We generalize an inequality of E. Heintze and H. Karcher for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for k-Ricci curvature. Even in the case of a pointwise bound this generalizes the classical inequality by replacing a sectional curvature bound with a k-Ricci bound. This work is motivated by the estimates of Petersen-Shteingold-Wei for the volume of tubes around a geodesic and generalizes their estimate. Using similar ideas we also prove a Hessian comparison theorem for k-Ricci curvature which generalizes the usual Hessian and Laplacian comparison for distance functions from a point and give several applications.

In first order logic, the Baldwin-Lachlan characterization of $\aleph_1$-categorical
theories implies that the notion is absolute between transitive models of set theory.  
Here, we seek a similar characterization for having a unique atomic model of size $\aleph_1$.
At present, we have several conditions that imply many non-isomorphic atomic models of size $\aleph_1$.
Curiously, even though the results are in ZFC, their proofs rely on forcing.
This is joint work with John Baldwin and Saharon Shelah.

This talk is aimed (mostly) at undergraduate students. 

Abstract: In the 1930’s and 1940’s, Murray and von Neumann developed a theory of operators on Hilbert spaces, which heuristically may be thought of as infinite matrices acting on infinite dimensional vector spaces. Their works include a procedure which starts with an infinite group, a discrete object, and generates a von Neumann algebra, an analytic object which is a continuous analog to the n × n matrices. Much of the active research in this field has been generated by the following question: are structural properties of groups able to classify the resulting algebras? Obtaining a satisfactory resolution to this problem has been surprisingly difficult since standard group invariants are often not invariants of the algebras. We give a brief survey of the evolution of this problem, the surprising broader impacts including the emergence Jones polynomial, and the recent rapid progress in this classification program due to the advent of S. Popas deformation/rigidity theory. We close by describing recent developments in this program which have been made by my collaborators and myself. 

About the speaker: Rolando de Santiago is currently an Assistant Adjunct Professor and a UC Presidential Postdoctoral Fellow at UCLA working under S. Popa. His work is in the classification of type II1 von Neumann algebras, a subfield of functional analysis, and his research interests extend into group theory, topology, fractal geometry, and mathematical physics. He was born and raised up in the South-Eastern part of Los Angeles with 6 of his siblings. He spent 27 years studying at numerous public institutions including Pasadena City College and Cal Poly Pomona. After approximately 8 years of undergraduate work, he finally earned his B.S. in Mathematics. He completed his Masters in Mathematics at Cal Poly Pomona shortly thereafter. His mentors at Cal Poly, J. Rock and R. Wilson, strongly suggested that he pursue his Ph.D. After a significant amount of convincing, he threw all his belongings into a U-Haul, moved to Iowa City, and started grad school at the University of Iowa. He worked under of I. Chifan, the advisor who would help his launch his research career.

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Earthquakes produce seismic waves. They provide a way to obtain information about the deep structures of our planet. The typical measurement is to record the travel time difference of the seismic waves produced by an earthquake. If the network of seismometers is dense enough and they measure a large number of earthquakes, we can hope to recover the wave speed of the seismic wave from the travel time differences. In this talk we will consider geometric inverse problems related to different data sets produced by seismic waves. We will state uniqueness results for these problems and consider the mathematical tools needed for the proofs. The talk is based on joint works with: Maarten de Hoop, Joonas Ilmavirta, Matti Lassas and Hanming Zhou.