To attend this seminar, you must register in advance, by clicking HERE.

The African Diaspora Joint Mathematics Workshop (ADJOINT) will take place at the Mathematical Sciences Research Institute in Berkeley, CA from June 15 to June 26, 2020.

ADJOINT is a two-week summer activity designed for researchers with a Ph.D. degree in the mathematical sciences who are interested in conducting research in a collegial environment.  

The main objective of ADJOINT is to provide opportunities for in-person research collaboration to U.S. mathematicians, especially those from the African Diaspora, who will work in small groups with research leaders on various research projects. 

Through this effort, MSRI aims to establish and promote research communities that will foster and strengthen research productivity and career development among its participants. The ADJOINT workshops are designed to catalyze research collaborations, provide support for conferences to increase the visibility of the researchers, and to develop a sense of community among the mathematicians who attend. 

The end goal of this program is to enhance the mathematical sciences and its community by positively affecting the research and careers of African-American mathematicians and supporting their efforts to achieve full access and engagement in the broader research community. 

During the workshop, each participant will: 

• conduct research at MSRI within a group of four to five mathematicians under the direction of one of the research leaders 
• participate in professional enhancement activities provided by the onsite ADJOINT Director 
• receive funding for two weeks of lodging, meals and incidentals, and one round-trip travel to Berkeley, CA 

After the two-week workshop, each participant will:

• have the opportunity to further their research project with the team members including the research leader 
• have access to funding to attend conference(s) or to meet with other team members to pursue the research project, or to present results 
• become part of a network of research and career mentors

To attend this seminar, you must register in advance, by clicking HERE.

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

On May 22 portions of the Critical Issues in Mathematics Education 2020: Today’s Mathematics, Social Justice, and Implications for Schools workshop will be streamed online via Zoom.

Friday 5/22: 12pm PST (3pm eastern time)

12:00 - 1:00
Rico Gutstein, Preparing Students Today for Whatever Tomorrow Brings

To attend this seminar, you must register in advance, by clicking HERE.

Consider a polynomial ring in n variables, together with the action of the symmetric group by coordinate permutations. In my talk I will describe many familiar notions in Commutative Algebra in the context of monomial ideals that are preserved by the action of the symmetric group. These include Castelnuovo-Mumford regularity, projective dimension, saturation, symbolic powers, or the Cohen-Macaulay property. My goal is to explain how changing focus from minimal resolutions to Ext modules can lead to a simplified picture of the homological algebra, and to provide concrete combinatorial recipes to determine the relevant homological invariants.

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

On May 15 portions of the Critical Issues in Mathematics Education 2020: Today’s Mathematics, Social Justice, and Implications for Schools workshop will be streamed online via Zoom.

Friday 5/15: 12pm PST (3pm eastern time)

12:00 - 1:00
Dan Reinholz, Preparing teachers to notice, name, and disrupt racial and gender inequity

To attend this seminar, you must register in advance, by clicking HERE.

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 5/8: 12pm PST (3pm eastern time)

12:00 - 1:00
Nathan Alexander, Mathematical Models in the Sociological Imagination
Lincoln Chandler, Pursuing Racial Equity within Schools 

Image drawn by Dr. Katy Craig

This workshop will be held online.  The link to join is: https://msri.zoom.us/j/92457794010. You must register for the workshop to receive the password.  The workshop is held in Pacific Standard Time.

Workshop Description:
The goal of the workshop is to explore the many emerging connections between the theory of Optimal Transport and models and algorithms currently used in the Machine Learning community. In particular, the use of Wasserstein metrics and the relation between discrete models and their continuous counterparts will be presented and discussed.

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 5/01: 12pm PST (3pm eastern time)

12:00 - 1:00 Hyman Bass, 'Mathematics and Social Justice': An undergraduate course. What could this be?

