Speaker: Nicolas Chavarria
"T-minimal Theories and Dimension II"
We continue looking at Will Johnson's work on the dimension of definable sets in t-minimal theories.
MC 5403
Speaker: Paul Marriott
"Statistics and Geometry: We don't talk any more."
George Bernard Shaw once said Britain and America are two counties separated by a common language. Perhaps the same can be said for Statistics and Geometry. This talk gives a high-level overview of a recent graduate course which explored the relationship between Statistics and Geometry. It looks at what the disciplines have in common but also where there are points of substantive difference. The talk will review the long history of geometric tools finding a place in statistical practice and will highlight modern developments using ideas from convex, differential and algebraic geometry and showing applications in Neuroscience.
MC 5417
Speaker: Cynthia Dai
"Global Heights"
In this talk, we will wrap up local heights with respect to a presentation of a line bundle, then define global heights. If time permits, we will also talk about Weil heights.
MC 5417
Speaker: Manuel Fernandez, Georgia Tech
"On the $\ell_0$-Isoperimetry of Measurable Sets"
Gibbs-sampling, also known as coordinate hit-and-run (CHAR), is a random walk used to sample points uniformly from convex bodies. Its transition rule is simple: Given the current point p, pick a random coordinate i and resample the i'th coordinate of p according to the distribution induced by fixing all other coordinates. Despite its use in practice, strong theoretical guarantees regarding the mixing time of CHAR for sampling from convex bodies were only recently shown in works of Laddha and Vempala, Narayanan and Srivastava, and Narayanam, Rajaraman and Srivastava. In the work of Laddha and Vempala, as part of their proof strategy, the authors introduced the notion of the $\ell_0$ isoperimetric coefficient of a measurable set and provided a lower bound for the quantity in the case of axis-aligned cubes. In this talk we will present some new results regarding the $\ell_0$ isoperimetric coefficient of measurable sets. In particular we pin down the exact order of magnitude of the $\ell_0$ isoperimetric coefficient of axis-aligned cubes and present a general upper bound of the $\ell_0$ isoperimetric coefficient for any measurable set. As an application, we will mention how the results give a moderate improvement in the mixing time of CHAR.
MC 5501
Cynthia Dai
Global Heights
In this talk, we will wrap up local heights with respect to a presentation of a line bundle, then define global heights. If time permits, we will also talk about Weil heights.
MC 5417
Speaker: Gauree Wathodkar, University of Mississippi
"Partition regularity in commutative rings."
Let A ∈ Mm×n(Z) be a matrix with integer coefficients. The system of equations A⃗x = ⃗0 is said to be partition regular over Z if for every finite partition Z \ {0} = ∪ri =1Ci, there exists a solution ⃗x ∈ Zn, all of whose components belonging to the same Ci. For example, the equation x + y − z = 0 is partition regular. In 1933 Rado characterized completely all partition regular matrices. He also conjectured that for any partition Z \ {0} = ∪ri =1Ci, there exists a partition class Ci that contains solutions to all partition regular systems. This conjecture was settled in 1975 by Deuber. We study the analogue of Rado’s conjecture in commutative rings, and prove that the same conclusion holds true in any integral domain.
MC5403
Gauree Wathodkar (University of Mississippi)
Partition regularity in commutative rings.
Let A ∈ Mm×n(Z) be a matrix with integer coefficients. The system of equations A⃗x = ⃗0 is said to be partition regular over Z if for every finite partition Z \ {0} = ∪ri =1Ci, there exists a solution ⃗x ∈ Zn, all of whose components belonging to the same Ci. For example, the equation x + y − z = 0 is partition regular. In 1933 Rado characterized completely all partition regular matrices. He also conjectured that for any partition Z \ {0} = ∪ri =1Ci, there exists a partition class Ci that contains solutions to all partition regular systems. This conjecture was settled in 1975 by Deuber. We study the analogue of Rado’s conjecture in commutative rings, and prove that the same conclusion holds true in any integral domain.
MC5403
Speaker: Aareyan Manzoor
"Metrics on Polish groups"
We investigate various compatible metrics that a Polish group admits. By Birkhoff-Kakutani, every Polish group admits a compatible left-invariant metric, and by being a Polish space, there is also a compatible metric which is complete. We discuss the relationship between these metrics, and introduce the class of CLI Polish groups, those Polish groups which admit a single compatible metric which is both left invariant and complete. We will mostly follow Section 2.2 of Gao's IDST book.
MC5403
Aareyan Manzoor
Metrics on Polish groups
We investigate various compatible metrics that a Polish group admits. By Birkhoff-Kakutani, every Polish group admits a compatible left-invariant metric, and by being a Polish space, there is also a compatible metric which is complete. We discuss the relationship between these metrics, and introduce the class of CLI Polish groups, those Polish groups which admit a single compatible metric which is both left invariant and complete. We will mostly follow Section 2.2 of Gao's IDST book.
MC5403
Speaker: William Gollinger
"The Adams Spectral Sequence"
After having a week off, in this talk we will review concepts in homotopy theory. This will include: fibrations and cofibrations; CW complexes; Eilenberg-MacLane spaces; homotopy fibers and related constructions; the Postnikov tower of a space.
MC5417