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Observance
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Begins for Winter & Spring according to enrollment appointments.

Roman Krutowski, UCLA
Hecke algebras via Morse theory of loop spaces

Higher-dimensional Heegaard Floer homology (HDHF) is defined by extending Lipshitz's cylindrical reformulation of Heegaard Floer homology from surfaces to arbitrary Liouville domains. The HDHF also serves as a model for Lagrangian Floer homology of symmetric products.
In this talk, I will present a Morse-theoretic model allowing for computations of the HDHF A_∞-algebra of k cotangent fibers in the cotangent bundle of a smooth manifold. We apply this model to get an explicit computation of this A_∞-algebra for the cotangent bundle of the 2-dimensional sphere. The result of this computation produces a differential graded algebra which may be regarded as the derived HOMFLYPT skein algebra of the sphere.

Observance
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Speaker:   Antal Jevicki
Title:   Emergent Hilbert Space in Collective Theories
Abstract:   We discuss the emergent,finite N Hilbert space of collective theories. Imposition of trace relations results in drastic reduction to: invariants of primary and secondary kind (a Hironaka decomposition). The former are in accordance with perturbative states ,while the later play a role at higher temperature, their number growing exponentially in powers of N.
Hashtag: #workshop

Mingyang Li, Stony Brook University
TBA

Title:   Discussion
Hashtag:#workshop

Michael Temkin, Hebrew University / IAS
New techniques in resolution of singularities

Since Hironaka's famous resolution of singularities in characteristics zero in 1964, it took about 40 years of intensive work of many mathematicians to simplify the method, describe it using conceptual tools and establish its functoriality. However, one point remained quite mysterious: despite different descriptions of the basic resolution algorithm, it was essentially unique. Was it a necessity or a drawback of the fact that all subsequent methods relied on Hironaka's ideas essentially?

The situation changed in the last decade, when a logarithmic, a weighted and a foliated analogues and generalizations were discovered in works of Abramovich-Temkin-Wlodarzcyk, McQuillan, Quek, Abramovich-Temkin-Wlodarzcyk-Belotto and others. At this stage we can already try to figure out general ideas and principles shared by all these methods and the picture is quite surprising -- it seems that each method is quite determined by its basic setting consisting of the class of geometric objects and basic blowings up one works with. In particular, the classical method is probably the only natural resolution (via principalization) method obtained by blowing up smooth centers in the ambient manifold.

In my talk I'll describe the settings and the methods on a very general level. If time permits, I will add some details about the simplest dream (or weighted) method, which has no memory and improves the singularity invariant by each weighted blowing up. Thus, the algorithm becomes simplest possible and the (modest) price one has to pay consists of extending the setting of varieties (or schemes) and blowings up along smooth centers to the setting of orbifolds and blowings up weighted centers.

Karola Meszaros, Cornell University
Knot polynomials via polytopes

A good way to understand the coefficients of a univariate polynomial with integer coefficients is to lift it to a “nice” multivariate polynomial with 0,1-coefficients. When the terms of the lift correspond to integer points of a magical polytope called a generalized permutahedron, a particularly nice story unfolds. I will illustrate the above by using it to prove a special case of Fox’s trapezoidal conjecture from 1962 that states that the absolute values of the coefficients of the Alexander polynomial of alternating links form a trapezoidal sequence. This talk is based on joint works with Hafner and Vidinas and with K\’alm\’an and Postnikov.

Title:   Discussion
Hashtag: #workshop

Paco Torres de lizaur, Universidad de Sevilla
TBA

Alex Kapiamba, Harvard University
TBA