in this talk we use the \emph{unit-graphs} and the \emph{special unit-digraphs} on matrix rings to show that every $n \times n$ nonzero matrix over $\bbb f_q$ can be written as a sum of two $\operatorname{sl}_n$-matrices when $n>1$. we compute the eigenvalues of these graphs in terms of kloosterman sums and study their spectral properties; and prove that if $x$ is a subset of $\operatorname{mat}_2 (\bbb f_q)$ with size $|x| > \frac{2 q^3 \sqrt{q}}{q - 1}$, then $x$ contains at least two distinct matrices whose difference has determinant $\alpha$ for any $\alpha \in \bbb f_q^{\ast}$. using this result we also prove a sum-product type result: if $a,b,c,d \subseteq \bbb f_q$ satisfy $\sqrt[4]{|a||b||c||d|}= \omega (q^{0.75})$ as $q \rightarrow \infty$, then $(a - b)(c - d)$ equals all of $\bbb f_q$. in particular, if $a$ is a subset of $\bbb f_q$ with cardinality $|a| > \frac{3} {2} q^{\frac{3}{4}}$, then the subset $(a - a) (a - a)$ equals all of $\bbb f_q$. we also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units.

studying invariant theory of commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. for a finite group acting on a polynomial ring, the remarkable chevalley-shephard-todd theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo-reflections. related questions try to find properties of the fixed ring under some special group actions. in recent years, progress was made in work of jing, jorgensen, kirkman, kuzmanovich, walton, zhang, and others to extend the theory to regular algebras which are a noncommutative generalization of polynomial rings. naturally, the question arises if the theory generalizes further to non-connected noncommutative algebras. this talk answers the question what conditions need to be satisfied by the fixed ring in order to make a rich theory possible. moreover, we construct a homological determinant for preprojective algebras and discuss how it being trivial for all elements of a finite group affects the related fixed ring.

in neuroscience, functional connectivity describes the connectivity between
brain regions that share functional properties. it is often characterized by a time
series of covariance matrices between functional measurements of distributed
neuron areas. an effective statistical model for functional connectivity and
its changes over time is critical for better understanding brain functions and
neurological diseases. to this end, we propose a matrix-log mean model with an
additive heterogeneous noise for modeling random symmetric positive definite
matrices that lie in a riemannian manifold. we introduce the heterogeneous
error terms to capture the curved nature of the nonlinear manifold. a scan
statistic is then developed for the purpose of multiple change point detection.
theoretically, we establish the sure coverage property. simulation studies and
an application to the human connectome project lend further support to the
proposed methodology.

in many binary classification applications, such as disease diagnosis and spam detection, practitioners commonly face the need to limit type i error (that is, the conditional probability of misclassifying a class 0 observation as class 1) so that it remains below a desired threshold. to address this need, the neyman-pearson (np) classification paradigm is a natural choice; it minimizes type ii error (that is, the conditional probability of misclassifying a class 1 observation as class 0) while enforcing an upper bound, alpha, on the type i error. although the np paradigm has a century-long history in hypothesis testing, it has not been well recognized and implemented in classification schemes. common practices that directly limit the empirical type i error to no more than alpha do not satisfy the type i error control objective because the resulting classifiers are still likely to have type i errors much larger than alpha. this talk introduces the speaker's work on np classification algorithms and their applications and raises current challenges under the np paradigm.

in this talk we investigate the erd\h{o}s-renyi model for random graphs, $g(n,p)$. our focus will be on determining the probability that $g(n,p)$ is connected, which was the first problem that erd\h{o}s and renyi considered in their original paper. as time permits we will also discuss the ``phase transition'' of $g(n,p)$.

for fixed degree $d$, one could ask for a meaningful compactification of the moduli space of smooth degree $d$ surfaces in $\mathbb{p}^3$. in other words, one could ask for a parameter space whose interior points correspond to [isomorphism classes of] smooth surfaces and whose boundary points correspond to degenerations of these surfaces. motivated by hacking's work for plane curves, i will discuss a ksba compactification of this space by considering a surface $s$ in $\mathbb{p}^3$ as a pair $(\mathbb{p}^3, s)$ satisfying certain properties. we will study an enlarged class of these pairs, including singular degenerations of both $s$ and the ambient space. the moduli space of the enlarged class of pairs will be the desired compactification and, as long as the degree $d$ is odd, we can give a rough classification of the objects on the boundary of the moduli space.