Watch on Youtube

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 4/24: 12pm PST (3pm eastern time)

12:00 - 1:00 Padmanabhan Seshaiyer, K-12 to Post-Secondary Viewpoint Critical Issues in Mathematics Education

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 4/17: 12pm PST (3pm eastern time)

12:00 - 1:00 Some unintended consequences of active learning

Sage Forbes-Gray, Sunset Park High School,  Brooklyn, NY, Mfa Master Teacher 
Sharon Collins - New Heights Academy Charter School, NYC, MfA Master Teacher; 
Kate Belin - Fannie Lou High School, NYC, MfA Master Teacher; 

Moderator: Courtney Ginsberg, MfA
Host: Katherine Stevenson, CSUN

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 4/10: 12pm PST (3pm eastern time)

12:00 - 1:00 Estrella Johnson, Some unintended consequences of active learning

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 3/27: Starting at 12pm PST (3pm eastern time)

12:00p - 1:00p
Nicol Turner Lee, Brookings Inst., Center for Tech Innov. -  Unconscious Bias
Saber Khan, Processing Foundation, leader of #EthicalCS -  Identity & Ethics

The link to this online workshop is: https://msri.zoom.us/j/999860976

This workshop will focus on recent developments in factorization homology, parametrized homotopy theory, and algebraic K-theory.  These seemingly disparate topics are unified by a common methodology, which leverages universal properties and unforeseen descent by way of higher category theory. Furthermore, they enjoy powerful and complementary roles in application to the cyclotomic trace.  This workshop will be a venue for experts in these areas to present new results, make substantive connections across fields, and suggest and contextualize outstanding questions and problems.  It will consist of 4 two-part lecture series and 10 one-hour talks. The lecture series will be given by Thomas Nikolaus, Akhil Mathew, David Ben-Zvi and a split Martina Rovelli and Viktoriya Ozornova.

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

Friday 3/20: Starting at 12pm PST (3pm eastern time)

12:00p - 12:45p Lisa Goldberg, Hot Hands: What Data Science Can (and Can't) Tell Us About Basketball Trends
12:45p - 1:00p Discussion with Lisa and Kate on:  What Bayes tells us about our ability to reason about randomness

Topological field theory studies the interplay of algebraic and topological structure (image credit Kevin Walker)

Link to stream workshop: https://msri.zoom.us/j/226801541

***Due to the COVID-19 virus outbreak, the 2020 Tensor categories and topological quantum field theories workshop will no longer be held onsite at MSRI, rather it will take place online from March 16-20 as scheduled***

The decision to move this workshop online is based on the available scientific data on COVID-19, and the strong advice from experts to avoid gatherings of large groups.

Announcement: Workshop Moved Online

Download Official Notice of Change

A formal Notice of Change letter is available here, which can be shared with your institution, funding agency, and others.

Due to the COVID-19 virus outbreak, the Critical Issues in Mathematics Education 2020 workshop was held online. The full workshop description and list of talks can be found HERE.

On March 12 and March 13, portions of the Critical Issues in Mathematics Education 2020: Today’s Mathematics, Social Justice, and Implications for Schools workshop will be streamed online via Zoom. Only the talks below will are scheduled at this time.  Further talks may be scheduled at a later date, and you will be notified when we know more.

Please see the schedule below, as well as links to the two sessions.
 

Thursday 3/12: Starting at 9am PST (noon eastern time)
9:00 - 9:10 Welcoming remarks
9:10 - 9:15 Introduction to CIME 2020 plan and speaker David Daley
9:15 - 9:55 David Daley, Why Your Vote Doesn't Count
9:55 - 10:00 Kate Stevenson, introduction of activity
10:00-10:30 Mathical Book Prize Announcement

Friday 3/13: Starting at 9am PST (noon eastern time)
9:00 - 9:05 Introduction of speaker Wesley Pegden
9:05 - 9:45 Wesley Pegden, Bringing Mathematics to the Courtroom
9:45 - 10:00 Q&A
 

Download Official Notice of Change

A formal Notice of Change letter is available here, which can be shared with your institution, funding agency, and others.

relations among 2-morphisms in the 2-dimensional unoriented bordism bicategory

This workshop will survey notable developments and applications of higher category theory; it will be a venue for end-users to share their vision of how to apply the theory, as well as developers to share technical advancements.  It will consist of 6 series of 3 lectures, each given by instrumental end-users & developers of higher category theory, together with a few question-answer sessions.  Each lecture series will be tailored to a diverse audience, accessible to graduate students and non-expert researchers with some background in homological also algebra.  The content of these lecture series will concern the following topics.

