In many Latin American countries, political instability, institutional weakness and a lack of government support for scientific research have hindered the development of mathematics. There have been signs of progress in recent years. In 2014, Brazilian mathematics received international recognition when Artur Avila became the first South American to be awarded a Fields Medal. In 2018, the International Congress of Mathematicians will be hosted in a Latin American country for the first time. Within the last five years, several major conferences, such as the Mathematical Congress of the Americas, the AGRA winter schools, and PRIMA 2017, have been organized with the specific aim of increasing mathematical activity in Latin American countries.

In spite of all of this progress, there is still room for improvement. Number theory research in South and Central America continues to be largely confined to geographically isolated pockets of activity, concentrated within a small number of subfields. Many of the strongest math students go abroad for their training, in some cases because they cannot find viable Ph.D. supervisors in the research areas that they hope to pursue in their home countries. In most areas, Latin American mathematicians continue to be poorly represented at major international conferences. The proposed workshop aims to address some of these issues. Our main objectives are as follows:

Facilitate collaboration between North, Central, and South American number theorists.

The primary aim of the proposed workshop is to promote collaboration between number theorists in North, Central, and South America. To do this, we will model our workshop after several other workshops that have been extremely successful at sparking new collaborations: the American Institute of Mathematics workshops, the AMS Mathematics Research Communities workshops, and the BIRS-sponsored Women In Numbers workshops. Participants will be divided into small project groups led by senior researchers. Most of the time during the workshop will be spent working on research in these project groups. The goal is for researchers to leave the workshop with the beginning of a research paper or, at least, with a list of good candidates for future collaborators and a deeper understanding of a timely subject.

Foster research in timely areas of number theory. All of our confirmed participants have impressive research track records, and several are leading mathematicians by world standards. All of our project groups are on areas central to current research in the field, and all of these areas can also be said to lie in the crossroads between number theory and other fields. In several cases, this requires little explanation: the study of the arithmetic of algebraic varieties lies in the intersection of number theory and algebraic geometry; the study of modular forms, which originated in complex analysis, has been essential to number theorists since Ramanujan. The Langlands program is inherently about building connections, particularly with representation theory.

Continuing with our list of topics: additive combinatorics is a relatively new name for an area that encompasses additive number theory, combinatorial arguments and probabilistic and ergodic ideas. The importance of analytical tools to number theory has been clear since Riemann, and the relevance of harmonic analysis and spectral theory has become clearer and clearer since the mid-20th century. Probabilistic arguments in number theory have been fruitful ever since Erd\H{o}s and Tur\'an. The relevance of ergodic theory and dynamical systems to number theory has been known at least since Furstenberg and Ratner. Geometry and number theory often give two different perspectives on arithmetic groups. In particular, spectral gaps and expanders are terrains where number theory, spectral theory and geometry meet.

Train young researchers. Rather than filling the workshop with invited participants, we will reserve some spaces for young researchers who can apply to work in project groups that match their interests. One of our aims is to provide specialized training for young researchers in Latin American countries and introduce them to interesting problems in areas that may not be well-represented in their home countries. In some cases, this will be their first experience with working on a collaborative project. We will take steps to create a supportive environment so that young researchers will feel encouraged by the experience. We will also hold several panel discussions on topics that will be of particular interest to young researchers (see the Overview for more details).

Provide mentoring opportunities for mathematicians who normally do not get to train young researchers. The project groups are designed to provide a vertical mentoring structure, enabling mathematicians at different stages of their careers to mentor one another. Some of our participants may be faculty members at institutions without Ph.D. programs, and some will come from countries where it is typical for the strongest students to go abroad for graduate school. Such participants will have an exceptional chance to mentor promising young researchers in their project groups.

Attract greater visibility for the work of Latin American number theorists. A growing number of Latin Americans are working in number theory. By assembling this group, we will demonstrate that there is, in fact, already a fair number of strong number theorists connected to Latin American countries. Holding our workshop at the CMO, and advertising it on the BIRS website, will lend them additional prominence.

Build a network of Spanish-speaking mathematicians. This workshop will provide the foundation for creating a global network of Spanish-speaking mathematicians. In particular, we plan to start an online community -- including a mailing-list and possibly a more visible database -- of self-identified Spanish-speaking number theorists from around the world, organized by research area, which we hope will be useful to future conference organizers. The defining criterion will be an ability and willingness to lecture and work in Spanish.

