# 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
9-9:50
10:00-10:30
Coffee Break
Coffee Break
Coffee Break
Coffee Break
10:30-11:20
11:40-12:30
LUNCH
Lunch
Lunch
Lunch
Lunch
15:00-15:50
16:10-17:00

# 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 ways---they 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 2-dimensional (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 nilpotent-free. A k-algebra is an algebra A over the complex numbers which is a k-module (with an evident compatibility between the algebra structure of A and the k-module structure of A). A is not required to have a unit and is not assumed to be commutative.
A k-algebra A is of finite type if as a k-module A is finitely generated. This talk will review Morita equivalence for k-algebras and will then introduce --- for finite type k-algebras --- 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 A-modules) 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 (Aubert-Baum-Plymen-Solleveld) conjecture applies the new equivalence relation within the setting of representation theory of reductive p-adic groups and the local Langlands conjecture.
• Jonathan Block:
Taking the D out of D-modules
When considering D-modules 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 2-disk. Choose k distinct points, say \{p_1, \cdots, p_k\} in the interior of S. A k-tangle 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 3-disk, 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-/F-Theory Compactfication
Discrete gauge groups naturally arise in F-theory compactifications on genus-one fibered Calabi-Yau manifolds. Such geometries appear in families that are parameterized by the Tate-Shafarevich group of the genus-one fibration. While the F-theory compactification on any element of this family gives rise to the same physics, the corresponding M-theory 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 Tate-Shafarevich group of the general cubic. We discuss how the different M-theory vacua and the associated discrete gauge groups can be obtained by Higgsing of a pair of five-dimensional U(1) symmetries. The Higgs fields arise from vanishing cycles in I2-fibers 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 n-fold covering space. In joint work in progress with Andy Neitzke we use a 3-dimensional spectral network to achieve a branched diagonalization of a rank 2 vector bundle over a 3-manifold, and use it to compute its complex Chern-Simons 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 (F-theory) and how his special realizations of SU(3) and SU(2) provide an alternative that in F-theory 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 four-manifolds
Given two simply-connected closed oriented topological four-manifolds 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 simply-connected oriented closed four-manifold 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 Landau-Ginzburg models associated with weighted homogeneous polynomials, elliptic genus of Calabi-Yau 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 Pfaffian-Grassmannian double mirrors which conjecturally are also related by a different type of phase transition (Hori-Tong 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 non-minimal 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 L2-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.