• K-theory: categorification, non-commutative motives, trace methods; 
• TQFT: functorial field theories, factorization homology.
• Parametrized higher category theory: stratifications, equivariant homotopy theory, operads, deformation theory and Koszul duality. 
• Synthetic higher category theory: model-independent characterizations, cosmoi.  

Picture of a Feynman graph.

This two-day workshop will survey notable developments in the foundations and applications of higher category theory. It will consist of two mini-courses given by emerging female leaders in the subject: Claudia Scheimbauer and Nathalie Wahl.  This will be paired with a problem sessions lead by selected "TA's", themselves experts in higher structures.  Each lecture series will be tailored to a diverse audience, accessible to graduate students and non-expert researchers with some background in homological algebra.  

The majority of the speakers and panelists for this event will be women and gender minorities, and members of these groups and of other underrepresented groups are especially encouraged to attend. This workshop is open to all mathematicians.

Jellyfish floating to the surface, as in the evaluation algorithm for certain planar algebras.

This workshop will consist of introductory minicourses on key topics in Quantum Symmetry: fusion categories, modular tensor categories, Hopf algebras, subfactors and planar algebras, topological field theories, conformal nets, and topological phases of matter.  These minicourses will be introductory and are aimed at giving semester participants exposure to the main ideas of subfields other than their own.

Photo by drmakete lab on Unsplash

This workshop will feature several talks by experts, along with numerous 5-minute presentations by junior mathematicians, on topics related to Quantum Symmetry. Such topics will include tensor categories, subfactors, Hopf algebras, topological quantum field theory and more. There will also be a panel discussion on professional development. The majority of the speakers and panelists for this event will be women and gender minorities, and members of these groups and of other underrepresented groups are especially encouraged to attend. This workshop is open to all mathematicians.

swallowtail identity

Though many of the ideas in higher category theory find their origins in homotopy theory — for instance as expressed by Grothendieck’s “homotopy hypothesis” — the subject today interacts with a broad spectrum of areas of mathematical research. Unforeseen descent, or local-to-global formulas, for familiar objects can be articulated in terms of higher invertible morphisms. Compatible associative deformations of a sequence of maps of spaces, or derived schemes, can putatively be represented by higher categories, as Koszul duality for E_n-algebras suggests. Higher categories offer unforeseen characterizing universal properties for familiar constructions such as K-theory. Manifold theory is natively connected to higher category theory and adjunction data, a connection that is most famously articulated by the recently proven Cobordism Hypothesis.
In parallel, the idea of "categorification'' is playing an increasing role in algebraic geometry, representation theory, mathematical physics, and manifold theory, and higher categorical structures also appear in the very foundations of mathematics in the form of univalent foundations and homotopy type theory. A central mission of this semester will be to mitigate the exorbitantly high "cost of admission'' for mathematicians in other areas of research who aim to apply higher categorical technology and to create opportunities for potent collaborations between mathematicians from these different fields and experts from within higher category theory.

MSRI will host a Symposium on the occasion of Julia Robinson’s 100th birthday on Monday, December 9, 2019 at MSRI. Julia Robinson (1919-1985) was an internationally renowned logician of the twentieth century. She was a trailblazer in mathematics as well as in many other ways: she was the first woman president of the American Mathematical Society, and the first woman mathematician elected to membership in the National Academy of Sciences.

Participating speakers in this day-long celebration of her work and of current mathematics insprired by her research include: Martin Davis, Kirsten Eisentrager, Yuri Matiyasevich, and  Lou van den Dries. Following the symposium, Lenore Blum will give a public lecture at UC Berkeley.

An example of a spectral network associated to the group SL(4).

Holomorphic differentials on Riemann surfaces have long held a distinguished place in low dimensional geometry, dynamics and representation theory. Recently it has become apparent that they constitute a common feature of several other highly active areas of current research in mathematics and also at the interface with physics. In some cases the areas themselves (such as stability conditions on Fukaya-type categories, links to quantum integrable systems, or the physically derived construction of so-called spectral networks) are new, while in others the novelty lies more in the role of the holomorphic differentials (for example in the study of billiards in polygons, special - Hitchin or higher Teichmuller - components of representation varieties, asymptotic properties of Higgs bundle moduli spaces, or in new interactions with algebraic geometry).