This workshop continues the investigation of the interface between number theory, geometry, and physics started in last year’s workshop. Some of the themes covered include Hodge theory and physics, arithmetic of black holes, mirror symmetry, arithmetic of scattering amplitudes, arithmetic gauge theories, and modularity of BPS states. The number of physicists around the world interested in number theory seems to be steadily increasing as are the number of activities. Over the past year, the organiser MK has spoken at the mathematical physics seminar in Heidelberg, the mathematics and physics colloquium in Amsterdam, and two Simons conferences on string theory and number theory. The current activity at KIAS attempts to present a coherent view of many of the most recent developments and insights for the benefit of participants and speakers.

A conference on the occasion of the 60th birthday of Vitaly Tarasov and the 70th birthday of Alexander Varchenko.

The goal of this summer school is to introduce Ph.D. students and Postdocs to exciting recent developments in the geometry of algebraic groups and in modular representation theory.

The school will consist of two morning lecture series and of afternoon lectures on topics directly related to the morning lecture series.

Michel Brion (Grenoble) will give one lecture series on the “Structure of algebraic groups and geometric applications”. Geordie Williamson (Sydney) will give the other lecture series “On the modular representation theory of algebraic groups.”

M. Brion’s abstract. The course will first give an overview of the “classical” structure theory of algebraic groups and of some related geometric developments and problems; for example, on automorphism groups of projective algebraic varieties. It will then address results and questions on algebraic groups over arbitrary fields, including the structure of pseudo-reductive groups (work of Conrad, Gabber and Prasad), and of pseudo-abelian varieties (Totaro). Five lectures, 75 minutes each.

G. Williamson’s abstract. This course will be about the modular (i.e. characteristic p) representation theory of reductive algebraic groups, like the general linear and symplectic groups. I will begin by reviewing the algebraic theory, where there are beautiful connections to classical Lie theory and finite group theory. I will then pass to the geometric theory (perverse and parity sheaves) which is behind recent breakthroughs in the subject. The theory is rich in mysteries and open conjectures, and I will try to outline potentially interesting research directions. Five lectures, 75 minutes each.

A finite free resolution over a commutative local ring is universal for its set of ranks if every other finite free resolution with the same set of ranks can be obtained from it via base change. The existence is formal, however the question remained whether the universal ring can be taken to be noetherian. Hochster established this in 1975 in the length 2 case. Recent advances have connected this story to Kac-Moody Lie algebras and representation theory and the goal of this workshop is to introduce this research area to graduate students, with special emphasis on the length 3 case. The structure of the universal ring controls the structure of free resolutions of a given rank, and this new link allows one to explore this with the use of representation theory.

We have travel and lodging support for students and young researchers. Please use the registration form if you are interested in attending.

The plan is to have 2 introductory lectures each morning (example topics are linkage, structure theorems for finite free resolutions, basic representation theory), with problem sessions and time to write Macaulay2 code in the afternoons. Prior experience with Macaulay2 will be very helpful and participants will be able to work on computer projects with a research component.

This is a workshop that aims to support new collaborations between female mathematicians. Before the workshop, each participant will be assigned to a working group according to her research interests. Prior to the conference, the project leaders will design projects and provide background reading and references for their groups.

Confirmed group leaders:

Irene Bouw (Ulm)
Rachel Newton (Reading) and Ekin Ozman (Bogazici)
Damaris Schindler (Utrecht) and Lilian Matthiessen (KTH)
Ramla Abdellatif (Picardie Jules Verne)
Cecilia Salgado (Rio de Janeiro)
Elisa Gorla (EPFL)
Eimear Byrne (Dublin) and Relinde Jurrius (Neuchâtel)
Kristin Lauter (Microsoft)
Marcela Hanzer (Zagrev)
Lejla Smajlovic (Sarajevo)

The purpose of the workshop is to support and expand research efforts by female mathematicians in the field of algebraic topology. The workshop will bring senior and junior researchers together with advanced graduate students to cooperate on research projects on topics of common interest. The focus of the workshop will be on active collaboration to advance the projects, rather than on knowledge communication through lectures.

Central European Number Theory Conference (CENT) is the successor of the traditional Czech and Slovak International Conference on Number Theory (NTC) which has been organized since 1972.

The aim of arithmetic geometry is to solve equations on integers by geometric methods. One of the most prominent achievements of this approach is certainly the Langlands program, which makes a connection between representations of the absolute Galois group of $\mathbb Q$ and certain adelic representations of reductive algebraic groups. In the early 2000's, Christophe Breuil suggested the existence of a purely $p$-adic version of the Langlands correspondence and supported his vision by numerous examples. Almost twenty years after, the $p$-adic Langlands correspondence has become a major topic in number theory.

