Brooke Shipley - US grants

9971953
Wilkerson
Wilkerson will continue his work with Dwyer on the extended Steenrod
problem of understanding spaces with properties at a prime similar to
those of the classifying spaces of connected compact Lie groups. He will
also work with Jeff Smith on homotopy automorphism groups of finite
nilpotent complexes and actions of finite groups on such complexes. In
addition, he will work with Jim Turner and Avramov on analogues of the
Serre theorem that establishes that non-trivial simply connected finite
complexes have nontrivial homotopy in an infinite number of dimensions.
McClure will work with Jeff Smith on higher centers of associative ring
spectra. He will also investigate a large class of spectral sequences
that should admit chain-complex models that facilitate calculation. A
step in this project that should be interesting in its own right is to
show that certain homotopy categories of module spectra can be modeled by
chain complexes. Shipley will develop a new model for equivariant stable
homotopy theory; this will provide algebraic models for rational
equivariant stable homotopy theory over compact Lie groups. She will
also study (with Dwyer) a cobar spectral sequence and its convergence
properties. With Rezk and Schwede she will study a way to replace
arbitrary model categories by simplicial model categories. Smith will
continue his work on using the theory of model categories as a tool
for the study of ring spectra and of commutative ring spectra. In
particular, he will study the homotopy theory of ring spectra over a
fixed ring R, the homotopy theory of coalgebras of R-bimodules, and the
bar construction. He will improve his theory of combinatorial model
categories to make it a more effective tool. In joint work with Adem he
will study the homotopy theory of group actions on homotopy products of
spheres.
As can be seen from this summary, this project will involve research
in many different directions within homotopy theory. Several of the
problems to be investigated have been important for a long time (for
example, the Steenrod problem and the joint work of Smith and Adem go
back to the 1960's), while others involve very recent developments
(especially the theory of strictly associative and commutative ring
spectra; Smith and Shipley were pioneers in this area, and McClure's
work makes heavy use of it). The joint work of McClure and Smith has
applications to mathematical physics, since it is closely related to
recent work of Kontsevich and Tamarkin on deformation quantization.
***

DMS-0134938
Brooke E. Shipley

The research component of this proposal involves several questions which fall into two overall projects. One project is to consider
when an equivalence between derived categories is induced by a
more richly structured equivalence. The goal here is to develop
an obstruction theory for realizing derived equivalences by
underlying structured equivalences. In particular, this may have applications to the question associated to Broue's Conjecture in
representation theory of which stable equivalences lift to derived equivalences. The second project is to develop homotopy theoretic algebraic models. One piece of this project is to continue work
with John Greenlees on algebraic models for rational equivariant cohomology theories. More generally, the investigator proposes
to develop algebraic models for various stable homotopy theories.

The proposed research projects involve the interplay beween the
study of algebraic structures and topology, the study of shapes
or spaces. Algebraic topologists use algebraic structures to
describe and simplify topological phenomena. In a project on
realizing derived equivalences, the investigator hopes to use techniques developed in algebraic topology to attack questions
which originate in algebra. These techniques include the use
of a theory of obstructions, which determines whether certain constructions are possible. In another project the
investigator plans on extending existing algebraic models to
include structures involving symmetries. The educational
component of this proposal also includes several parts.
The investigator will be involved in the summer research
programs for undergraduates at the University of Chicago
and Purdue University. With Lucho Avramov, she will organize
a workshop on topics related to derived categories of interest
to a broad range of algebraically related fields. With
Jim McClure and Guershon Harel, the investigator will develop
a new course for mathematics education majors based on Euler's ``Elements of Algebra". The goals of this course are to help mathematical education students master skills in algebra,
develop confidence for teaching algebra, and understand the
motivation for material contained in a standard abstract
algebra course. The investigator will also participate in several aspects of the Women in Science Program at Purdue University.

The goal of the Women in Science and Engineering System Transformation (WISEST)
project, at the University of Illinois at Chicago (UIC), is to increase the number, participation,
and leadership status of women, majority and minority, in eleven science and engineering
(STEM) departments through institutional transformation. WISEST will use an innovative
approach of a network of faculty facilitators from all STEM departments working with
department heads and an executive committee of key administrators and a social scientist.
This network will carry out five integrated and mutually reinforcing strategies: warm the
climate and decrease the isolation of women STEM faculty; recruit minority women faculty
through an unique mentored postdoctoral program; transform STEM departments to foster
diversity and womens leadership; promote womens scholarship and teaching; and improve
the ability to track and report on gender equity. Proposed outcomes for STEM women
faculty include: increased numbers of majority and minority faculty; improved retention
rate; salary equity with men of similar accomplishments and productivity; increased
percentage of leadership positions; improved job satisfaction; and increased national visibility
for our scholars. The intellectual merit of WISEST is that it will assess the impact of
systemic change to erase gender stereotyping rather than individual remediation and it will
specifically extend the focus of action to include the postdoctoral level to recruit faculty. Its
broader impact will be the creation of a life-friendly work climate for all UIC faculty.
WISEST will share its experiences nationally, and serve as an exemplary model for fostering
gender equity and diversity in academe.

