Stratified spaces in geometric and computational topology and physics
Schedule of Talks
All talks will be in Van Vleck Hall, Room B239.
Coffee breaks and registration will be in the 9th floor lounge, Room 911, in Van Vleck Hall.
Please register on the morning of March 30, before the first talk of the conference.

March 30 
March 31 
April 1 
April 2 
99:50 




10:0010:30 
Coffee Break 
Coffee Break 
Coffee Break 
Coffee Break 
10:3011:20 




11:4012:30 




LUNCH 
Lunch 
Lunch 
Lunch 
Lunch 
15:0015:50 




16:1017:00 




Titles and Abstracts
 Paolo Aluffi:
Degrees of projections of determinantal varieties
We provide formulas for the degrees of the projections of the locus of square matrices with given
rank from linear spaces spanned by a choice of matrix entries. The motivation for these computations
stem from geometric complexity, specifically applications to 'matrix rigidity'. Also, these degrees
are combinatorially significant in somewhat unexpected waysthey appear to match the numbers of
Kekulé structures of certain benzenoid hydrocarbons, and arise in many other contexts with no apparent direct connection to the enumerative geometry of rank conditions.
 Javier Arsuaga:
Identification of Copy Number Aberrations in Breast Cancer
Subtypes using Persistence Topology
DNA copy number aberrations (CNAs) are of biological and medical interest
because they help identify regulatory mechanisms underlying tumor initiation and evolution.
Identification of tumor driving CNAs (driver CNAs) however remains a challenging task
because they are frequently hidden by CNAs that are the product of random events that take
place during tumor evolution. Experimental detection of CNAs is commonly accomplished
through array Comparative Genomic Hybridization (aCGH) assays followed by supervised
and/or unsupervised statistical methods that combine the segmented profiles of all patients
to identify driver CNAs. Here we extend a previously presented supervised algorithm for
the identification of CNAs that is based on a topological representation of the data. Our
method associates a 2dimensional (2D) point cloud to each CGH profile and generates a
sequence of simplicial complexes, mathematical objects that generalize the concept of a
graph, that segments the data at different resolutions. Identification of CNAs is achieved
by interrogating the topological properties of these simplicial complexes. We tested our
approach on a published data set with the goal of identifying breast cancer CNAs associated
with specific molecular subtypes. Our results confirmed all regions found in the original
publication except for 17q in LuminalB and detected 30 additional regions. Most of the
additional regions have been reported in other independent studies. Two not previously
reported CNAs, gain of 1p, gain of 2p were found in the basal subtype and validated on a
second published data set. We therefore suggest that topological approaches that incorporate
multiresolution analyses and that interrogate topological properties of the data can help in
the identification of copy number changes in cancer.
 Paul Baum:
Morita Equivalence Revisited
Let X be a complex affine variety and k its coordinate algebra.
Equivalently, k is a unital algebra over the complex numbers which
is commutative, finitely generated, and nilpotentfree.
A kalgebra is an algebra A over the complex numbers which is a kmodule
(with an evident compatibility between the algebra structure of A and the kmodule structure of A).
A is not required to have a unit and is not assumed to be commutative.
A kalgebra A is of finite type if as a kmodule A is finitely generated.
This talk will review Morita equivalence for kalgebras and will
then introduce  for finite type kalgebras  a weakening of Morita equivalence called geometric equivalence.
The new equivalence relation preserves the primitive ideal space (i.e. the set of equivalence classes of irreducible Amodules)
and the periodic cyclic homology of A. However, the new equivalence relation permits a tearing apart of
strata in the primitive ideal space which is not allowed by Morita equivalence.
The ABPS (AubertBaumPlymenSolleveld) conjecture applies the new equivalence relation within
the setting of representation theory of reductive padic groups and the local Langlands conjecture.
