The Mathematical and Computational Sciences at Western are represented by four separate departments: Applied Mathematics, Computer Science, Mathematics, and Statistical & Actuarial Sciences. The Department of Mathematics has established research groups in several areas of contemporary mathematics including algebra, analysis and analytic geometry, homotopy theory, and noncommutative geometry.
Each week the Department of Mathematics hosts a wide variety of seminars and events. For a comprehensive list of events, please consult our departmental calendar.
In this work we consider the Galois point of view in determining the structure of a space of orderings of fields via considering small Galois quotients of absolute Galois
groups G_F of Pythagorean formally real fields. Galois theoretic, group theoretic and combinatorial arguments are used to reduce the structure of W-groups.
It is classical that the topological K-theory KU(X)
of a space X agrees with maps from X to KU, and that
cohomology operations correspond to maps from KU to itself.
Dual to this is the structure of "co-operations", i.e.,
the KU-homology of KU relative to the ring KU(point).
This data has a structure, dubbed Hopf algebroid, which is
related to combinatorics and numerical polynomials.
In joint work with Pelaez, we determine the analogous structure
for algebraic K-theory KGL, regarded as a motivic object.
Applying the motivic slice filtration, we solve a problem of Voevodsky.
Although large social and information networks are often thought of as
having hierarchical or tree-like structure, this assumption is rarely
tested. We have performed a detailed empirical analysis of the
tree-like properties of realistic informatics graphs using two very
different notions of tree-likeness: Gromov's δ-hyperbolicity, which is
a notion from geometric group theory that measures how tree-like a
graph is in terms of its metric structure; and tree decompositions,
tools from structural graph theory which measure how tree-like a graph
is in terms of its cut structure. Although realistic informatics
graphs often do not have meaningful tree-like structure when viewed
with respect to the simplest and most popular metrics, e.g., the value
of δ or the treewidth, we conclude that many such graphs do have
meaningful tree-like structure when viewed with respect to more
refined metrics, e.g., a size-resolved notion of δ or a closer
analysis of the tree decompositions. We also show that, although these
two rigorous notions of tree-likeness capture very different tree-
like structures in the worst-case, for realistic informatics graphs they
empirically identify surprisingly similar structure. We interpret this
tree-like structure in terms of the recently-characterized "nested
core-periphery" property of large informatics graphs; and we show that
the fast and scalable k-core heuristic can be used to identify this
I will introduce some fundamental concepts of lattice theory (unimodular
lattices, lattice gluing) and explain why we expect them to naturally
appear on a homological algebra level. We will discuss the example of the
cut and flow lattices of a graph, and a categorical realisation which
serves as an example for lifting lattice theoretic concepts as mentioned
above. We end with a number of open questions and directions.
Joint work with Anthony Licata.
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