Minors and topological minors are two closely related graph containment relations that have attracted extensive attentions. Though giant breakthroughs on graph minors have been made over decades, several questions about these two relations remain open, especially for topological minors. This talk addresses part of our recent work in this direction, including a proof of Robertson's conjecture on well-quasi-ordering graphs by the topological minor relation, a complete characterization of the graphs having the Erdos-Posa property with respect to topological minors which answers a question of Robertson and Seymour, and a proof of Thomas' conjecture on half-integral packing. More open questions, such as Hadwiger's conjecture on graph coloring and its variations and relaxations, will be discussed in this talk.

Given a tuple $A=(A_1,\ldots,A_g)$ of symmetric matrices of the same size, the affine linear matrix polynomial $L(x):=I-\sum A_j x_j$ is a monic linear pencil. The solution set $S_L$ of the corresponding linear matrix inequality (LMI), consisting of those $x$ in $\mathbb R^g$ for which $L(x)$ is positive semidefinite (PsD), is called a spectrahedron. It is a convex semialgebraic subset of $\mathbb R^g$, and LMIs are ubiquitous in many areas: mathematical optimization, control theory, statistics, etc.

We study the question whether inclusion holds between two spectrahedra. Most of our results concern the case where the included spectrahedron is a hypercube, an NP-hard problem introduced and studied by Ben-Tal and Nemirovskii, who identified a tractable relaxation of the original problem. This relaxation is obtained by considering the inclusion problem for the corresponding ``matricial'' spectrahedra.

To estimate the error inherent in the relaxation we employ probabilistic methods and an old result of Rev. Simmons on flipping biased coins to obtain an elegant scalar optimization formula.

This is based on joint work with Bill Helton, Scott McCullough and Markus Schweighofer.

An intriguing feature of the explicit charged (Reissner-Nordstrom) or spinning (Kerr) black hole spacetimes is the existence of a regular Cauchy horizon, beyond which the Einstein equation loses its predictive power. The strong cosmic censorship conjecture of Penrose is a bold claim that, nevertheless, such a pathological behavior is nongeneric.

In this lecture, I will give a short introduction to general relativity and the strong cosmic censorship conjecture. Then I will describe my recent joint work with J. Luk, where we rigorously establish a version of this conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry, which has long been studied by physicists and mathematicians as a useful model for this problem.

The talk concerns (matrices of) noncommutative polynomials $f=f(x)$ from the perspective of free real algebraic geometry. There are several natural notions of a ``zero set'' of $f$. The one we study is the \textbf{free locus} of $f$, $(f)$, which is defined to be the union of hypersurfaces
$$\left\{X\in \operatorname{M}_n(k)^g:\det f(X)=0\right\}$$
over all $n\in\mathbb{N}$. The talk will describe a recent advance on characterizing when $(f_1) \subseteq (f_2)$ holds, which was mainly achieved using linear matrix pencils.

The latter have been for decades an important tool in noncommutative algebra and other areas, e.g. linear systems, automata theory and computational complexity. Given a monic matrix pencil $L=I+\sum_jA_jx_j$, we can evaluate it at an arbitrary tuple of matrices $X$ as
$$L(X)=I\otimes I+\sum_jA_j\otimes X_j.$$
The talk will be mostly concerned with singularity of these evaluations.

First we will give an algebraic certificate for $(L_1)\subseteq (L_2)$ to hold. Then we will consider a fundamental irreducibility theorem for $(L)$ which is obtained with the aid of invariant theory. Next we will apply the preceding results to factorization in the free algebra. Lastly, smooth points on $(L)$ will be related to one-dimensional kernels of $L(X)$, which leads to the free version of Kippenhahn's conjecture and improves existing Positivstellens\''{a}tze on free semialgebraic sets.

Iwasawa theory is a bridge between algebraic and analytic invariants attached to an arithmetic object,
for a given prime p. When this arithmetic object is an elliptic curve or a modular form, the primes
come in two flavors -- ordinary and supersingular. When p is ordinary, the theory has historically
been relatively well behaved. When p is supersingular, there are several difficulties, and we explain
how to address the difficulties involved in the case of elliptic curves, culminating in the proof of the
Main Conjecture. If time permits, we will sketch joint work in progress with Castella, Ciperiani, and
Skinner concerning main conjecture for weight-two modular forms.

Very often in algebraic geometry, the totality of geometric objects under some natural setting becomes again an algebraic variety, which is one the main object of study in algebraic geometry.
Hilbert scheme is one of such space parametrizing families of projective algebraic varieties having the same fixed Hilbert polynomial, i.e. sharing certain basic extrinsic attributes and intrinsic invariants.
In this talk we will start with the basic construction of the Hilbert scheme of projective algebraic curves due to Alexander Grothendieck.
We then proceed further and discuss about the current state of affairs especially on the irreducibility and the rigidity of the Hilbert scheme of smooth projective curves after reviewing a brief history of the study since the era of Italian school.