Abstract

Award: DMS-0706877
Principal Investigator: Brooke E. Shipley

The PI proposes to study algebraic derived equivalences using
methods from homotopy theory. Specifically, the main project
proposed here is to consider when an equivalence between derived
categories can be realized by an underlying richly structured
equivalence. The PI and Dugger have shown that certain derived
equivalences of differential graded algebras (DGAs) can only be
realized by "topological equivalences" (involving tilting
spectra) which arise by considering DGAs in the broader context
of ring spectra. One goal of this project is to produce simple
invariants for detecting topological equivalences of DGAs.
Another goal is to extend the study of topological equivalences
to DG-categories. In on-going joint work with Dugger, the PI is
also investigating an exotic example of two derived equivalent
DGAs with no underlying structured equivalence.

The proposed research projects involve the interplay beween the
study of algebraic structures and topology, the study of shapes
or spaces. Algebraic topologists use algebraic structures to
describe and simplify topological phenomena. In a project on
realizing derived equivalences, the investigator hopes to use
techniques developed in algebraic topology to attack questions
which originate in algebra. The PI also plans to organize
conferences which will help to train graduate students as well as
disseminate results. In addition, the PI will continue to be
involved with several organizations which promote the
participation of women in science.

This project will support travel to the conference "Homotopical Group Theory and Topological Algebraic Geometry,"
at the University of Copenhagen and at the Max Planck Institut in Bonn in June 2008. Homotopical Group Theory and Topological Algebraic Geometry involve the interaction at the highest level of formerly distant areas of mathematics. They are bound together by a common toolbox from modern homotopy theory. Homotopical Group Theory has shed light on fundamental issues such as the structure of classifying spaces and group actions, and the nature and classification of compact Lie groups. It has very recently shown the potential of shedding new light on the classification of finite simple groups. Topological Algebraic Geometry brings together ideas from algebraic geometry and homotopy theory. It has contributed to significant advances in both
areas, including elliptic cohomology and the relationship between the chromatic stable homotopy theory and the Langlands program.

Homotopical Group Theory and Topological Algebraic Geometry are new areas of mathematics, and they have recently enjoyed explosive development by researchers around the world. The conference supported by this proposal will begin at the University of Copenhagen, focused on two series of lectures by Paul Goerss of Northwestern University and Bill Dwyer of the University of Notre Dame and aimed at beginning researchers. The following week at the Max Planck Institut, leading researchers will report on the latest developments. The idea is to bring together current leaders and new researchers, and to enable interaction and cross-fertilization of ideas among a broad group of mathematicians.

The PI continues to study when derived equivalences can be realized by underlying richly structured equivalences. Examples of this arise in the PI's long term project with John Greenlees of constructing algebraic models for rational G-equivariant spectra for compact Lie groups G. The PI and Greenlees plan to extend their model for connected groups in the free case to non-connected groups. The PI and Greenlees also plan to extend their model for tori to provide an algebraic model for rational G-equivariant commutative and associative ring spectra. As a step towards the general case, the specific case of SO(3) is also being considered. The PI and Kathryn Hess propose to develop model categories for coalgebras over various comonads in various settings. The main motivation here is to provide a homotopical setting for studying Hopf-Galois extensions of ring spectra.

The proposed research projects involve the interplay between the study of algebraic structures and topology, the study of shapes or spaces. Algebraic topologists use algebraic structures to describe and simplify topological phenomena. Spectra, which represent cohomology theories, are algebraic structures built out of topological spaces and hence are useful for translating from one field to the other. In one project, the PI and Greenlees develop algebraic models for certain types of spectra which allow complete calculations. The PI also continues to train graduate students and disseminate research results. In addition, the PI is involved with several organizations which promote the participation of women and underrepresented minorities in science.

This proposal from the Association for Women in Mathematics (AWM) includes two workshops for early-career women, the first at the Joint Mathematics Meeting (JMM) in San Antonio, January 10-13, 2015, and the second at the SIAM Conference on Computational Science and Engineering (CSE) in Salt Lake City, March 14-18, 2015. It also includes a named lectureship, the Emmy Noether Lecture, to be held at the San Antonio JMM meeting. The workshops are designed to create sustainable networks, encourage mentoring relationships, and promote research collaborations in key fields of mathematics. The Noether Lecture highlights outstanding contributions by a female mathematician and provides an inspiring role model to more junior women. As such, these events address some of the key issues that frequently inhibit women's participation in the field.