Besides, following the rapid development of computer science throughout the 20th century, a large panel of algorithmical tools has been deployed and are now quite performant, in particular for attacking questions in Number Theory. A computational approach to the (classical) Langlands correspondence has been already investigated in recent times as well. We believe that the time has come to begin to extend it to the $p$-adic Langlands correspondence.

This conference is a first step towards this perspective. It will bring together the most internationally recognized experts in $p$-adic Langlands correspondence on the one hand and effective aspects of the Langlands correspondence on the other hand. Young researchers, and more generally researchers who are familiar with one side (either the abstract one or the effective one) and are willing to learn the other side, are particularly encouraged to attend our event: a enthousiastic program with 2 mini-courses, a bunch of short lectures and an introduction to the mathematical software SageMath is specially designed for them.

An inverse problem refers to a situation where the quantity of interest cannot be measured directly, but only through an action of a nontrivial operator of which it is a parameter. The corresponding operator, also called forward operator, stems from a physical application modelling. Prominent examples include: Radon and Fourier transforms for X-ray CT and MRI, respectively or partial differential equations e.g. EIT or DOT.

The prevalent characteristics of inverse problems is their ill-posedness i.e. lack of uniqueness and/or stability of the solution. This situation is aggravated by the physical limitations of the measurement acquisition such as noise or incompleteness of the measurements. Inverse problems are ubiquitous in applications from bio-medical, science and engineering to security screening and industrial process monitoring. The challenges span from the analysis to efficient numerical solution. This conference will bring together mathematicians and statisticians, working on theoretical and numerical aspects of inverse problems, as well as engineers, physicists and other scientists, working on challenging inverse problem applications. We welcome industrial representatives, doctoral students, early career and established academics working in this field to attend. Topic list: • Imaging • Inverse problems in partial differential equations (Memorial Lecture for Slava Kurylev) • Model and data driven methods for inverse problems • Optimization and statistical learning • Statistical inverse problems Invited Speakers Julie Delon (MAP5, Paris Descartes University) Markus Haltmeier (University of Innsbruck) Mike Hobson (University of Cambridge) Matti Lassas (University of Helsinki) Gabriel Peyre (DMA, École Normale Supérieure) Michael Unser (EPFL)

Call for Papers Papers will be accepted for the conference based on a 100 word abstract for oral or poster presentation. We welcome abstracts to be submitted by 1 June via https://my.ima.org.uk. Please indicate whether your title is intended for oral “presentation” or “poster” presentation. Please send your abstracts in plain text format (no equations). Note: If you are an IMA Member or you have previously registered for an IMA conference, then you are already on our database. Please “request a new password” using the email address previously used, to log in.

This workshop, sponsored by AIM, EPSRC, and the NSF, will introduce participants to the L-functions and Modular Forms Database (LMFDB) as a tool for research and teaching.

The LMFDB contains a wealth of information on L-functions, modular forms of several types, elliptic curves and genus 2 curves, number fields, and much more. In addition to detailed information about individual objects, the LMFDB also includes information about connections between objects, including the connections described by the Langlands Program.

The workshop will involve a mixture of demonstrations, explorations, discussions about mathematical content, and discussions about the future of the LMFDB.

The goal of this school is to give a detailed and example-based introduction, accessible to PhD students and post-docs in the field, to the Emerton-Gee stack: its construction, its properties and some of its applications.

In this school, we intend to understand connections between the arithmetic theory of modular forms and new developments in p-adic Hodge theory that grew from the breakthrough work of Peter Scholze on perfectoid spaces (see P. Scholze "Perfectoid spaces" Publ. Math. de l’IHES 116 (2012)).

p-adic methods play a key role in the study of arithmetic properties of modular forms. This theme takes its origins in Ramanujan congruences between the Fourier coeffcients of the unique eigenform of weight 12 and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number B12. After the work of Deligne on Ramanujan's conjecture it became clear that congruences between modular forms reflect deep properties of corresponding p-adic representations. The general framework for the study of congruences between modular forms is provided by the theory of p-adic modular forms developed in fundamental papers of Serre, Katz, Hida and Coleman (1970's-1990's).