It is remarkable how widely scattered are the motivating questions in these areas, and how diverse are the backgrounds of the researchers pursuing them. Bringing together experts in this wide variety of fields to explore common interests and discover unexpected connections is the main goal of our program. Our workshop will be of interest to those working in many different fields, including low-dimensional dynamical systems (via the connection to billiards); differential geometry (Higgs bundles and related moduli spaces); and different types of theoretical physics (electron transport and supersymmetric quantum field theory).

As part of the Mathematical Sciences Collaborative Diversity Initiatives, six mathematics institutes are pleased to host their annual SACNAS pre-conference event, the 2019 Modern Math Workshop (MMW). The Modern Math Workshop is intended to encourage minority undergraduates to pursue careers in the mathematical sciences and to assist undergraduates, graduate students and recent PhDs in building their research networks.

Elwyn Berlekamp (1937-2019) was a pioneering contributor to combinatorial game theory, greatly advancing the subject over the course of a more than five-decade career. Along with his coauthors, John Conway and Richard Guy, Berlekamp invented the modern form of the theory, with the publication of Winning Ways for Your Mathematical Plays in 1982. His later work substantially advanced our understanding of the mathematical structure of well-known games such as Go, Amazons, and Dots-and-Boxes. More information about his life can be found at www.msri.org/elwyn.

This workshop will be an informal two-day mini-conference honoring Berlekamp's work and the subject he helped create. The event will consist of talks, afternoon workshops, and a combinatorial games tournament.

http://web.stanford.edu/~andras/rbmconf.html

Microlocal analysis provides tools for the precise analysis of problems arising in areas such as partial differential equations or integral geometry by working in the phase space, i.e. the cotangent bundle, of the underlying manifold. It has origins in areas such as quantum mechanics and hyperbolic equations, in addition to the development of a general PDE theory, and has expanded tremendously over the last 40 years to the analysis of singular spaces, integral geometry, nonlinear equations, scattering theory, hyperbolic dynamical systems, probability… As this description shows microlocal analysis has become a very broad area. Due to its breadth, it is a challenge for researchers to be aware of what is happening in other parts of the field, and the impact this may have in their own research area. The purpose of this workshop is thus to bring together researchers from different parts of microlocal analysis and its applications to facilitate the transfer of new ideas. 

The objective of the meeting is to bring theorists and theoretically-motivated experimentalists together to discuss promising theoretical frameworks for understanding cognitive processes and how these may be brought to bear on interpreting neural data or formulating new experiments. We hope that this meeting will be a chance to discuss future goals for theory in neuroscience: what are missing areas and emerging approaches that might help the field to make real progress in developing theories of brain function.

Microlocal analysis provides tools for the precise analysis of problems arising in areas such as partial differential equations or integral geometry by working in the phase space, i.e. the cotangent bundle, of the underlying manifold. It has origins in areas such as quantum mechanics and hyperbolic equations, in addition to the development of a general PDE theory, and has expanded tremendously over the last 40 years to the analysis of singular spaces, integral geometry, nonlinear equations, scattering theory… This workshop will provide a comprehensive introduction to the field for postdocs and graduate students as well as specialists outside the field, building up from standard facts about the Fourier transform, distributions and basic functional analysis.

This workshop will provide a gentle introduction to a selection of applications of microlocal analysis.  These may be drawn from among geometric microlocal analysis, inverse problems, scattering theory, hyperbolic dynamical systems,  quantum chaos and relativity.  The workshop will also provide  a panel discussion, a poster session and an introduction/research session. 

This workshop is open to all mathematicians.

Some holomorphic differentials on a genus 2 surface, with close up views of singular points, image courtesy Jian Jiang.

Holomorphic differentials on Riemann surfaces have long held a distinguished place in low dimensional geometry, dynamics and representation theory. Recently it has become apparent that they constitute a common feature of several other highly active areas of current research in mathematics and also at the interface with physics. In this introductory workshop, we will bring junior and senior researchers from this diverse range of subjects together in order to explore common themes and unexpected connections.

Some holomorphic differentials on a genus 2 surface, with close up views of singular points, image courtesy Jian Jiang.

This two-day workshop will consist of various talks given by prominent female mathematicians on topics of new developments in the role of holomorphic differentials on Riemann surfaces. These will be appropriate for graduate students, post-docs, and researchers in areas related to the program.  

