In this talk I'll introduce the Zoom for Thought seminar and then
teach the basics of making a website and how to connect with the UCSD
server. If you want to have a working website by the end of the week, it
will help to (1) talk to Saul about getting your login information for the
UCSD math servers, (2) have a text editor to make the website (I like
notepad++), and (3) have a way to SSH to the math server (I like WinSCP).
To stall for time I'll also talk about other resources that might be of
use to a math PhD student. Other people are more than welcome to
contribute ideas in this direction.

Given an $r$-uniform hypergraph $H$ and an $r$-uniform hypergraph $F$, define the relative Tur\'an number $ex(H,F)$ to be the maximum number of edges in an $F$-free subgraph of $H$. In this talk we discuss bounds for $ex(H,F)$ when $F$ is a loose cycle, Berge cycle, and a complete $r$-partite $r$-graph when the host $H$ is either arbitrary or when it is the random hypergraph $H_{n,p}^r$.
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This is joint work with Jiaxi Nie and Jacques Verstraete.

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.
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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.
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The schedule of the lecture series will be approximately as follows:
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1. Heat Kernel and entropy estimates on Ricci flow backgrounds and related geometric bounds.
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2. Continuation of Lecture 1 + Synthetic definition of Ricci flows (metric flows) and basic properties
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3. Convergence and compactness theory of metric flows
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4. Partial regularity of limits of Ricci flows

Brakke flow is a measure-theoretic generalization of the mean curvature flow which exploits the flexibility of geometric measure theory in order to describe the evolution by (generalized) mean curvature of surfaces exhibiting singularities, such as, for instance, a planar network with multiple junctions. In the first part of this talk, I will discuss the proof of the following result: given any $n$-dimensional rectifiable subset $\Gamma_0$ of a strictly convex bounded domain $U \subset \mathbb{R}^{n+1}$ such that $U \setminus \Gamma_0$ is not connected, there exists a Brakke flow of surfaces (possibly weighted with integer multiplicities) starting from $\Gamma_0$ and with the additional property that their boundary coincides with $\partial \Gamma_0$ at all times. Furthermore, the flow subconverges, as $t \to \infty$ and in the sense of varifolds, to a (generalized) minimal surface in $U$ with the prescribed boundary $\partial \Gamma_0$, thus providing a dynamical solution to Plateau's problem. In the second part, I will discuss recent developments concerning the relationship between the singularities of a stationary initial surface $\Gamma_0$ and the (non) uniqueness of the flow. This investigation leads to the definition of a class of \emph{dynamically stable} stationary varifolds, which seems worthy of further study. Based on joint works with Yoshihiro Tonegawa (Tokyo Institute of Technology).

Abstract: Consider an $n\times n$ deterministic matrix $A$ and a random matrix $M$ with independent standard Gaussian entries. In this talk I will discuss recent results that state that, if $||A||\leq 1$, for any $\delta \textgreater 0$, with high probability $A+\delta M$ has eigenvector condition number of order poly$(n/\delta)$ and eigenvalue gaps of order poly$(\delta/n)$, which implies that the randomly perturbed matrix has a stable spectrum. This has useful applications to numerical analysis and was used to obtain the fastest known provable algorithm for diagonalizing an arbitrary matrix.

This is joint work with Jess Banks, Archit Kulkarni and Nikhil Srivastava.

In this talk I will discuss the optimal transport problem with ``storage fees.'' This is a variant of the semi-discrete optimal transport (a.k.a. Monge-Kantorovich) problem, where instead of transporting an absolutely continuous measure to a fixed discrete
measure and minimizing the transport cost, one must choose the weights of the target measure, and minimize the sum of the transport cost and some given ``storage fee function'' of the target weights. This problem arises in queue penalization and semi-supervised
data clustering. I will discuss some basic theoretical questions, such as existence, uniqueness, a dual problem, and characterization of solutions. Then, I will present a numerical algorithm which has global linear and local superlinear convergence for a subcase
of storage fee functions.
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All work in this talk is joint with M. Bansil (UCLA).

The exceptional group $E_{7,3}$ has a symmetric space with
Hermitian tube structure. On it, Henry Kim wrote down low weight
holomorphic modular forms that are ``singular'' in the sense that their
Fourier expansion has many terms equal to zero. The symmetric space
associated to the exceptional group $E_{8,4}$ does not have a Hermitian
structure, but it has what might be the next best thing: a quaternionic
structure and associated ``modular forms''. I will explain the
construction of singular modular forms on $E_{8,4}$, and the proof that
these special modular forms have rational Fourier expansions, in a
precise sense. This builds off of work of Wee Teck Gan and uses key
input from Gordan Savin.

How does one measure area? As an example, how can one determine
the area of a region on a map for the purpose of real estate appraisal?
Wouldn't it be great if there were an instrument that would measure the
area of a region by simply tracing its boundary? It turns out that there
is such an instrument: it is called a planimeter. In this talk we will
discuss a particular type of planimeter called the compensating polar
planimeter. There will be a little bit of history and some analysis
involving line integrals and Green's theorem. Finally, there will be a
chance to (virtually) see actual examples of these fascinating instruments
from the speaker's collection.

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