It is a classical, but also difficult, problem to determine the defining equations of the blowup of P^n along a given subvariety X. In this talk, I will focus on the cases where X is defined by the ideal of maximal minors of a 2xn matrix of linear forms. I will explain how to determine the defining equations and the singularities of their blowup algebras. This relies on a stratification of the Hilbert scheme of determinantal ideals and the combinatorics of square-free degenerations. This is joint work with Alessio Sammartano.

It has been said that working with non-compact spaces is like trying to hold change in your pocket with a hole in it. One of the central examples of non-compact spaces in algebraic geometry are moduli spaces. Broadly speaking, the points of a moduli space represent equivalence classes of algebraic varieties of a given type, and its geometry reflects the ways these varieties deform in algebraic families. The classification of algebraic varieties of a given type is tantamount to understanding the geometry of the corresponding moduli space. The goal of this talk is to discuss recent progress on compactifying moduli spaces of higher dimensional varieties, focusing on the interplay between compactifications of moduli spaces and singular degenerations of the objects they classify.

Abstract: Revisiting Kronecker's claim that "God created the integers, all else is the work of man", let's take a look at just how much of mathematics can be formalised in arithmetic. Since a lot of mathematics requires forming sets, we'll allow ourselves some set existence axioms, but we'll be suspicious of them, because, well, we're not sure if they have divine approval. In the interest of time, we won't talk about all of mathematics, but we'll focus on a couple of theorems about the existence of ideals in commutative rings.

Quasi-Gorenstein rings are, roughly speaking,"Gorenstein rings which possibly fail the Cohen-Macaulay property”. They are much more than Gorenstein rings, and in some situation they are more natural: For example, the Stanley-Reisner ring of an orientable manifold is quasi-Gorenstein, while the only orientable manifolds whose Stanley-Reisner ring is Gorenstein are (homology) spheres. In this talk I will discuss some features of quasi-Gorenstein rings, a liaison theory by quasi-Gorenstein ideals generalizing the classical one by Gorenstein ideals, and two applications of the latter: one on the combinatorics of the minimal prime ideals of a quasi-Gorenstein ring, and the other one explaining a connection with the topological Lefshetz duality. All this is based on a joint work with Hongmiao Yu.

A pervading question in the study of stochastic PDE is how small-scale random forcing in an equation combines to create nontrivial statistical behavior on large spatial and temporal scales. I will discuss recent progress on this topic for several related stochastic PDEs - stochastic heat, KPZ, and Burgers equations - and some of their generalizations. These equations are (conjecturally) universal models of physical processes such as a polymer in a random environment, the growth of a random interface, branching Brownian motion, and the voter model. The large-scale behavior of solutions on large scales is complex, and in particular depends qualitatively on the dimension of the space. I will describe the phenomenology, and then describe several results and challenging problems on invariant measures, growth exponents, and limiting distributions.

Combining elements from a long line of research on the tree property, we aim to prove that it is consistent that every regular cardinal between aleph_2 and aleph_{omega^2+3} has the tree property while aleph_{omega^2} is strong limit. In this talk, I'll give some background and give a sampling of some of the many ideas that go into the proof along with their connections to other research. This is joint work in progress with James Cummings, Yair Hayut, Menachem Magidor, Itay Neeman and Dima Sinapova.

For the past 90 years, a fundamental question in classical homotopy theory is to understand the stable homotopy groups of spheres. The most modern method to study these groups is to compare them with the ``motivic stable homotopy groups of spheres". Motivic homotopy theory has its roots in algebraic geometry. As a result of the recent advances, there is a reintegration of algebraic topology and algebraic geometry, with close connections to equivariant homotopy theory and number theory. In this talk, I will introduce the classical and motivic stable homotopy categories and the connections between the two. I will then talk about the rich properties and extra structures that are present in the motivic stable homotopy category. The presence of these extra structures gives new computational tools that dramatically improve our understanding of the classical stable homotopy groups. Moreover, the flow of information can be reversed as well, producing new results in motivic stable homotopy theory for general fields.

I will discuss some ways that triangulations of the two-dimensional sphere connect to various areas of mathematics: (1) how modular forms and the Siegel-Weil formula relate to the enumeration of buckyballs, (2) how representation theory of the symmetric group arises in generalizing these enumerative results, and (3) how triangulated affine structures arise when compactifying moduli spaces of K3 surfaces.

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