network data analysis has wide applications in computational social science, computational biology, online social media, and data visualization. for many of these network inference problems, the brute-force (yet statistically optimal) methods involve combinatorial optimization, which is computationally prohibitive when we are faced with large scale networks. therefore, it is important to understand the effect of computational constraints on statistical inference.

in this talk, we will discuss three closely related statistical models for different network inference problems. these models answer inference questions on cliques, communities, and ties, respectively. for each particular model, we will describe the statistical model, propose new computationally efficient algorithms, and study the theoretical properties and numerical performance of the algorithms. further, we will quantify the computational optimality through describing the intrinsic barrier for certain efficient algorithm classes, and investigate the computational-to-statistical gap theoretically. a key feature shared by our studies is that, as the parameters of the model changes, the problems exhibit different phases of computational difficulty.

the {\itshape zimmer program} is a collection of conjectures and questions regarding actions of lattices in higher-rank simple lie groups on compact manifolds. for instance, it is conjectured that all non-trivial volume-preserving actions are built from algebraic examples using standard constructions. in particular, on manifolds whose dimension is below the dimension of all algebraic examples, {\itshape zimmer's conjecture} asserts that every action is finite.

i will present some background, motivation, and selected previous results in the zimmer program. i will then explain two of my own results within the zimmer program:
(1) a solution to zimmer's conjecture for actions of cocompact lattices in $sl(n,r), n>=3$ (joint with d. fisher and s. hurtado);
(2) a classification (up to topological semiconjugacy) of lattice actions on tori whose induced action on homology satisfies certain criteria (joint with f. rodriguez hertz and z. wang).

cohomology jump loci are generalizations of usual cohomology groups
of a topological space. in the first part of the talk, i will give a
survey on the recent development of the theory of cohomology jump loci
of complex algebraic varieties. in the second part of the talk, i will
use some concrete examples of (real) 6-dimensional symplectic-complex
calabi-yau manifolds to illustrate how cohomology jump loci can give
new constraints on the topology of compact kahler manifolds.

one of the main goals of any resource theory such as entanglement, quantum thermodynamics, quantum coherence, and asymmetry, is to find necessary and sufficient conditions (nsc) that determine whether one resource can be converted to another by the set of free operations. in this talk i will present such nsc for a large class of quantum resource theories which we call affine resource theories (arts). arts include the resource theories of athermality, asymmetry, and coherence, but not entanglement. remarkably, the nsc can be expressed as a family of inequalities between resource monotones (quantifiers) that are given in terms of the conditional min entropy. the set of free operations is taken to be (1) the maximal set (i.e. consists of all resource non-generating (rng) quantum channels) or (2) the self-dual set of free operations (i.e. consists of all rng maps for which the du al map is also rng). as an example, i will discuss the applications of the results to quantum thermodynamics with gibbs preserving operations, and several other arts. finally, i will discuss the applications of these results to resource theories that are not affine.

the birch and swinnerton-dyer conjecture is a central open problem in mathematics, and euler systems have been at the source of much of the progress to date in this direction. in my talk, i will give a motivated introduction to the bsd conjecture, survey what we know about it, and highlight some recent advances arising from the construction of new euler systems with a bearing on the arithmetic of elliptic curves.

the truncated moment problem seeks conditions on an n-dimensional multisequence of degree $m$,
$y \equiv (y_i)_{|i| ≤ m}$, such that there exists a positive borel measure $\mu$ on $\mathbb{r}^n$ satisfying $y_i = \int \xi d \mu \, (|i| ≤ m)$ (where $x = (x_1, \ldots, x_n)$, $i = (i_1, \ldots, i_n)$). in previous work we associated to $y$ an algebraic variety in $\mathbb{r}^n$ , the core variety $v = v(y)$, and showed that if $v$ is nonempty, then the riesz functional $l$ corresponding to $y$ is strictly v-positive, i.e., if $p(x) := \sigma a_i x_i \, (|i| ≤ m)$ is nonnegative on $v$, and $p|_v$ is not identically $0$, then $l(p) := \sigma a_i y_i > 0$. in current work with g. blekherman, we prove that if $l$ is strictly $k$-positive for any closed subset $k$ of $\mathbb{r}^n$, then $y$ has a representing measure $\mu$ (as above) whose support is contained in $k$. as a consequence, we prove that $y$ has a representing measure if and only if $v(y)$ is nonempty, in which case $v(y)$ coincides with the union of the supports of all representing measures. as a corollary, we obtain a new proof of the bayer-teichmann theorem on multivariable cubature.

it is a theorem of de bruijn and erdos that $n$ points in the plane
determine at least $n$ lines, unless all the points lie on a line. this
is one of the earliest results in enumerative combinatorial geometry.
we will present a higher dimensional generalization of this theorem,
which confirms a “top-heavy” conjecture of dowling and wilson in 1975.
i will give a sketch of the key idea of the proof, which uses the hard
lefschetz theorem and the decomposition theorem in algebraic geometry.
i will also talk about a log-concave conjecture on the number of
independent sets. this is joint work with june huh.

let $g$ be uniformly distributed on the set of all simple $d$-regular graphs on $n$ vertices, and assume $d$ is bigger than some (small) power of $n$. we show that the second largest eigenvalue of $g$ is of order $\sqrt{d}$ with probability close to one. combined with earlier results covering the case of sparse random graphs, this settles the problem of estimating the magnitude of the second eigenvalue, up to a multiplicative constant, for all values of $n$ and $d$, confirming a conjecture of van vu. joint work with pierre youssef.

let $e$ be an elliptic curve defined over ${q}$. the torsion subgroup
of $e$ over the compositum of all quadratic extensions of ${q}$ was
studied by michael laska, martin lorenz, and yasutsugu fujita. laska
and lorenz described a list of $31$ possible groups and fujita proved
that the list of $20$ different groups is complete.

in this talk, we will generalize the results of laska, lorenz and
fujita to the elliptic curves defined over a quadratic cyclotomic
field i.e. $q(i)$ and $q(\sqrt{-3})$.

while the topic of geometric incidences has existed for
several decades, in recent years it has been experiencing a
renaissance due to the introduction of new polynomial methods. this
progress involves a variety of new results and techniques, and also
interactions with fields such as algebraic geometry and harmonic
analysis.

a simple example of an incidences problem: given a set of $n$ points and
set of n lines, both in $r^2$, what is the maximum number of point-line
pairs such that the point is on the line. while this may seem as a
simple problem, incidence problems often have a deep underlying
theory, which may involve the uncovering of hidden structure and
symmetries.

in this talk we introduce and survey the topic of geometric
incidences, focusing on the recent polynomial techniques and results
(some by the speaker). we will see how various algebraic and analytic
tools can be used to solve such combinatorial problems.

in joint work with g. blekherman we proved that a truncated multisequence $y$ of degree $m$ has a representing measure in the truncated moment problem if and only if its core variety $v(y)$ is nonempty, in which case $v(y)$ coincides with the union of the supports of all representing measures. in general, for a given numerical sequence $y$, it may be quite difficult to compute $v(y)$ or even to determine if it is nonempty. we illustrate some cases where we can compute $v(y)$ or can otherwise describe it concretely.

hyperkahler (hk) manifolds appear in many fields of mathematics, such
as differential geometry, mathematical physics, representation theory,
and algebraic geometry. compact hk manifolds are one of the building
blocks for algebraic varieties with trivial first chern class and
their role in algebraic geometry has grown immensely over the last 20
year. in this talk i will give an overview of the theory of compact hk
manifolds and then focus on some of my work, including a recent joint
work with r. laza and c. voisin.