p-adic Hodge theory was developed in pioneering papers of Fontaine in 80’s as a theory classifying p-adic representations arising from algebraic varieties over local fields. It culminated with the proofs of Fontaine's de Rham, crystalline and semistable conjectures (Faltings, Fontaine-Messing, Kato, Tsuji, Niziol,...). In order to classify all p-adic representations of Galois groups of local fields, Fontaine (1990) initiated the theory of (φ, Г)-modules. This gave an alternative approach to classical constructions of the p-adic Hodge theory (Cherbonnier, Colmez, Berger). The theory of (φ, Г)-modules plays a fundamental role in Colmez's construction of the p-adic local Langlands correspondence for GL2. On the other hand, in their famous paper on L-functions and Tamagawa numbers, Bloch and Kato (1990) discovered a conjectural relation between p-adic Hodge theory and special values of L-functions. Later Kato discovered that p-adic Hodge theory is a bridge relating Beilinson-Kato Euler systems to special values of L-functions of modular forms and u sed it in his work on Iwasawa-Greenberg Main Conjecture. One expects that Kato’s result is a particular case of a very general phenomenon.

The mentioned above work of Scholze represents the main conceptual progress in p-adic Hodge theory after Fontaine and Faltings. Roughly speaking it can be seen as a wide generalization, in the geometrical context, of the relationship between p-adic representations in characteristic 0 and characteristic p provided by the theory of (φ,Г)-modules. As an application of his theory, Scholze proved the monodromy weight conjecture for toric varieties in the mixed characteristic case. On the other hand, in a series of papers, Scholze applied his theory to the study of the cohomology of Shimura varieties. In particular to the construction of mod p Galois representations predicted by the conjectures of Ash (see P. Scholze “On torsion in the cohomology of locally symmetric space" (Ann. Of Math. 182 (2015)). Another striking application of this theory is the geometrization of the local Langlands correspondence in the mixed characteristic case. Here the theory of Fontaine—Fargues plays a fundamental role.

This goal of the proposed summer school is twofold:

1. Give an advanced introduction to Scholze's theory.
2. To understand the relation between perfectoid spaces and some aspects of arithmetic of modular (or, more generally, automorphic) forms such as representations mod p, and lifting of modular forms, completed cohomology, local Langlands program and special values of L-functions.

We wish to bring together experts in the area of arithmetic geometry that will felicitate future research in the direction. We strongly encourage participation of young researchers.

There will be about 15 mini-courses of introductory nature, related to the Geometry-Topology research subjects. Graduate students, recent Ph.D.s and under-represented minorities are especially encouraged to the summer school. Partial financial support is available.

It is a pleasure to announce the meeting:

Algebraic Combinatorics in Genova

which will take place in Genova, Italy, from the 11th to the 13th of September, 2019.

Invited speakers:

Bruno Benedetti (University of Miami)

Riccardo Biagioli (Université de Lyon)

Francesco Brenti (Università di Roma Tor Vergata)

Fabrizio Caselli (Università di Bologna)

Giulia Codenotti (Freie Universität Berlin)

Alessio D'Ali (University of Warwick)

Emanuele Delucchi (Université de Fribourg)

Lukas Katthän (Goethe-Universität, Frankfurt am Main)

Christian Krattenthaler (Universität Wien)

Martina Juhnke-Kubitzke (Universität Osnabrück)

Martina Lanini (Università di Roma Tor Vergata)

Luca Moci (Università di Bologna)

Leonardo Patimo (Albert-Ludwigs-Universität Freiburg)

Volkmar Welker (Philipps-Universität Marburg)

More information can be found below and on the conference website:

https://sites.google.com/view/alg-comb-genova-2019/home

The Organising Commitee

Aldo Conca Emanuela De Negri Mario Marietti

Invited Speakers Prof Michael Bruen, University College Dublin, Ireland Prof Guoqing Wang, Nanjing Hydraulic Research Institute, China

As in the recent publication of the IPCC AR5 report, flood risks have been again highlighted as one of the most perceivable indicators when it comes to the understanding of climate change impact. Research on flood risk has never been ampler, yet we still witness the growing causalities of severe flooding around globe on a yearly basis. The stark resemblance of the consequences between the two catastrophic flooding events in both Japan and Indonesia 2018, exposes a bitter reality that there are still knowledge gaps where both the academia and the practitioners need to fill, with no exception for even the developed world.

Continuing from the successful series of previous IMA Conferences on Flood Risk, the IMA is planning its 4th International Conference on this topic in 2019 to bring together engineers, mathematicians and statisticians working in the field. The conference intends to be used as a forum for participants to meet and exchange their views on important technical issues, new and emerging methods and technologies in assessing flood risks in a world that is being altered by the climate change and moving towards an uncertain future. The emphasis will be on new developments in mathematical modelling methods, statistical techniques in assessing flood risks, especially on quantification of flood risks with a nonstationary climate and modelling uncertainty. Methods and application of assimilating new data from cutting-edge new type of observations will be among the topics for discussion.