Each workshop will focus on an area of mathematics of current interest. The 2015 AWM-JMM workshop will focus on homotopy theory. Homotopy theory, an area of algebraic topology, has close ties to algebraic geometry, geometric topology, number theory, representation theory, and mathematical physics. It is a highly active area and has seen significant progress in recent years. The 2015 AWM-SIAM workshop will focus on mathematical modeling and high-performance computing for multiphysics and multiscale problems. Advances in science and engineering - in disciplines ranging from health to energy and the environment to defense - rely on predictive measurement and analysis of multiscale multiphysics systems. Topics to be addressed at the workshop include physical modeling, mathematical analysis, numerical analysis, algorithms, implementation, performance, and scalability.

https://sites.google.com/site/awmmath/programs/workshops

The research projects involve the interplay between the study of algebraic structures and topology, the study of shapes or spaces. Algebraic topologists use algebraic structures to describe and simplify topological phenomena. Spectra, which represent cohomology theories, are algebraic structures built out of topological spaces and hence are useful for translating from one field to the other. In one project, the PI and Greenlees develop algebraic models for certain types of spectra with symmetries that allow complete calculations. The PI continues to train graduate students and disseminate research results. In addition, the PI is involved with several organizations that promote the participation of women and underrepresented minorities in math and science.

The PI continues to study when derived equivalences can be realized by underlying richly structured equivalences. Examples of this arise in the PI's long term project with John Greenlees of constructing algebraic models for rational G-equivariant spectra for compact Lie groups G. More specifically, the PI and Greenlees plan to extend their model for tori to provide an algebraic model for rational G-equivariant commutative and associative ring spectra. The PI and Kathryn Hess continue to develop the homotopical setting for coalgebras with motivations coming from Rognes' Hopf-Galois extensions of ring spectra and Hess' homotopical framework for descent. The PI and Birgit Richter are developing a simple algebraic model for homotopically commutative differential graded algebras.

A research program is to be held at the Centre de Recerca Matematica (CRM) in Barcelona, Spain in the Spring semester of 2015. The focus will be on the intersection between representation theory, algebraic topology, and commutative algebra. There is a long tradition of interaction between commutative algebra and algebraic topology, and commutative algebra and representation theory. Techniques and concepts that arose and were developed in one field have driven developments in the other. Regrettably, opportunities for building and fostering collaborations, especially among graduate students and younger researchers, from the three subjects (or indeed any two of them) have been all too rare, especially in the U.S. The three subjects have witnessed remarkable changes and developments in the last decade, and the CRM program comes at an opportune moment. This NSF award will fund the travel of younger researchers (graduate students, recent PhDs, and junior faculty) at U.S. institutions to participate in some of the key events of the program. This will provide them with a unique opportunity to learn about current research in these fields and the interactions between fields; discuss ideas with one another, and begin collaborations on future research.

The topics to be covered by the CRM program, and in particular the advanced courses and workshops: matrix factorizations, maximal Cohen-Macaulay modules; cluster algebras and cluster categories; cohomological theory of support varieties; and tilting theory, to name a few, are at the intersection of several important and rapidly developing areas of mathematics, each subject benefitting from advances in the others. The CRM program will bring together leading researchers in each of these subjects, and one can be optimistic that this will lead to the new insights and developments in them. The highlights of the program include two, week-long, advanced courses, one in February 2015, titled (Re)emerging methods in Commutative Algebra and Representation Theory, and another in April 2015, titled Building bridges between Algebra and Topology. Each will be followed by a week-long workshop. Additionally, a conference is scheduled for 25--29th May, 2015.

This project supports the travel and local expenses of early-career US-based mathematicians attending the international conference "Equivariant Topology and Derived Algebra (ETDA)" to be held from July 29 to August 2, 2019 at the Norwegian University of Science and Technology in Trondheim. These mathematical research areas have seen a dramatic expansion in recent years and have attracted increased interest at an international level, especially amongst young researchers, who are taking these fields in new directions. The direct impact of this funding will be to augment the training and career development of approximately twenty junior US researchers, who will gain the opportunity to participate in a major conference in Europe. A secondary impact is to further develop collaboration between emerging research groups in algebraic topology, derived algebra, and derived algebraic geometry in the US and Europe.

The conference topics are centered on the areas of equivariant stable homotopy theory, derived commutative algebra, and derived algebraic geometry. The conference program features invited talks on all three areas by leading researchers and time for contributed talks, with a focus on young researchers. The conference will be an ideal venue for results and new questions arising from recent research progress to be announced and discussed. It will also provide a place to reinforce the networks of connections between mathematicians and add junior researchers as new nodes. Moreover, the particular topic of equivariant symmetric monoidal structures, together with the derived algebra that realizes those structures, is experiencing rapid development and will be a focal point of the conference talks. The conference website is: https://sites.google.com/view/etda2019/home

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.