Point configurations appear naturally in different contexts, ranging from the study of the geometry of data sets to questions in commutative algebra and algebraic geometry concerning determinantal varieties and invariant theory. In this talk we bring these perspectives together: we consider the scheme \(X_{r,d,n}\) parametrizing \(n\) ordered points in projective space \(\mathbb{P}^r\) that lie on a common hypersurface of degree \(d\). We show that this scheme has a determinantal structure and, if \(r\geq2\), we prove that it is irreducible, Cohen-Macaulay, and normal. Moreover, we give an algebraic and geometric description of the singular locus of \(X_{r,d,n}\) in terms of Castelnuovo-Mumford regularity and \(d\)-normality. This yields a characterization of the singular locus of \(X_{2,d,n}\) and \(X_{3,2,n}\). This is joint work with Alessio Caminata and Han-Bom Moon.

Given any toric vector bundle, we may construct its parliament of polytopes. This is a generalization of the Newton polytope (or moment polytope) of a toric line bundle. This object contains a huge amount of information about the original bundle: notably on its global sections and its positivity. We can also easily know if the toric bundle is (semi-/poly-)stable with respect to any polarisation. I will give a combinatorial visualisation of stability of toric vector bundles.

This is a remote talk on WebEx. contact the organizer for the meeting link.

A distinct covering system of congruences is a list of congruences
\[
a_i \bmod m_i, \qquad i = 1, 2, ..., k
\]
whose union is the integers. Erdős asked if the least modulus \(m_1\) of a distinct covering system of congruences can be arbitrarily large (the minimum modulus problem for covering systems, $1000) and if there exist distinct covering systems of congruences all of whose moduli are odd (the odd problem for covering systems, $25). I'll discuss my proof of a negative answer to the minimum modulus problem, and a quantitative refinement with Pace Nielsen that proves that any distinct covering system of congruences has a modulus divisible by either 2 or 3. The proofs use the probabilistic method and in particular use a sequence of pseudorandom probability measures adapted to the covering process. Time permitting, I may briefly discuss a reformulation of our method due to Balister, Bollobás, Morris, Sahasrabudhe and Tiba which solves a conjecture of Schinzel (any distinct covering system of congruences has one modulus that divides another) and gives a negative answer to the square-free version of the odd problem.

TBA

In this talk, I consider a hyperelliptic curve \(C\) over a \(p\)-adic field \(K\). Under a particular condition called "total degeneracy", the curve C can be uniformized, that is, realized as the quotient of the projective line over \(K\) by a (free) subgroup of the projective linear group \(\mathrm{PGL}_2(K)\). In the case that \(C\) is an elliptic curve, this is the well-known theory of Tate uniformization. I will briefly explain Tate uniformization, go on to describe how this may be generalized for totally degenerate hyperelliptic curves, and discuss my current progress on some open questions relating to how to find a subgroup of \(\mathrm{PGL}_2(K)\) under which the quotient of the projective line over \(K\) can be realized as a hyperelliptic curve.

TBA

TBA

Finite tensor categories are important generalizations of the categories of finite dimensional modules of finite dimensional Hopf algebras. There are two support theories for them, the cohomological one and one based on the noncommutative Balmer spectrum of the corresponding stable module category. We will describe general results linking the two types of support via a new notion of categorical center of the cohomology ring of a finite tensor category and will state a conjecture giving the exact relation. The construction and results will be illustrated with various examples. This is a joint with Daniel Nakano (Univ Georgia) and Kent Vashaw (MIT).

We will talk about two related problems: counting polynomials with small Galois groups, and counting number fields of a certain degree up to a certain discriminant bound. The former problem is related to Hilbert's irreducibility theorem, and the second is a classical (and very hard) problem. I aim for this to be a largely accessible talk.

Positivity of line bundles is a classical topic in projective algebraic geometry. For a vector bundle \(\mathcal E\) of higher rank, positivity is usually imported from the analogous positivity property of \(\mathcal O_{\mathbb P(\mathcal E)}(1)\). Here we explore a stronger generalization via base loci and also investigate their properties. This is in joint work with Nabanita Ray.

TBA

TBA

The Seshadri constants of vector bundles are important local positivity invariants. We compute them for tangent bundles of toric varieties. In particular we confirm the toric case of a conjecture of Fulger and Murayama.

TBA