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
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
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
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
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
Speaker: Filip Milidrag, University of Waterloo
"The Classification of Irreducible Discrete Reflection Groups"
In this talk we will make a correspondence between irreducible discrete reflection groups and associated connected Coxeter diagrams. Then we will use this to classify all connected Coxeter diagrams and by extension every irreducible discrete reflection group.
MC 5501
Utkarsh Bajaj
Klein's icosahedral function
Can we define a rational function on the sphere? Sure we can. Can we define a rational function on the sphere so that it is invariant under the rotational symmetries under the icosahedron? Yes - by embedding the icosahedron in the Riemann sphere (and then doing some algebra). We then show how this beautiful function reveals connections between the symmetries of the icosahedron and the E8 lattice - the lattice that gives the most efficient packing of spheres in 8 dimensions!
MC 5501
Keira Gunn (University of Calgary)
Some Problems on the Dynamics of Positive Characteristic Tori.
The real (or characteristic zero) torus is simply R/Z, or the "decimal part" of any real number with operations of addition and integer multiplication. With the positive characteristic integers defined to be polynomials with coefficients from a finite field, and the positive characteristic reals their Laurent series counterparts, we can similarly construct the positive characteristic tori (each torus dependent on the choice of field). At first glance there are many similarities to how operations work in both positive and zero characteristic, but these similarities break down quickly upon further inspection, particularly from a view of dynamics on the tori.
In this talk, we will discuss results on some orbital sets and dynamics formulae on the positive characteristic tori, including the Artin-Mazur zeta function and analogous hypothesis for Furstenberg's Orbital Theorem.
MC5403