Bilevel optimization problem is a two-level optimization problem, where a subset of its variables is constrained in the optimizer set of another optimization problem parameterized by the remaining variables. In this talk, we introduce a Lagrange multiplier expression method for bilevel polynomial optimization based on polynomial optimization relaxations. Each relaxation is obtained from the Kurash-Kuhn-Tucker (KKT) conditions for the lower level optimization and the exchange technique for semi-infinite programming. The global convergence of the method is proved under some general assumptions. And some numerical examples will be given to show the efficiency of the method.

Let $A$ be a $C^*$-algebra, $f \colon \mathbb{R} \to \mathbb{C}$ be a continuous function, and $\tilde{f} \colon A_{\text{sa}} \to A$ be the functional calculus map $A_{\text{sa}} \ni a \mapsto f(a) \in A$. It is elementary to show that $\tilde{f}$ is continuous, so it is natural to wonder how the differentiability properties of $f$ relate/transfer to those of $\tilde{f}$. This turns out to be a delicate, complicated problem. In this talk, I introduce a rich class $NC^k(\mathbb{R}) \subseteq C^k(\mathbb{R})$ of noncommutative $C^k$ functions $f$ such that $\tilde{f}$ is $k$-times differentiable. I shall also discuss the interesting objects, called multiple operator integrals, used to express the derivatives of $\tilde{f}$.

Report on work in progress, joint with Paul Arne Oestvaer.
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A cellular motivic invariant is a special type of functor from the
category of commutative rings (or the opposite of schemes, say) to
spectra. Examples include algebraic K-theory, motivic cohomology, \'e{}tale
cohomology and algebraic cobordism. Dwyer-Friedlander observed that for
2-adic \'e{}tale K-theory and certain related invariants, the value on
Z[1/2] can be described in terms of a fiber square involving the values
on the real numbers, the complex numbers, and the field with three elements.
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I will explain a generalization of this result to arbitrary 2-adic
cellular motivic invariants. As an application, we show that $\pi_0$ of the
motivic sphere spectrum over Z[1/2] is given by the Grothendieck-Witt
ring of Z[1/2], up to odd torsion.

Consider a town of $n$ people and suppose this town wants to impose the following rules on its clubs (formally subsets of the towns $n$ residents). First, each club must have an odd number of members. Second, each distinct pair of clubs in the town must have an even number of members in common. What is the maximum number of clubs this town can have while adhering to the rules?

The last decade has seen tremendous progress in applying random matrix methods to adjacency matrices or Laplacians of random graphs, in order to understand their spectra and be able to apply the new results to algorithms in machine learning, coding theory, data science, etc. Nevertheless, many problems remain. I will present some of the most interesting tools and new results and mention some (still) open problems.

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension $n \geq 4$, we find a family of $Z2$ $\times$ $O(n - 1)$-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.

I will talk about some results and open questions related
to the moduli of maps of curves to K3 surfaces, sheaves
on K3 surfaces, and moduli of K3 surfaces themselves.