Georgia Institute of Technology College of Sciences

The Plücker embedding exhibits the Grassmannian Gr(r, n) as a closed subvariety of projective space. A theorem of Hodge shows that its homogeneous ideal has as a quadratic Gröbner basis the so-called multiple-exchange relations between Plücker coordinates. Since the set of these polynomials is quite large and unwieldy, it is often preferable to work with a smaller set of single-exchange Plücker relations. An even smaller set of polynomials is the collection of local (or 3-term) exchange relations. We will recall and clarify the relationships between these three. We go on to examine the situation over hyperfields. In their pioneering work, Baker and Bowler showed that the theories of matroids, oriented matroids, valuated matroids etc. can be collectively understood under a common banner as the theory of Grassmanians over hyperfields. Their work gives a good accounting of the relationship between single- and local-exchange relations in this generalized setting. We will discuss what can be said about the multiple-exchange relations. This leads to considerations of elementary linear-algebraic facts in the hyperfield setting. All results may be suitably extended to the flag setting---which we will discuss, time permitting. The talk is based on joint work with Nathan Bowler and Changxin Ding.

We study approximation properties of Gaussian reproducing kernel Hilbert spaces restricted to low-dimensional manifolds embedded in Euclidean space. Using only ambient Gaussian kernels, and without assuming any smooth ambient extensions or estimating geometric quantities of the manifold, we show that intrinsically defined Hölder functions on the manifold can be approximated at rates governed by intrinsic dimension and smoothness. The construction is based on a small-scale expansion in real space rather than a spectral representation. As an application, we obtain adaptive nonparametric convergence rates for Gaussian process regression on manifolds, where the regression procedure itself is unchanged and intrinsic adaptivity results from the approximation analysis.

In the past few years there have been a host of remarkable topological results arising from considering "real" versions of various gauge and Floer-theoretic invariants of three- and four-dimensional manifolds equipped with involutions. Recently Guth and Manolescu defined a real version of Lagrangian Floer theory, and applied it to Ozsváth and Szabó's three-manifold invariant Heegaard Floer homology, producing an invariant called real Heegaard Floer homology associated to a 3-manifold together with an orientation-preserving involution whose fixed set is codimension two (for example a branched double cover). We review the construction of real Heegaard Floer theory and use tools from equivariant Lagrangian Floer theory, originally developed by Seidel-Smith and Large in a somewhat different context, to produce a spectral sequence from the ordinary to real Heegaard Floer homologies in their simplest "hat" version, in particular proving the existence of a rank inequality between the theories. Our results apply more generally to the real Lagrangian Floer homology of exact symplectic manifolds with antisymplectic involutions. Along the way we give a little history and context for this kind of result in Heegaard Floer theory. This is a series of two talks; the first "prep" talk will discuss some background and context that might be helpful to (for example) graduate students in attendance.

Link Floer homology is a powerful invariant of links due to Ozsváth and Szabó. One of its most striking properties is that it detects each link's Thurston norm, a result also due to Ozsváth and Szabó. In this talk I will discuss generalizations of this result to the context of 4-ended tangles, as well as some tangle detection results. This is joint work in progress with Subhankar Dey and Claudius Zibrowius.

The tropical Grassmannian Trop G(k,n), introduced by Speyer and Sturmfels, parametrizes tropical linear spaces in tropical projective space. For k=2, it can be identified with the space of phylogenetic trees. Beyond applications to mathematical biology, it has seen striking new connections in physics to generalized scattering amplitudes via the CEGM framework.

Despite this, constructing a combinatorial model for the positive tropical Grassmannian at higher k has remained an open problem. I will describe such a model built from the planar basis, a distinguished basis of the space of tropical Plücker vectors whose elements are rays of the positive tropical Grassmannian, together with a duality between tropical u-variables and noncrossing tableaux, which provides an explicit inverse to the Speyer–Williams parameterization. For k=3, the model connects to SL(3) representation theory via a cross-ratio formula that computes tropical invariants directly from non-elliptic webs, and to CAT(0) geometry via diskoids in affine buildings.

Based on joint work with Thomas Lam.

In the mid 1990s, Thomassen proved that planar graphs are 5-list-colourable, and that planar graphs of girth at least five are 3-list-colourable. Moreover, it can be shown via a simple degeneracy argument that planar graphs of girth at least four are 4-list-colourable.  In 2021, Postle and I unified these results, showing that if $G$ is a planar graph and $L$, a list assignment for $G$ where all vertices have size at least three; vertices in 4-cycles have list size at least four; and vertices in triangles have list size at least five, then $G$ is $L$-colourable. In this talk, I will discuss a strengthening of this latter result: that it also holds for correspondence colouring, a generalization of list colouring. In fact, it holds even in the still stronger setting of weak degeneracy. I will also speak briefly on some other weak degeneracy results in the area.

No prior knowledge of correspondence colouring nor list colouring will be assumed.  (Ft. joint work with Ewan Davies, and with Anton Bernshteyn and Eugene Lee.)

The Chebotarev density theorem is a powerful tool in number theory, in part because it guarantees the existence of primes whose Frobenius lies in a given conjugacy class in a fixed Galois extension of number fields.  However, for some applications, it is necessary to know not just that such primes exist, but to additionally know something about their size, say in terms of the degree and discriminant of the extension.  In this talk, I'll discuss recent work with Peter Cho and Asif Zaman on a closely related problem, namely determining the least prime with a given cycle type.  We develop a new, comparatively elementary approach for thinking about this problem that nevertheless frequently yields the strongest known results.  We obtain particularly strong results in the case that the Galois group is the symmetric group $S_n$ for some $n$, where determining the cycle type of a prime is equivalent to Chebotarev.