The stochastic block model (SBM) is a generative model for random graphs
with a community structure, which has been one of the most fruitful
research topics in community detection and clustering. A phase
transition behavior for detection was conjecured by Decelle el al.
(2011), and was confirmed by Mossel et al. (2012,2013) and Massouli\'e
(2013). We consider the community detection problem in random
hypergraphs. Angelini et al. (2015) conjectured a phase transition for
community detection in sparse hypergraphs generated by a hypergraph
stochastic block model (HSBM). We confirmed the positive part of the
phase transition for the 2-block case by a generalization of the method
developed in Massouli\'e (2013). We introduced a matrix which counts
self-avoiding walks on hypergraphs, whose leading eigenvectors give us a
correlated reconstruction. This is joint work with Soumik Pal.

We work out an explicit theory for the shtuka
function of rank 1 sign-normalized Drinfeld modules over the function
field of an elliptic curve. Using these explicit formulas, we obtain a
product formula for the fundamental period of the exponential function
associated to the Drinfeld module. We also find identities for
deformations of reciprocal sums and as a result prove special value
formulas for L-series over the function field.

A main goal of algebraic geometry is the
classification of algebraic varieties and a central tool in this
endeavor is the study of moduli spaces. I will discuss the moduli
space of plane curves of degree d: a parameter space where each point
corresponds to an isomorphism class of a certain curve. There are many
techniques to compactify this space, including GIT, the minimal model
program, and a differential geometric approach called K stability. In
joint work with K. Ascher and Y. Liu, we interpolate between these
different compactifications and study the problem via wall crossings.

Given $s$ finite sets $A_1, \ldots, A_s$, determining the size of the
union of the $s$ sets is an easy problem. Determining the maximimum number
of size $k$ subsets of an $n$ element set for which there does not exist
$s$ sets which union has size $q$ is a very hard problem in general. Many
problems in extremal set theory can be restated in this language for
particular choices of $s,k,q$. For instance, the case where $s=2$ is
equivalent to the complete intersection theorem, and when $sk=q$, this is
equivalent to the Erd{\H o}s matching conjecture; one of the biggest open
problems in the field. This talk is based off a recent paper of Peter
Frankl and Andrey Kupavskii.

Hacon and McKernan have proved that there exist integers $r_n$ such that if $X$ is a smooth variety of general type and dimension $n$, then the pluricanonical maps $|rK_X|$ are birational for all $r\geq r_n$. These values are typically very large: for example $r_3\geq 27$ and $r_4\geq 94$. In this talk we will show that the $r^{\textup{th}}$ canonical maps of smooth threefolds and fourfolds of general type have birationally bounded fibers for $r\geq 2$ and $r\geq 4$ respectively. Furthermore, we will generalize these results to higher dimensions in terms of the constants $r_n$ and we will discuss recent progress on a conjecture of Chen and Jiang.

A famous theorem of Y. T. Siu states that plurigenera of projective complex manifolds are invariant under deformation. The only known proof of this result uses deep techniques from complex analysis, which are not available in the algebraic category. In this talk, we will illustrate some recent progress toward an algebraic proof of Siu's result, and explain how these methods can be used to prove analogous results in positive and mixed characteristic