This workshop is open to all mathematicians.

Some holomorphic differentials on a genus 2 surface, with close up views of singular points, image courtesy Jian Jiang.

Holomorphic differentials on Riemann surfaces have long held a distinguished place in low dimensional geometry, dynamics and representation theory. Recently it has become apparent that they constitute a common feature of several other highly active areas of current research in mathematics and also at the interface with physics. In some cases the areas themselves (such as stability conditions on Fukaya-type categories, links to quantum integrable systems, or the physically derived construction of so-called spectral networks) are new, while in others the novelty lies more in the role of the holomorphic differentials (for example in the study of billiards in polygons, special - Hitchin or higher Teichmuller - components of representation varieties, asymptotic properties of Higgs bundle moduli spaces, or in new interactions with algebraic geometry).

It is remarkable how widely scattered are the motivating questions in these areas, and how diverse are the backgrounds of the researchers pursuing them. Bringing together experts in this wide variety of fields to explore common interests and discover unexpected connections is the main goal of our program. Our program will be of interest to those working in many different elds, including low-dimensional dynamical systems (via the connection to billiards); differential geometry (Higgs bundles and related moduli spaces); and different types of theoretical physics (electron transport and supersymmetric quantum field theory).

Microlocal analysis provides tools for the precise analysis of problems arising in areas such as partial differential equations or integral geometry by working in the phase space, i.e. the cotangent bundle, of the underlying manifold. It has origins in areas such as quantum mechanics and hyperbolic equations, in addition to the development of a general PDE theory, and has expanded tremendously over the last 40 years to the analysis of singular spaces, integral geometry, nonlinear equations, scattering theory… This program will bring together researchers from various parts of the field to facilitate the transfer of ideas, and will also provide a comprehensive introduction to the field for postdocs and graduate students.

The Complementary Program has a limited number of memberships that are open to mathematicians whose interests are not closely related to the core programs; special consideration is given to mathematicians who are partners of an invited member of a core program. 

This simplicial fan in 3-dimensional space

Toric varieties are algebraic varieties defined by combinatorial data, and there is a wonderful interplay between algebra, combinatorics and geometry involved in their study. Many of the key concepts of abstract algebraic geometry (for example, constructing a variety by gluing affine pieces) have very concrete interpretations in the toric case, making toric varieties an ideal tool for introducing students to abstruse concepts.

The image of a large sphere isometrically embedded into a small space through a C^1 embedding. (Attributions: E. Bartzos, V. Borrelli, R. Denis, F. Lazarus, D. Rohmer, B. Thibert)

This two week summer school will introduce graduate students to the theory of h-principles.  After building up the theory from basic smooth topology, we will focus on more recent developments of the theory, particularly applications to symplectic and contact geometry, fluid dynamics, and foliation theory.

Popular visualization of the MNIST dataset

Learning theory is a rich field at the intersection of statistics, probability, computer science, and optimization. Over the last decades the statistical learning approach has been successfully applied to many problems of great interest, such as bioinformatics, computer vision, speech processing, robotics, and information retrieval. These impressive successes relied crucially on the mathematical foundation of statistical learning.

Recently, deep neural networks have demonstrated stunning empirical results across many applications like vision, natural language processing, and reinforcement learning. The field is now booming with new mathematical problems, and in particular, the challenge of providing theoretical foundations for deep learning techniques is still largely open. On the other hand, learning theory already has a rich history, with many beautiful connections to various areas of mathematics (e.g., probability theory, high dimensional geometry, game theory). The purpose of the summer school is to introduce graduate students (and advanced undergraduates) to these foundational results, as well as to expose them to the new and exciting modern challenges that arise in deep learning and reinforcement learning.

Fluid-flow stream function color-coded by vorticity in 3D flat torus calculated by K. Nakai (The University of Tokyo)

The purpose of the workshop is to introduce graduate students to fundamental results on the Navier-Stokes and the Euler equations, with special emphasis on the solvability of its initial value problem with rough initial data as well as the large time behavior of a solution. These topics have long research history. However, recent studies clarify the problems from a broad point of view, not only from analysis but also from detailed studies of orbit of the flow.

from distinct distances in the plane to line incidences in R^3

In the past eight years, a number of longstanding open problems in combinatorics were resolved using a new set of algebraic techniques. In this summer school, we will discuss these new techniques as well as some exciting recent developments.

