All talks happen in Math 520 in the Mathematics building on Columbia campus.  There is also a zoom link:


01/20, 10,30am,  Kyle Hayden, Rutgers University 

Title: Surfaces and 4-manifolds 

Abstract: The topology of smooth manifolds is governed largely by geometry in low dimensions and by algebraic topology in high dimensions. The phase transition occurs in dimension four, where continuous and differential topology split apart and "exotic" phenomena emerges. I will begin by describing how this phase transition can be studied via embedded surfaces in4-manifolds, then I will survey recent developments in this area. In particular, I will explain how quantum invariants (such as Khovanov homology)have recently been used to address existence and uniqueness questions about surfaces in 4-manifolds -- and what this might imply for foundational questions about 4-manifolds themselves.


01/25, 4,30pm,  Eric Ramos, Bowdoin College

Title: The many kinds of uniformity in graph configuration spaces.

Abstract: For a given topological space X, the (unordered) configuration space of n points on X, F_n(X), is the space of n-element subsets of X. Much of the work on these spaces has considered cases where the underlying space X is a manifold of dimension higher than two. For instance, one famous result of McDuff states that if X is the interior of a compact manifold of dimension at least two with boundary, then for any i the isomorphism class of the homology group H_i(C_n(X)) is independent of n whenever n is big enough. Put more succinctly, if X is a “sufficiently nice” manifold of dimension at least 2, then the configuration spaces C_n(X) exhibit homological stability.

In this talk, we will consider configuration spaces in the cases where X is a graph. That is, when X is 1-dimensional. In this setting we will find that the homology groups H_i(C_n(X)) exhibit extremely regular behaviors in two orthogonal ways. The first, similar to the classical setting, is when X is fixed and n is allowed to grow. In this case we will see that rather than stabilizing, the Betti numbers grow as polynomials in n. The second kind of regular behavior is observed when one fixes n and allows X to vary. In this case we will use extremely powerful structural theorems in graph theory to discover features of the homology groups H_i(C_n(X)) that must be common across all graphs X.


01/30 4,30pm, Alisa Knizel, University of Chicago 

Title: The Universality Phenomenon for Log-Gas Ensembles

Abstract: Though exactly solvable systems are very special, their asymptotic properties are believed to be representative for larger families of models. In this way, besides being interesting in their own right, exactly solvable systems are examples of their conjectured universality classes and can be used to build intuition and tools, as well as to make predictions. I will illustrate the phenomenon of universality with the examples from my work on log-gas ensembles.


Faculty Offices

Barnard's tenured mathematics faculty are also Columbia University professors. Their offices are located in the Mathematics Building on Columbia's campus across the street from Barnard's campus on Broadway. Click here to see Columbia's campus map.