In doing mathematics, we often encounter beautiful identities and proofs, shining like a full moon in the night sky. Some of them have combinatorial flavors. This talk is an introduction to combinatorial proof methods using bijections and weight-preserving-sign-reversing involutions, with examples including Franklin's bijective proof of the Euler pentagonal number theorem, a combinatorial proof of the Cayley-Hamilton theorem, the Robinson-Schensted correspondence and recent combinatorial proofs of some identities involving secant and tangent numbers.

A (positive definite and integral) quadratic form $f$ is called irrecoverable (from its subforms) if there is a quadratic form $F$ that represents all proper subforms except for $f$ itself, and such a quadratic form $F$ is called an isolation of $f$. In this talk, we present recent advances on irrecoverable quadratic forms and discuss their possible generalizations.

Geometric evolution equations describe how geometric objects such as curves, surfaces, or metrics evolve toward more symmetric or optimal shapes. Among the most fundamental examples are the mean curvature flow and the Ricci flow, which have played central roles in modern differential geometry and topology. In this talk, I will give an introduction to these flows, explaining how curvature acts as a driving mechanism that smooths and reshapes geometry. I will also outline the key ideas behind Perelman’s proof of the Poincaré conjecture, focusing on the role of singularity formation and the classification of canonical neighborhoods. Finally, I will discuss the problem of classifying singularity models arising under geometric flows and present some recent progress on the classification of ancient oval solutions, together with possible further directions.

Associated to a group action on a bifoliated plane, satisfying some reasonable conditions, one can associate a combinatorial object known as a veering triangulation. Since their introduction by Agol (in a very different setting), these triangulations have recently played an interesting role in studying pseudo-Anosov flows, the structure of fibered 3-manifolds, algorithmic properties of mapping class groups, and fixed points of surface homeomorphisms, to name just a few (from my own biased perspective). This talk will be an overview of these applications, starting with the most basic properties from the initial bifoliated plane.

In this talk, we introduce the concept of t-core partitions. We discuss the generating function and modularity, along with some results and applications of t-core partitions. Recent results on simultaneous core partitions will also be presented. Toward the end of the talk, we introduce numerical semigroups and explore connections between numerical semigroups (or numerical sets) and partitions. Additionally, we present some open problems related to these topics.

We study the partial dimensional semi-classical Weyl’s laws, describing the quantum subband structures for two-dimensional electron gases (2DEGs). As a simple application, we derive lowest free energy states for the subband models describing non-interacting 2DEGs.