Polynomial optimization is a class of optimization programs whose objective and constraining functions are polynomials. The core task in polynomial optimization is to compute global optimizers when the optimization is nonconvex. Tensor computation is about optimization and decompositions of tensors, such as tensor norms, tensor eigenvalues and tensor decompositions. All these problems are connected to each other by the theory nonnegative polynomials and moment problems. This talk will give an introduction about classical backgrounds, currently existing results and remaining challenges for the research of these topics.

The minimal genus problem is a fundamental question in smooth 4-manifold topology. Every 2-dimensional homology class can be represented by a surface. But how small can this surface be? A generation ago, techniques from gauge theory were used to solve this in a large class of 4-manifolds with extra geometric structure, namely symplectic 4-manifolds. Recent work on trisections if 4-manifolds has revealed a deep connection with symplectic geometry and gives a new perspective on this result.

If $\operatorname{M}_n(\mathbb{C})$ is the set of $n \times n$ complex
matrices and $A \in \operatorname{M}_n(\mathbb{C})$, then we write
$\sigma(A) \subseteq \mathbb{C}$ for the set of eigenvalues of $A$. If $A$
is diagonalizable and $f \colon \sigma(A) \to \mathbb{C}$ is any function,
then one can define $f(A) \in \operatorname{M}_n(\mathbb{C})$ in a
reasonable way. Now, let
$\operatorname{M}_n(\mathbb{C})_{\operatorname{sa}}$ be the set of $n
\times n$ Hermitian matrices, which are unitarily diagonalizable and have
real eigenvalues. If $f \colon \mathbb{R} \to \mathbb{C}$ is a continuous
function, then one can fairly easily show that the map $\tilde{f} \colon
\operatorname{M}_n(\mathbb{C})_{\operatorname{sa}} \to
\operatorname{M}_n(\mathbb{C})$ defined by $A \mapsto f(A)$ is also
continuous. In this talk, we shall discuss the less elementary fact that
if $f$ is $k$-times continuously differentiable, then so is $\tilde{f}$.
Time permitting, we shall also discuss the much more complicated
infinite-dimensional case -- where instead of matrices, one considers
linear operators on a Hilbert space -- which is still an active area of
research.

This is an example-based introduction to the characteristic zero representation theory of p-adic GL(n). Highlights include Reeder's construction of the ``simple supercuspidal representations'', the proof that these are irreducible, and a discussion of their Langlands parameters. We assume a few basic definitions from representation theory and p-adic arithmetic.

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds.
\\
\\
As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.
\\
\\
The schedule of the lecture series will be approximately as follows:
1. Heat Kernel and entropy estimates on Ricci flow backgrounds and related geometric bounds.
2. Continuation of Lecture 1 + Synthetic definition of Ricci flows (metric flows) and basic properties
3. Convergence and compactness theory of metric flows
4. Partial regularity of limits of Ricci flows

The theory of RCD spaces has seen a huge development in the last teen years. They are metric measure structures satisfying a synthetic notion of Ricci bounded below. This class includes several spaces with boundary, such as Gromov-Hausdorff limits of manifolds with convex boundary and Ricci bounded below in the interior. In this talk we will present new stability and regularity results for boundaries of RCD spaces. We will focus mostly on a new epsilon-regularity theorem which is new even in the setting of smooth Riemannian manifolds. It is based on a work in progress joint with Aaron Naber and Daniele Semola.

Convolutional Neural Networks are highly successful tools for image recovery and restoration. A major contributing factor to this success is that convolutional networks impose
strong prior assumptions about natural images - so strong that they enable image recovery without any training data. A surprising observation that highlights those prior assumptions is that one can remove noise from a corrupted natural image by simply fitting
(via gradient descent) a randomly initialized, over-parameterized convolutional generator to the noisy image.
\\
\\
In this talk, we discuss a simple un-trained convolutional network, called the deep decoder, that provably enables image denoising and regularization of inverse problems such as compressive sensing with excellent performance. We formally characterize the dynamics
of fitting this convolutional network to a noisy signal and to an under-sampled signal, and show that in both cases early-stopped gradient descent provably recovers the clean signal.
\\
\\
Finally, we discuss our own numerical results and numerical results from another group demonstrating that un-trained convolutional networks enable magnetic resonance imaging from highly under-sampled measurements, achieving results surprisingly close to trained
networks, and outperforming classical untrained methods.

I will present a vast generalization of Taelman's 2012
celebrated class-number formula for Drinfeld modules to the setting of
(rigid analytic) L-functions of Drinfeld module motives with Galois
equivariant coefficients. I will discuss applications and potential
extensions of this formula to the category of t-modules and t-motives.
This is based on joint work with Ferrara, Green and Higgins, and a
result of meetings in the UCSD Drinfeld Module Seminar.

Recently, Marian-Oprea-Pandharipande proved Lehn's conjecture for
Segre numbers associated to Hilbert schemes of points on surfaces.
They also provided a conjectural correspondence between Segre and
Verlinde numbers. For surfaces with holomorphic 2-form, we propose
conjectural generalizations of their results to moduli spaces of
stable sheaves of any rank. We provide several verifications by using
Mochizuki's formula. Joint work with Gottsche.

Percy MacMahon introduced the ``major index'' of a permutation a century
ago and proved that it has the same distribution as the ``inversion
number''. Such statistics are now called ``Mahonian''. Baxter and
Zeilberger considered the joint distribution of inversions and the major
index on permutations and showed they are jointly independently
asymptotically normally distributed. However, Baxter-Zeilberger's argument
has no hope of giving a more precise ``local limit theorem'' and
Zeilberger subsequently offered a \$300 reward for a different proof. This
is the story of that reward.