 Jonathan Block:
Taking the D out of Dmodules
When considering Dmodules on stacks it has been easier to use a Koszul dual approach to
them using modules over the de Rham algebra. We develop
this idea further and apply it to yet more general contexts.
 Sylvain Cappell:
The Unitary Representation Extension Theorem for Tangles
Let S be a compact oriented surface with boundary, often a 2disk. Choose
k distinct points, say
\{p_1, \cdots, p_k\} in the interior of S.
A ktangle in S \times [0, 1] is a proper embedding, say F,
of the union of k disjoint intervals, say \cup_{j=1}^k [0, 1]_j,
into S \times [0, 1]
such that the embedding F sends the beginning of the intervals, \cup_{j=1}^k \{0\}_j,
bijectively to the
k points \{p_j\} \times 0, and the ends of the intervals,
\cup_{j=1}^k \{1\}_j,
bijectively to the k points \{p_j\} \times 1.
If S = D^2, so S \times [0, 1] is a 3disk, the tangle is called a disk tangle.
For convenience, let F(0_j ) = p_j \times 0 and F(1_j ) = p_{\sigma(j)} \times 1
where \sigma is a permutation of \{1, \cdots , k\}, and let
\gamma_j \in \pi_1(S \times 0 \setminus \{p_1, \cdots , p_k\} \times 0),
\gamma'_j \in \pi_1(S \times 1 \setminus \{p_1, \cdots , p_k\} \times 1) be meridional generators
which arise by starting at the base point,
tracing out to a neighborhood of p_j \times 0, p_j \times 1 in S \times 0, S \times 1,
respectively, then tracing around this point and returning by the
reversed path to the base point.
Theorem: Every unitary representation \Phi : \pi_1((S \setminus \{p_1, \cdots , p_k\}) \times 0) \to U(n)
extends to some unitary representation
\Phi' : \pi_1((S \times [0, 1]) \setminus F(\cup_{j=1}^k [0, 1]_j )) \to U(n).
This is joint work with Edward Y. Miller.
 Mirjam Cvetic:
Discrete Gauge Symmetries and Multiple Phases of M/FTheory Compactfication
Discrete gauge groups naturally arise in Ftheory compactifications on genusone fibered CalabiYau manifolds.
Such geometries appear in families that are parameterized by the TateShafarevich group of the genusone fibration.
While the Ftheory compactification on any element of this family gives rise to the same physics, the corresponding
Mtheory compactifications on these geometries differ and are obtained by a fluxed circle reduction of the former.
We focus on an element of order three in the TateShafarevich group of the general cubic. We discuss how the different
Mtheory vacua and the associated discrete gauge groups can be obtained by Higgsing of a pair of fivedimensional U(1)
symmetries. The Higgs fields arise from vanishing cycles in I2fibers that appear at certain codimension two loci
in the base. We explicitly identify all three curves that give rise to the corresponding Higgs fields.
 Dan Freed:
Diagonalization with stratification
A geometric version of diagonalization expresses a rank n vector
bundle as the direct image of a line bundle on an nfold covering
space. In joint work in progress with Andy Neitzke we use a
3dimensional spectral network to achieve a branched
diagonalization of a rank 2 vector bundle over a 3manifold, and
use it to compute its complex ChernSimons invariant in terms of
dilogarithms.
 Antonella Grassi:
Algebraic geometry, topology, strings
Algebraic geometry, topology and strings manifest the structure of Lie algebra of
the Dynkin diagrams associated to some singularities, and more.
Based on joint work with Jim Halverson and Julius Shaneson.
 James Halverson:
Kodaira's Alternative to Grand Unification
In Kodaira's 1960's classification of singular fibers in elliptic surfaces,
he found an infinite class of singular fibers related to the Lie algebra SU(N),
and two other special fibers related only to SU(3) and SU(2). About ten years later,
physicists discovered that particle interactions are governed at low energies by
SU(3) × SU(2) × U(1), and many theoretical attempts have been made since then to
embed the particle interactions into higher rank groups such as SU(5) or SO(10),
an idea known as grand unification. In this talk I will explain how Kodaira's
mathematics finds a natural home in string theory (Ftheory) and how his special
realizations of SU(3) and SU(2) provide an alternative that in Ftheory seems
more natural than grand unification, as it can solve a serious moduli problem.
