Michigan State University

Let $(M,\varphi)=(M_1,\varphi_1)*(M_2,\varphi_2)$ be a free product of non-trivial von Neumann algebras equipped with arbitrary faithful normal states. In 2011, Ueda showed that the diffuse summand of $M$ is a factor if and only if the maximum vector space dimension of $M_1$ and $M_2$ is at least 3, and in this case one further has that the diffuse summand has no asymptotically central sequences. Now consider the von Neumann algebra $N$ generated by non-trivial subalgebras $N_i\leq M_i$ with $\varphi_i$-preserving expectations for $i=1,2$, and let $z \in N$ be the central projection supporting the diffuse summand of $N$. Ueda's result characterizes when $Nz$ is a (full) factor, and, in fact, several cases treated in his proof yield the stronger property that $zMz$ lacks asymptotically $(Nz)$-central sequences. Thus, is natural to wonder: is this stronger property equivalent to the maximum vector space dimension of $N_1$ and $N_2$ being at least 3? In this talk, I will make use of amalgamated free products to answer this question in the affirmative. These same techniques can also be used to analyze the centralizer $M^\varphi = \{ x \in M \colon \varphi(xy) = \varphi(yx)\ \forall y\in M\}$ of the free product state, generalizing a separate 2011 result of Ueda from almost periodic states to arbitrary states. This is based on joint work with Aldo Garcia Guinto, Fehmi Ekin Giritlioglu, Yoonkyeong Lee, and Rahul K. Ramachandran.

(Joint work with Brent Nelson) In this talk we investigate the anticoarse space of the von Neumann algebras generated by the kernel and the domain of a closable derivation. We show that when a tuple (x_i)_{i\in I} admits a conjugate system, then for any proper subset J \subset I  with |J| \geq 2 the inclusion W*(x_j :j \in j) \subset W*(x_i: i \in I) is irreducible, infinite index and non-regular.

Gauge theory excels at solving minimal genus problems for 3- and 4-manifolds.  A notable triumph is its resolution of the Thom conjecture, asserting that the genus of a smooth complex curve in the complex projective plane is no larger than any smooth submanifold homologous to it.  Gauge theoretic techniques have also been used to verify analagous conjectures for Kähler surfaces or, more generally, symplectic 4-manifolds.  One can formulate versions of these conjectures for surfaces with boundary lying in a 3-manifold, and I'll discuss work in progress with Katherine Raoux which attempts to extend these "relative" Thom conjectures outside the complex (or even symplectic) realm using tools from Floer homology.

Singularities are measured in different ways in characteristic zero, positive characteristic, and mixed characteristic. However, classes of singularities usually form analogous groups with similar properties, with an example of such a group being klt, strongly F-regular and BCM-regular.  In this talk we shall focus on newly introduced mixed characteristic counterparts of Du Bois and log canonical singularities and discuss their properties. 

This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker and Jakub Witaszek. 

Co-occurrence networks formed by bipartite projection are widely studied in many contexts, including politics (bill co-sponsorship), bibliometrics (paper co-authorship), ecology (species co-habitation), and genetics (protein co-expression). It is often useful to focus on the backbone, a binary representation that includes only the most important edges, however many different backbone extraction models exist. In this talk, I will demonstrate the "backbone" package for R, which implements many such models. I will also use it to compare two promising null models: the fixed degree sequence model (FDSM) that imposes hard constraints, and the stochastic degree sequence model (SDSM) that imposes soft constraints, on the bipartite degree sequences. While FDSM is more statistically powerful, SDSM is more efficient and offers a close approximation.

Topological data analysis (TDA) is a field with tools to quantify the shape of data in a manner that is concise and robust using concepts from algebraic topology. Persistent homology, one of the most popular tools in TDA, has proven useful in applications to time series data, detecting shape that changes over time and quantifying features like periodicity. In this talk, I will present two applications using tools from TDA to study time series data: the first using zigzag persistence, a generalization of persistent homology, to study bifurcations in dynamical systems and the second, using the shape of weighted, directed networks to distinguish periodic and chaotic behavior in time series data.

We give a brief introduction to Floer homotopy, from the Seiberg-Witten point of view.  We will then discuss Manolescu's version of finite-dimensional approximation for rational homology spheres.  We prove that a version of finite-dimensional approximation for the Seiberg-Witten equations associates equivariant spectra to a large class of three-manifolds.  In the process we will also associate, to a cobordism of three-manifolds, a map between spectra.  We give some applications to intersection forms of four-manifolds with boundary. This is joint work with Hirofumi Sasahira. 

We study dynamic backward problems, with the computation of conditional expectations as a special objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the Functional Ito calculus to our more general framework. In particular, unlike in the Functional Ito calculus, where one needs only to consider stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths.  We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing and hedging in a financial market with rough volatility is presented.

Joint work with JianFeng Zhang (USC).