A Holomorphic Curve

Symplectic topology is a fast developing branch of geometry that has seen phenomenal growth in the last twenty years. This two weeks long summer school, organized in the setting of the Séminaire de Mathématiques Supérieures, intends to survey some of the key directions of development in the subject today thus covering: advances in homological mirror symmetry; applications to hamiltonian dynamics; persistent homology phenomena; implications of flexibility and the dichotomy flexibility/rigidity; legendrian contact homology; embedded contact homology and four-dimensional holomorphic techniques and others. With the collaboration of many of the top researchers in the field today, the school intends to serve as an introduction and guideline to students and young researchers who are interested in accessing this diverse subject. 

Rips's δ-thin triangle condition for Gromov hyperbolicity of metric spaces (Stomatapoll)

Geometric group theory studies discrete groups by understanding the connections between algebraic properties of these groups and topological and geometric properties of the spaces on which they act. The aim of this summer school is to  introduce graduate students to specific central topics and recent developments in geometric group theory. The school will also include students presentations to give the participants an opportunity to practice their speaking skills in mathematics.  Finally, we hope that this meeting will help connect Latin American students with their American and Canadian counterparts in an environment that encourages discussion and collaboration. 

An illustration of an adaptation of Quillen's classical homological stability spectral sequence argument

This summer school will give an introduction to representation stability, the study of algebraic structural properties and stability phenomena exhibited by sequences of representations of finite or classical groups -- including sequences arising in connection to hyperplane arrangements, configuration spaces, mapping class groups, arithmetic groups, classical representation theory, Deligne categories, and twisted commutative algebras.  Representation stability incorporates tools from commutative algebra, category theory, representation theory, algebraic combinatorics, algebraic geometry, and algebraic topology. This workshop will assume minimal prerequisites, and students in varied disciplines are encouraged to apply. 

The MSRI-UP summer program is designed to serve a diverse group of undergraduate students who would like to conduct research in the mathematical sciences.

In 2019, MSRI-Up will focus on the application of combinatorial arguments and techniques to enumerate, examine, and investigate the existence of discrete mathematical structures with certain properties. The areas of interest for these applications encompass a wide range of mathematical fields and will include algebra, number theory, and graph theory, through weight multiplicity computations, the study of vector partition functions, and graph domination problems, respectively. The research program will be led by Dr. Pamela E. Harris, Assistant Professor of Mathematics at Williams College.

The purpose of the MSRI's program, Summer Research for Women in Mathematics, is to provide space and funds to groups of women mathematicians to work on a research project at MSRI. Research projects can arise from work initiated at a Women's Conference, or can be freestanding activities.

The figure represents a blow-up. The so called blow-up algebras or Rees rings are the algebraic realizations of such blow-ups.

Linkage is a method for classifying ideals in local rings. Residual intersections is a generalization of linkage to the case where the two `linked' ideals  need not have the same codimension. Residual intersections are ubiquitous: they play an important role in the study of blowups, branch and multiple point loci, secant varieties, and Gauss images; they appear naturally in intersection theory; and they have close connections with integral closures of ideals. 

Commutative algebraists have long used the Frobenius or p-th power map to study commutative rings containing a finite field. The theory of tight closure and test ideals has widespread applications to the study of symbolic powers and to Briancon-Skoda type theorems for equi-characteristic rings.

Numerical conditions for the integral dependence of ideals and modules have a wealth of applications, not the least of which is in equisingularity theory. There is a long history of generalized criteria for integral dependence of ideals and modules based on variants of the Hilbert-Samuel and the Buchsbaum-Rim multiplicity that still require some remnants of finite length assumptions.

The Rees ring and the special fiber ring of an ideal arise in the process of blowing up a variety along a subvariety. Rees rings and special fiber rings also describe, respectively, the graphs and the images of rational maps between projective spaces. A difficult open problem in commutative algebra, algebraic geometry, elimination theory, and geometric modeling is to determine explicitly the equations defining graphs and images of rational maps.

The school will consist of the following four courses with exercise sessions plus a Macaulay2 workshop

• Linkage and residual intersections
• Characteristic p methods and applications
• Blowup algebras
• Multiplicity theory