These special realizations of SU(3) and SU(2) also lead naturally toward the
spectrum of particles observed in nature.
This talk is based on recent work with Julius Shaneson, in collaboration also with Antonella Grassi and Wati Taylor.
 Alexandra Kjuchukova:
On the classification of branched covers of fourmanifolds
Given two simplyconnected closed oriented topological fourmanifolds X and Y, we ask:
can Y be realized as a branched cover of X? For B a surface embedded in X with an isolated singularity,
we will prove a necessary condition for the existence of an irregular dihedral branched covering map
f:Y → X with branching set B.
Conversely, given a simplyconnected oriented closed fourmanifold X, we will outline a construction realizing
as irregular dihedral covers of X infinitely
many of the manifolds Y afforded by the necessary condition.
 Connie Leidy:
Searching for structure in the knot concordance group
The set of knots up to concordance has been studied for over fifty years.
Since the late 1960's, much of the focus has been on the structure of this as an abelian group.
Recent work has shown that this perspective may be too limited.
We will describe some alternative structures on this set. We will present some joint work with Cochran and Harvey
that provides evidence that the set of knot concordance classes has a "fractal" structure.
We will also consider the structure of the set of knot concordance classes as an algebra over the "generalized doubling operad".
 Anatoly Libgober:
Elliptic genus of phases of N=2 theories
I will describe a construction from my recent preprint with
the same title which assigns elliptic genus
to Witten's phases of N=2 theories which include
the elliptic genus of LandauGinzburg models
associated with weighted homogeneous polynomials,
elliptic genus of CalabiYau manifolds and elliptic genus of
hybrid models. In particular, using this construction
one can derive LG/CY correspondence
for elliptic genus from McKay correspondence.
In the second part of the talk I discuss a recent
joint paper with L. Borisov on
PfaffianGrassmannian double mirrors which
conjecturally are also related by a different type of
phase transition (HoriTong type glop).
 David Morrison:
Singularities in elliptic fibrations and minimality of Weierstrass models
In complex algebraic geometry, elliptic fibrations with section are known to have Weierstrass models,
which are birational models of the total space with an equation of a particular "Weierstrass" form.
However, Weierstrass models are not unique: different models are related by birational transformations
of the total space. Under suitable conditions, there is a "minimal" Weierstrass model for the original fibration.
We will discuss the types of singular fibers which can occur in nonminimal Weierstrass models,
and criteria for recognizing the properties of the associated minimal Weierstrass model.
The monodromy representation of the elliptic fibration plays an important role in our discussion.
 Monica Nicolau:
Tackling the topology and geometry underlying big data
The recent onslaught of data has brought about profound changes in understanding a range of phenomena
as dynamic, high complexity processes. New technology has provided an unprecedented wealth of information,
but it has generated data that are hard to analyze mathematically, thereby making an interpretation difficult.
These challenges have given rise to a myriad novel exciting mathematical problems and have provided an impetus
to modify and adapt traditional mathematics tools, as well as develop novel techniques to tackle the data analysis problems.
I will discusses a general approach to address some of these computational challenges by way of a combination
of geometric data transformations and topological methods. In essence geometric transformations deform the data
to focus intensity on relevant questions, and topological methods identify statistically significant shape
characteristics of the data. These methods have been applied in a range of settings, in particular for the
study of the biology of disease. I will discuss some concrete applications of these methods, including their
use to discover a new type of breast cancer, identify disease progression trends, and highlight the driving
mechanisms in acute myeloid leukemia. In keeping with the focus of this conference, in will conclude with
recent work adapting the setting of stratified spaces to the study of large data.
While the specifics of the work are focused on biological data analysis, the general approach addresses
computational challenges in the analysis of any type of large data.
