A graph $G$ is \textit{$k$-locally sparse} if for each vertex $v \in V(G)$, the subgraph induced by its neighborhood contains at most $k$ edges. Alon, Krivelevich, and Sudakov showed that for $f > 0$ if a graph $G$ of maximum degree $\Delta$ is $\Delta^2/f$-locally-sparse, then $\chi(G) = O\left(\Delta/\log f\right)$. We introduce a more general notion of local sparsity by defining graphs $G$ to be \textit{$(k, F)$-locally-sparse} for some graph $F$ if for each vertex $v \in V(G)$ the subgraph induced by the neighborhood of $v$ contains at most $k$ copies of $F$. Employing the R\"{o}dl nibble method, we prove the following generalization of the above result: for every bipartite graph $F$, if $G$ is $(k, F)$-locally-sparse, then $\chi(G) = O\left( \Delta /\log\left(\Delta k^{-1/|V(F)|}\right)\right)$. This improves upon results of Davies, Kang, Pirot, and Sereni who consider the case when $F$ is a path. Our results also recover the best known bound on $\chi(G)$ when $G$ is $K_{1, t, t}$-free for $t \geq 4$, and hold for list and correspondence coloring in the more general so-called ``color-degree'' setting.

Given a smooth Fano variety and a smooth curve B, let Hom(B,X) be the moduli space of maps from B to X. Let M be a component of Hom(B,X). If a general curve parameterized by M is free, it means M has the expected dimension and good deformation behavior. Components M consisting entirely of curves that are not free are more mysterious. We give a geometric characterization of which curves can be nonfree, explaining that roughly they come from fibrations. We show how this result can be seen as the analogue of Manin's Conjecture, which predicts the number of rational on a variety of bounded height. This is joint work with Brian Lehmann and Sho Tanimoto.

Partial differential equations and random fields have been used as successful models in various areas of applied mathematics, statistical mechanics, theoretical physics, theoretical neuroscience, theory of complex chemical reactions, fluid dynamics, hydrology, cosmology, mathematical finance, and other scientific areas. In this talk I will consider non-linear space-time fractional (stochastic) heat type equations. These types of time fractional (stochastic) heat type equations are attractive models that can be used to model phenomenon with random effects with thermal memory. I will review my most recent work on (i) continuous time random walk limits; (ii) heat type Cauchy problems with fractional time derivatives; and (iii) stochastic fractional equations. In particular, I will talk about the asymptotic behavior of the solution with respect to time and a parameter $\lambda$; intermittency. These results are our recent joint work with Jebessa B Mijena, Mohammud Foondun, Sunday Asogwa and Guerngar Ngartelbaye.

Recently Yates (2021) introduced the "version age of information" metric for studying communication networks where packets are sent between users via independent Poisson processes. For any network G, this produces a graph invariant vAoI(G) characterizing the worst-case version age of a node in the network. In this talk I'll discuss the combinatorics of the vAoI metric, and in particular how this metric varies as one interpolates between the empty graph and complete graph. I'll give the vAoI for the Erdos-Reyni random graph, random d-regular graph, and the family of complete bipartite graphs $K_{n,n-m}$. This is based on my work accepted to ISIT 2024.

We consider the algorithmic problem of finding large balanced independent sets in sparse random bipartite graphs, and more generally the problem of finding independent sets with specified proportions of vertices on each side of the bipartition. In a bipartite graph it is trivial to find an independent set of density at least half (take one of the partition classes). In contrast, in a random bipartite graph of average degree d, the largest balanced independent sets (containing equal number of vertices from each class) are typically of density (2 + o_d(1)) log d/d . Can we find such large balanced independent sets in these graphs efficiently? By utilizing the overlap gap property and the low-degree algorithmic framework, we prove that local and low-degree algorithms (even those that know the bipartition) cannot find balanced independent sets of density greater than (1 + ε) log d/d for any ε > 0 fixed and d large but constant. This factor 2 statistical–computational gap between what exists and what local algorithms can achieve is analogous to the gap for finding large independent sets in (non-bipartite) random graphs. Our results therefore suggest that this gap is pervasive in many models, and that hard computational problems can lurk inside otherwise tractable ones. A particularly striking aspect of the gap in bipartite graphs is that the algorithm achieving the lower bound is extremely simple and can be implemented as a 1-local algorithm and a degree-1 polynomial (a linear function).

