In the theory of projective modules over a commutative ring, a fundamental problem is the existence of a unimodular element. The problem is well understood when the rank of the projective module equals the dimension of the ring, but is much harder when the rank is smaller. It is typically studied via obstructions arising from $K$-theory, Chow(-Witt) groups, Grothendieck-Witt groups, and related invariants. Recent work of Asok, Bachmann, and Hopkins shows that in the corank $1$ case the Chern class in Chow groups gives the correct obstruction for smooth affine algebras over an algebraically closed field of characteristic $0$. In this talk, we consider the corank $1$ situation for an arbitrary ring of dimension $3$. We discuss a result of J. Fasel, where the vanishing of a certain Witt group defined via antisymmetric forms, together with a condition on some orbit space of unimodular elements, provides a positive answer to our question. We also see a few examples.

Imagine searching for a single sentence in a library of three billion letters — and finding it in seconds. This is exactly what proteins do every time they locate their target on DNA to switch a gene on, repair damage, or copy the genome. This talk tells the physics story behind that search: how proteins cleverly combine random 3D diffusion with sliding and hopping along the DNA strand itself, turning an impossible search problem into a fast, efficient one. We'll explore how the environment inside a cell — DNA's twisting and bending, the crowding of countless other molecules, and DNA's packaging into chromatin — all shape and even accelerate this search. Along the way we'll see how proteins adapt their shape and motion to read the DNA landscape, how they cooperate with each other, and how damaged or unusual DNA changes the rules of the game.

Every cell in our body carries the same two meters of DNA, folded down to fit inside a nucleus a few microns wide — roughly like packing a thread the length of a football field into a tennis ball, without ever tangling it. Yet this folding isn't random: it determines which genes get switched on, and errors in it are linked to disease and developmental disorders. This talk builds that picture from the ground up. We'll start at the scale of a single gene, where physics-guided models — informed directly by experimental data — let us reconstruct the detailed 3D shape of specific genomic loci with striking precision. Then we'll zoom out to the scale of an entire chromosome, where a different kind of physics takes over. We'll see how physics principles reveal a hidden "backbone" that holds chromosomes together, a common blueprint for genome folding that holds across cell types and species, and how it shifts in predictable ways as cells develop, specialize, or go awry in disease.

In this thesis, we investigate lower bounds for the Weil height of algebraic numbers and the canonical height of points on elliptic curves. We establish three principal results. First, we show a connection between lower bounds of height of an algebraic number $\alpha$ to the low-lying zeros of the Dedekind zeta-function of $Q(\alpha)$. Second, we formulate a $p$-adic criterion for Lehmer’s conjecture. Finally, we prove the Bogomolov property for certain infinite extensions, referred to as asymptotically positive extensions and establish the corresponding Bogomolov property for the canonical height of points on elliptic curves defined over such extensions.

Youth Astronomy and Space Congress