Adaptive cubic regularization methods have several favorable properties for nonconvex optimization. In particular, under mild assumptions, they are globally convergent to a second-order stationary point. In this talk, I will introduce an adaptive cubic regularization method for unconstrained optimization. Methods analogous to those used to solve the trust-region subproblem will be discussed for solving the local cubic model. Some numerical results will be presented that compare a cubic regularized Newton's method, a standard trust-region method and a trust-search method.

A unitary representation $\pi\colon G\to B(H)$ of a locally compact group $G$ is an \emph{$L^p$-representation} if $H$ admits a dense subspace $H_0$ so that the matrix coefficient

$ G\ni s\mapsto \langle \pi(s)\xi,\xi\rangle$

belongs to $L^p(G)$ for all $\xi\in H_0$. The \emph{$L^p$-C*-algebra} $C^*_{L^p}(G)$ is the C*-completion $L^1(G)$ with respect to the C*-norm

$ \|f\|_{C^*_{L^p}}:=\sup\{\|\pi(f)\| : \pi\textnormal{ is an }L^p\textnormal{-representation of $G$}\}\qquad (f\in L^1(G)).$

Surprisingly, the C*-algebra $C^*_{L^p}(G)$ is intimately related to the enveloping C*-algebra of the Banach $*$-algebra $PF^*_p(G)$ ($2\leq p\leq \infty$). Consequently, we characterize the states of $C^*_{L^p}(G)$ as corresponding to positive definite functions that ``almost'' belong to $L^p(G)$ in some suitable sense for ``many'' $G$ possessing the Haagerup property, and either the rapid decay property or Kunze-Stein phenomenon. It follows that the canonical map

$$ C^*_{L^p}(G)\to C^*_{L^{p'}}(G)$$

is not injective for $2\leq p' \leq p \leq \infty$ when $G$ is non-amenable and belongs to the class of groups mentioned above. As a byproduct of our techniques, we give a near solution to a 1978 conjecture of Cowling.

This is primarily based on joint work with E. Samei.

For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic (i) cycle $C_{2n}$ with $2n$ vertices, (ii) cycle $C_{\geq 2n}$ with at least $2n$ vertices, (iii) path $P_{2n}$ with $2n$ vertices, and (iv) path $P_{2n+1}$ with $2n+1$ vertices. This implies a generalization of the conjecture by Gy\' arf\' as, Ruszink\' o, S\' ark\H ozy and Szemer\' edi that for every $2$-edge-coloring of the complete $3$-partite graph $K_{n,n,n}$ there is a monochromatic path $P_{2n+1}$.

An important tool is our recent stability theorem on monochromatic connected matchings (A matching $M$ in $G$ is connected if all the edges of $M$ are in the same component of $G$). We will also talk about exact Ramsey-type bounds on the sizes of monochromatic connected matchings in $2$-colored multipartite graphs. Joint work with J\' ozsef Balogh, Alexandr Kostochka and Mikhail Lavrov.

For each regular coadjoint orbit of a compact group, we construct an exhaustion by symplectic embeddings of toric domains. As a by-product we arrive at a conjectured formula for the Gromov width of coadjoint orbits. Our method combines ideas from Poisson-Lie groups and from the geometric crystals of Berenstein-Kazhdan. We also prove similar results for multiplicity-free spaces. This is joint work with A. Alekseev, J. Lane, and Y. Li.

The classical uniformization theorem transforms the study of moduli spaces of marked Riemann surfaces into the study of constant curvature metrics with singularities. I will give a survey on constant curvature metrics with cusp and conical singularities, including works joint with Richard Melrose, Rafe Mazzeo and Bin Xu, where new analytic tools have been developed to understand the uniformization. ``Resolution of singularities'' is the key idea in the analysis, which can be seen as an analogue of the Deligne--Mumford compactification of Riemann moduli spaces.

In this talk, we will discuss recent results on stable self-similar singularity formation for the 2D Boussinesq and singularity formation for the 3D Euler equations in the presence of the boundary with $C^{1,alpha}$ initial data for the velocity field that has finite energy. The blowup mechanism is based on the Hou-Luo scenario of a potential 3D Euler singularity. We will also discuss some 1D models for the 3D Euler equations that develop stable self-similar singularity in finite time. For these models, the regularity of the initial data can be improved to $C_c^{infty}$. Some of the results are joint work with Thomas Hou and De Huang.

Contact manifolds, which arise naturally in mechanics, dynamics, and geometry, carry natural Riemannian and sub-Riemannian structures and it was shown by Gromov that the latter can be obtained as a limit of the former. Subsequently, Rumin found a complex of differential forms reflecting the contact structure that computes the singular cohomology of the manifold. He used this complex to describe the behavior of individual eigenvalues of the Riemannian Hodge Lapacians in the sub-Riemannian limit but was unable to determine the behavior of global spectral invariants. I will report on joint work with Hadrian Quan in which we determine the global behavior of the spectrum by explaining the structure of the heat kernel along this limit in a uniform way.

In this talk we shall look at an overview of discrete Morse theory in the context of simplicial complexes. Discrete Morse theory, based on the work by R. Forman, provides a framework to study the ``shape'' (i.e. the topology) of a simplicial complex via discrete Morse functions which are real valued functions defined on the simplices of the complex. The critical simplices, which are determined by the respective discrete Morse function, reveal key topological features of the simplicial complex. This is, in essence, a discrete adaptation of Morse theory in differential topology which allows us to study the topology of a manifold by looking at the differentiable functions on the manifold. The talk will cover the basics of discrete Morse theory with multiple examples and will also discuss possible applications in the context of persistent homology.

In this talk, we will explore various $q$ analogs of previous results from the seminar. The main
result will be the vector space analog of a Theorem of Frankl and Wilson. We will also discuss
some applications to constructive lower bounds on Ramsey numbers. The talk will be based off a
paper with the same title written by Peter Frankl and Ronald Graham.

An ordinary Gushel-Mukai fourfold $X$ is a smooth quadric section of a linear section of the Grassmannian $G(2,5)$. Kuznetsov and Perry proved that the bounded derived category of $X$ admits a semiorthogonal decomposition whose non-trivial component is a subcategory of K3 type. In this talk I will report on a joint work in progress with Alex Perry and Laura Pertusi, in which we construct Bridgeland stability conditions on the K3 subcategory of $X$. Then I will explain some applications concerning the existence of a homological associated K3 surface, and related algebraic constructions in hyperkaehler geometry.

Ideas from theoretical physics have had a
profound impact on geometry, topology, and representation theory over
the last several decades. An early high point of this interaction was
Witten's quantum field theoretic interpretation of the celebrated
Donaldson invariants, which in turn opened the door to his discovery
of the even-more-celebrated Seiberg-Witten invariants. In this talk,
we'll explain how more recently this interaction has made possible
dramatic advances in geometric representation theory, with a focus on
joint work with Sabin Cautis revealing the structure of the coherent
Satake category of a complex Lie group. This is an intricate cousin of
the constructible Satake category appearing in the geometric Satake
equivalence, a cornerstone of the geometric Langlands program. The
coherent Satake category turns out to have rich connections to the
Fomin-Zelevinsky theory of cluster algebras, as well as to the
representation theory of quantum groups and quiver Hecke
algebras. However, while these connections can be stated in purely
mathematical terms, their discovery hinged crucially on first
understanding how to interpret the coherent Satake category in terms
of physics --- in fact, the very same physics (4d N=2 supersymmetric
Yang-Mills theory) behind the Donaldson and Seiberg-Witten invariants.