The Kawamata–Morrison cone conjecture is a long-standing problem in birational geometry. Totaro generalized the conjecture and proved it for klt Calabi–Yau pairs in dimension two. The conjecture predicts that such a pair has only finitely many birational contractions modulo its automorphism group. I will explain that the finiteness of the targets of these contractions follows once they admit polarizations of bounded degree. In dimension two, this provides a new proof of the generalized Kawamata–Matsuki conjecture on the finiteness (up to log isomorphism) of weak log canonical models within a birational class.

We'll explore what it means to be an author, your responsibilities, ethical considerations, and the core principles of integrity in research dissemination. Gain insights on what editors look for in high-quality submissions and how to improve your chances of getting published. We’ll also cover open access options, open data practices, and the importance of reproducibility. Discover how to maximise the visibility and impact of your research, including how to ethically and responsibly use new tools like Generative AI to support your writing and research.

Given a graph $H$ and a (weighted) graph $G$, let $t(H, G)$ denote the density of $H$ in $G$. Given a fixed set of graphs $H_1, H_2, \dots, H_k$, what can we say about the possible tuples $(t(H_1, G), t(H_2, G), \dots, t(H_k, G))$ as $G$ ranges over all (weighted) graphs? What inequalities of the form $\sum c_i t(H_i, G) \geq 0$ are always valid? Can all such inequalities be verified by sum of squares? When are such inequalities optimized when $G$ is a random graph? These questions are connected to several long-standing open problems in extremal combinatorics. In this talk, we will discuss recent progress on these problems.

TBA

It is well known that asset returns usually do not follow a normal distribution, rather, they have long and fat tails. This paper focuses on the quantile portfolio methodology, which considers the whole distribution of asset returns and employs expected loss as a risk measurement. In particular, we explore statistical properties of tau risk and propose related theories of quantile portfolio optimization. We also introduce portfolio performance terms for the quantile portfolio framework.