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Jimmy He, Ohio State University

The study of quantum groups and integrable systems have long been connected through the Yang-Baxter equation. Recently, these ideas have been introduced in probability to study integrable stochastic models. I will discuss a similar story for models with boundary, which are connected to quantum symmetric spaces through the reflection equation, as well as some applications to probability. The focus will be on illustrating the connections through illustrative examples and applications for a specific model called the six vertex model.

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Date/Time : 30 Oct 2024 16:45
Venue : S17 #04-06 (SR1)
Speaker : Desmond Lau,

We develop a forcing framework based on the idea of amalgamating language fragments into a theory with a canonical term model. We then demonstrate the usefulness of this framework by applying it to variants of the extended Namba problem, as well as to the analysis of models of certain theories with constraints in interpretation (TCIs). Separately, we look at small extensions of V as generalised degrees of computability over V. Using TCIs, we formalise and investigate the complexity of certain methods one can use to define, in V, subclasses of degrees over V. Finally, we give a characterisation of the complexity of forcing.

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