Curvature pinching theorems restrict the topology of smooth manifolds satisfying suitable curvature assumptions. In some situations the Ricci flow can transform initial integral curvature bounds into later pointwise bounds and thereby extend pointwise to integral pinching results. I will first review $ L^p$ integral pinching theorems of Gursky, Hebey--Vaugon, Dai--Wei--Ye, and others, which all rely on supercritical powers p greater than n/2 or on Chern--Gauss--Bonnet in dimension four. Then I will discuss how stronger control of the Sobolev inequality obtained using Perelman's mu-functional can be used to address the critical case p=n/2, leading both to a generalization of previous results as well as to a separate pinching result in the asymptotically flat setting. Some of the work presented is joint with Guofang Wei and Rugang Ye.

Multiscale substitution tilings are a new family of tilings of Euclidean space that are generated by multiscale substitution rules. Unlike the standard setup of substitution tilings, which is a basic object of study within the aperiodic order community and includes examples such as the Penrose and the pinwheel tilings, multiple distinct scaling constants are allowed, and the defining process of inflation and subdivision is a continuous one. Under a certain irrationality assumption on the scaling constants, this construction gives rise to a new class of tilings, tiling spaces and tiling dynamical systems, which are intrinsically different from those that arise in the standard setup. In the talk I will describe these new objects and discuss various structural, geometrical, statistical and dynamical results. Based on joint work with Yaar Solomon.

In MIP*$=$RE the authors settle the CEP in the negative by showing that two computational complexity classes are equal. But the heart of their argument is the construction of a synchronous game with certain properties. In this talk we will describe the theory of synchronous games, and show how our construction of the algebra of a game leads more directly to the CEP. This approach to the CEP still depends on their construction of a synchronous game with particular properties, but stays within the context of operator algebras and games.

Given the ubiquity of vector bundles, it is perhaps surprising that there are so many open questions about them -- even on projective spaces. In this talk, I will outline some results about vector bundles on projective spaces, including my ongoing work on complex rank 3 topological vector bundles on $CP^5$. In particular, I will describe a classification of such bundles which involves a connection to topological modular forms; a concrete, rank-preserving additive structure which allows for the construction of new rank 3 bundles on $CP^5$ from ``simple'' ones; and future directions related to this project.

As an independent mathematician, you want to determine which of your axioms are internally consistent. Since you only have a finite amount of time, you may choose to come to this talk! I will be giving a countably long survey of results related to the Axiom of Choice. Well, hopefully you find it a maximally useful way to spend your Tuesday afternoon. (Or maybe you won't not find it useful, depending on what you decide.)

Let $V$ be a vector bundle over a smooth projective curve of rank $n$. We are interested in understanding the set of sub-bundles of rank $r$ with maximal degree. When we impose some numerical conditions, this set happens to be finite. In this talk, we will go over chronological developments in finding the number of maximal sub-bundles. Surprisingly, these numbers are related to the Gromov-Witten invariants of Grassmannian which are computed by the Vafa-Intriligator formula.

In this talk I will discuss certain singular (and degenerate) solutions to the Kaehler Ricci flow (KRF) on smooth compact complex manifolds. Algebraically this will correspond to solving the Kahler Ricci flow on a projective varieties with so called log canonical singularities. Analytically this will correspond to solving a complex parabolic Monge Ampere equation on a smooth manifild, with degeneracies and singularities in the equation and possibly the initial condition. Settings for this study include the analytic minimal model program via KRF, pluri-potential theory and KRF, the conical KRK, and the flow of complete unbounded curvature metrics. Our results will be discussed within each of these contexts.

This talk discusses multiple methods for clustering high-dimensional data, and explores the delicate balance between utilizing data density and data geometry. I will first present path-based spectral clustering, a novel approach which combines a density-based metric with graph-based clustering. This density-based path metric allows for fast algorithms and strong theoretical guarantees when clusters concentrate around low-dimensional sets. However, the method suffers from a loss of geometric information, information which is preserved by simple linear dimension reduction methods such as classic multidimensional scaling (CMDS). The second part of the talk will explore when CMDS followed by a simple clustering algorithm can exactly recover all cluster labels with high probability. However, scaling conditions become increasingly restrictive as the ambient dimension increases, and the method will fail for irregularly shaped clusters. Finally, I will discuss how a more general family of path metrics, combined with MDS, give low-dimensional embeddings which respect both data density and data geometry. This new method exhibits promising performance on single cell RNA sequence data and can be computed efficiently by restriction to a sparse graph.

The mod p cohomology of locally symmetric spaces for definite
unitary groups at infinite level is expected to realize the mod p local
Langlands correspondence for $GL_n$. In particular, one expects the
(component at p) of the associated Galois representation to be
determined by cohomology as a smooth representation. I will describe how
one can establish this expectation in many cases when the local Galois
representation is Fontaine-Laffaille.
\\
\\
This is joint work with D. Le, S. Morra, C. Park and Z. Qian.

Deciding nonnegativity of real polynomials is a key
question in real algebraic geometry with crucial importance in
polynomial optimization. It is well-known that in general this problem
is NP-hard, therefore one is interested in finding sufficient conditions
(certificates) for nonnegativity, which are easier to check. Since the
19th century, sums of squares (SOS) are a standard certificate for
nonnegativity, which can be detected by using semidefinite programming
(SDP). This SOS/SDP approach, however, has some issues, especially in
practice if the problem has many variables or high degree.
In this talk I will introduce sums of nonnegative circuit polynomials
(SONC). SONC polynomials are certain sparse polynomials having a special
structure in terms of their Newton polytopes and supports and serve as a
nonnegativity certificate for real polynomials, which is independent of
sums of squares. I will present some structural results of SONC
polynomials and I will provide an overview about polynomial optimization
via SONC polynomials.

We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$,
for $i\in\{0,\ldots, n-1\}$.
\\
\\
The problem of realizing $A$ by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an \emph{angle graph}.
\\
\\
In 2D, we characterize sequences $A$ for which every generic polygon $P\subset \mathbb{R}^2$ realizing $A$ has at least $c$ crossings, for every $c\in \mathbb{N}$, and describe an efficient algorithm that constructs, for a given sequence $A$, a generic polygon $P\subset \mathbb{R}^2$ that realizes $A$ with the minimum number of crossings.
\\
\\
In 3D, we describe an efficient algorithm that tests whether a given sequence $A$ can be realized by a (not necessarily generic) polygon $P\subset \mathbb{R}^3$, and for every realizable sequence the algorithm finds a realization.