 Kent Orr:
A transfinite filtration of compact orientable three manifolds
In some of his earliest work, John Milnor filtered links by comparing the lower central series
quotients of link complement groups. Over a decade ago, Cochran, Orr, and Teichner filtered knots
using derived subgroups and associated Whitney tower constructions to resolve some old conjectures
on knots. One new tool they developed, higher order Alexander modules of knots, helped inspire
the later work of Leidy and Maxim in their investigation of plane curve complements and associated
L^{2}Betti numbers. Extending these tools by applying transfinite filtrations remains an open and
potentially promising direction for further investigation. With Jae Choon Cha, we explore this
idea in the context of Milnor's original work, and extend Milnor's work to the transfinite lower
central series of a closed orientable three manifolds.
 Raul Rabadan:
The Topology of Evolution
Phylogenies are popularly used to represent evolutionary relationships between organisms, species or other taxa.
However, phylogenetic representations can be misleading when applied to genomic data, particularly during reticulate
evolution mediated by nonvertical exchange of genetic material between different organisms. Such events can lead
to different phylogenetic histories for each gene and even different sections within a single gene. In this talk
I present a mathematical structure able to capture and represent largescale properties of evolution.
Persistent homology aims to extract global topological features from sequence data by reconstructing simplicial complexes,
which at a particular scale of genetic distance represents the relation between different genomes.
We show that there exist topological obstructions to the use of phylogeny for certain genomic datasets.
In particular, we identify a set of topological equalities that, if unsatisfied, invalidates phylogenetic representations.
To illustrate how persistent homology can be used to infer global evolutionary properties, we have selected a set of
RNA viruses with distinct modes of exchanging genomic material: clonal evolution, reassortment and recombination.
Beyond detecting reticulate evolution, we succinctly recapitulate the history of complex genetic exchanges involving
more than two parental strains, such as the triple reassortment of H7N9 avian influenza and the formation of circulating HIV1 recombinants.
In addition, we identify recurrent, largescale patterns of reticulate evolution, including frequent PB2PB1PANP cosegregation
during avian influenza reassortment. Finally, we applied topological approaches to characterize recombinations in humans.
 Jörg Schürmann:
Singular Todd classes of tautological sheaves on Hilbert schemes
of points on a smooth surface
Let X be a quasiprojective, smooth complex algebraic surface, with
X^{[n]}
the Hilbert scheme of n points on X, so that the (rational)
cohomology of all these Hilbert schemes together can be generated by the
cohomology of X in terms of Nakajima creation operators. Given an algebraic
vector bundle V on X, there exist universal formulae for the
characteristic classes of the associated tautological vector bundles
V^{[n]}
on X^{[n]}
in terms of the Nakajima creation operators and the
corresponding characteristic classes of V. But in general the corresponding
coefficients are not known.
Based on the derived equivalence of BridgelandKingReid and work of Haiman
and Scala, we give an explicit formula in case of the singular Todd classes,
but in terms of Nakajima creation operators of the
delocalized equivariant cohomology of all X^{n} with its natural S_{n}action.
 Lisa Traynor:
Lagrangian Cobordisms between Legendrian Submanifolds
Smooth cobordisms play an important role in topology. Lagrangian cobordisms between Legendrian submanifolds are smooth
cobordisms that satisfy additional geometric conditions imposed by symplectic and contact structures. From a qualitative perspective,
Lagrangian cobordisms have more topological rigidity than smooth cobordisms: for example, any Lagrangian cobordism between a Legendrian
unknot and a Legendrian trefoil must have genus equal to 1. From a quantitative perspective, Lagrangian cobordisms are at times very
flexible while at other times have some rigidity: between some pairs of Legendrian submanifolds it is possible to find arbitrarily short
Lagrangian cobordisms, while between other pairs of Legendrians there is a positive lower bound to the length of a Lagrangian cobordism.
This is joint work with Joshua Sabloff.