By now it is quite classical that one can understand singularities in prime characteristic local algebra/algebraic geometry, through properties of the Frobenius endomorphism. A foundational result illustrating this is the celebrated theorem of Kunz characterizing the regularity of a noetherian scheme (in prime characteristic) in terms of whether a Frobenius pushforward on that scheme is flat. In this talk, I'll discuss a structural explanation of, that also recovers, the theorem of Kunz and other theorems of this ilk. Namely, I’ll discuss recent joint work with Ballard, Iyengar, Lank, and Mukhopadhyay where we show that over an F-finite noetherian scheme of prime characteristic high enough Frobenius pushforwards generate the bounded derived category.

Co-Higgs bundles on a complex manifold $M$ are given by pairs $(E,\phi)$ consisting of a holomorphic vector bundle $E$ on $M$ together with a Higgs field $\phi \in H^0(M,\rm{End}(E) \otimes TM)$ such that $\phi \wedge \phi = 0$. They correspond to generalized holomorphic bundles on complex manifolds. They also give rise to a special class of holomorphic Poisson structures on the projective bundles $\mathbb{P}(E)$. Co-Higgs bundles were first studied by S. Rayan on Riemann surfaces and $\mathbb{CP}^2$. M. Matviichuk has more recently found necessary conditions on the Higgs fields $\phi$ for the corresponding holomorphic Poisson structures on $\mathbb{P}(E)$ to be integrable. In this talk, we consider co-Higgs bundles on compact holomorphic Poisson surfaces. We describe some of their moduli and discuss how the properties of the Higgs fields reflect those of corresponding holomorphic Poisson structures. This is joint work with Eric Boulter and Brady Ali Medina.

The Statistical Laboratory will present the 2023-2024 Statistics Graduate Student Service Award, Consulting Award, and Research Award.

We present an analogue of Gödel’s second incompleteness theorem. Whereas Gödel showed that sufficiently strong theories that are $\Pi^0_1$-sound and $\Sigma^0_1$-definable do not prove their own $\Pi^0_1$-soundness, we prove that sufficiently strong theories that are $\Pi^1_1$-sound and $\Sigma^1_1$-definable do not prove their own $\Pi^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis. We will then turn to characterizations of ordinal analysis itself. One of the main goals of ordinal analysis is measuring the “strength” of theories by calculating their proof-theoretic ordinals. But in what sense do proof-theoretic ordinals measure the strength of theories? What is the attendant notion of strength? We provide some abstract answers to this question.

Several undergrads participated in this semester's Directed Reading Program. They will each give final short presentations about what they have learned over the course of this semester. A link to the titles, abstracts can be found here: https://drive.google.com/file/d/1WBEz-XY35B-aOqrArlzKOxcKRJoha_gv/view?usp=sharing

In a celebrated work Mazur classified which torsion subgroups can occur for elliptic curves defined over the rationals. A natural analogue is to consider surfaces with geometric endomorphisms by a quaternion order (PQM surfaces), since the associated moduli space is 1-dimensional. In this talk I will discuss progress towards classifying which torsion subgroups are possible for these surfaces. We also give a description of the moduli problem for PQM surfaces. This is joint work with Eran Assaf, Jef Laga, Freddy Saia, Ari Shnidman, Jacob Swenberg and John Voight.

A holomorphic symplectic manifold is a complex manifold $X$ together with a closed, non-degenerate holomorphic 2-form $\Omega$. The top power of $\Omega$ gives a trivialisation of the canonical bundle so that $X$ has trivial first Chern class. In the context of K\”ahler geometry, such manifolds play a very important role due to the Bogomolov covering theorem, which states that any compact Kahler manifold with vanishing first Chern class has a covering that splits as the product of Calabi–Yau manifolds, complex tori and irreducible holomorphic symplectic manifolds. Among these, the last two are, in fact, compact holomorphic symplectic manifolds. Furthermore, irreducible holomorphic symplectic manifolds correspond to compact hyperk\”ahler manifolds in the K\”ahler setting. In general, finding compact holomorphic symplectic manifolds is very difficult. In this talk, I will present some classical constructions of holomorphic symplectic manifolds in various geometric contexts and discuss some recent developments in finding compact examples in the non-Kahler setting.

In this talk, we give an overview of several recent results where quantitative estimates play a key role. In the first part of the talk, we discuss new convex integration constructions for fluid systems with external forcing. We will then discuss a novel application of these ideas to the surface quasi-geostrophic (SQG) equation. Moving forward with the theme of quantitative estimates, in the second part of the talk we will describe new bounds for the defocusing energy-supercritical Nonlinear Schrödinger equation (NLS) and use these to give a universal blow-up criteria which goes below the scaling invariant threshold. These results are in line with a recent breakthrough construction of finite-time blow-up solutions, and in particular give the first generic result distinguishing potential defocusing blow-up phenomena from many of the known examples of blow-up in the focusing setting. At the end of the talk, we will briefly describe applications to related models.

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