ICTS

Jigsaw puzzles are a lot of fun: Smaller pieces which fit together to make a beautiful picture. But sometimes it is not just play and can relate to a wide variety of deep questions ranging from logic to statistical physics to crystallography. We will have a look at some of these questions and try to see many facets of it before moving on to things I actively think about.

Sarting with the historical need to fit a model we outline the issues involved in fitting. Next we explain the set-up of forecast and illustrate with some examples and applications. Issues, some unresolved, in fitting are described.

Some questions are long and verbose but have very short answers. Can we find correct answers without reading the whole question? How much can we skip?

Pinpointing this is the challenge in query complexity. This talk will examine a specific setting where diagonalisation is the star technique for algorithmic upper bounds, and nifty combinatorics provides matching lower bounds.

Mathematicians often speak of thinking algebraically, analytically, or geometrically, each with its own instincts and habits of mind. In this talk, we ask what it means to think combinatorially. Through a handful of elementary problems and puzzles from combinatorics and graph theory, we will try to get a feel for the distinctive flavour of the subject. No background beyond first-year undergraduate mathematics will be assumed.

I will give exposure to the subject "knot theory" which is a branch of topology and mention some interesting areas where knot theory has been useful.

We study the problem of selecting large language models (LLMs) for user queries in settings where multiple LLM providers submit the cost of solving a query. From the user’s perspective, choosing an optimal model is a sequential, query-dependent decision problem: high-capacity models offer more reliable outputs but are costlier, while lightweight models are faster and cheaper. We formalize this interaction as a reverse auction design problem with contextual online learning, where the user adaptively discovers which model performs best while eliciting costs from competing LLM providers. Our framework unifies mechanism design and adaptive learning, enabling efficient, truthful, and query-aware LLM selection.

Plant–animal interactions are central to plant reproduction, and community stability. This talk examines how mutualistic and antagonistic interactions are structured across individual, species, and community levels. Drawing on field studies, experiments, trait-based analyses, and ecological network approaches, I will discuss how plant traits, animal behavior, spatial context, and anthropogenic change shape interaction outcomes and influence biodiversity persistence and ecosystem resilience under environmental change.