The conference will be of interest to flood defence practitioners; flood defence managers; statisticians, mathematicians, civil engineers.

As in the recent publication of the IPCC AR5 report, flood risks have been again highlighted as one of the most perceivable indicators when it comes to the understanding of climate change impact. Research on flood risk has never been ampler, yet we still witness the growing causalities of severe flooding around globe on a yearly basis. The stark resemblance of the consequences between the two catastrophic flooding events in both Japan and Indonesia 2018, exposes a bitter reality that there are still knowledge gaps where both the academia and the practitioners need to fill, with no exception for even the developed world.

Continuing from the successful series of previous IMA Conferences on Flood Risk, the IMA is planning its 4th International Conference on this topic in 2019 to bring together engineers, mathematicians and statisticians working in the field. The conference intends to be used as a forum for participants to meet and exchange their views on important technical issues, new and emerging methods and technologies in assessing flood risks in a world that is being altered by the climate change and moving towards an uncertain future. The emphasis will be on new developments in mathematical modelling methods, statistical techniques in assessing flood risks, especially on quantification of flood risks with a nonstationary climate and modelling uncertainty. Methods and application of assimilating new data from cutting-edge new type of observations will be among the topics for discussion.

The conference will be of interest to flood defence practitioners; flood defence managers; statisticians, mathematicians, civil engineers.

Invited Speakers Prof Michael Bruen, University College Dublin, Ireland Prof Guoqing Wang, Nanjing Hydraulic Research Institute, China

We look forward to contributions drawing upon diverse conceptual bases, including reliability theory, statistical analysis, mathematical modelling of flood flows, forecasting systems and assimilation techniques, modelling of flood risk communication and decision making, and of course mathematics. Papers will be organised under four principal themes: • Climate change, hydro-climatic extremes and non-stationarity • Computational methods in flood flow modelling and the emerging AI techniques • Emerging observation technology and data assimilation method for flood modelling • Mathematical methods for evaluating flood risk perception, communication and human reaction Contributions that span these while addressing the overall conference theme are most welcome. Contributed papers will be accepted on the basis on an abstract (300-500 words) which should be submitted via https://my.ima.org.uk/ by 3 May 2019. If you are an IMA Member or you have previously registered for an IMA conference, then you are already on our database. Please “request a new password” using the email address previously used, to log in.

International Workshop on Applied Nonlinear Analysis (IWANA2019)

September 12-14, 2019, The Tide Hotel, Bangsean, Chonburi, Thailand

Organizing Committee of IWANA is pleased to invite you to the 2nd International Workshop on Applied Nonlinear Analysis.

The purpose of the workshop is to bring together leading experts and researchers in nonlinear analysis, in particular, fixed point theory and to assess new developments, ideas, and methods in this important and dynamic field. A special emphasis will be put on applications in related areas, as well as other sciences, such as the natural sciences, medicine, economics, and engineering.

The first meeting of this series, IWNAA 2018 was successfully done in Granada, Spain:

https://internationalworkshoponnonlinearanalysis.wordpress.com/fo

We are looking forward to welcoming you in Bangsean Chonburi, Thailand.

The annual British Topology Meeting will this year be held at the University of Warwick. Speakers include:

• Keiran Fleming (University of Leicester)
• Jelena Grbic (University of Southampton)
• Lennart Meier (University of Utrecht)
• Luca Pol (University of Sheffield)
• Constanze Roitzheim (University of Kent)
• Bernadette Stolz (University of Oxford)
• James Walton (University of Glasgow)
• Ittay Weiss (University of Portsmouth)
• Ana Garcia-Pulido (University of Liverpool)

An autumn school on computations in motivic homotopy theory with lecture series by Marc Hoyois, Oliver Röndigs, Kirsten Wickelgren, and Paul Arne Østvær. Please register at the school's website. Limited financial support is available.

This is part of the annual school organized by the Institute of Mathematics and Related Areas (IMCA) in various fields of Mathematics, in which invited specialists meet at IMCA to share their research work through several mini-courses and/or conferences.

The present IMCA School is devoted to the research progress on algebraic cycles and related topics in algebraic and arithmetic geometry.

The venue of the School is Instituto de Matematica y Ciencias Afines. Calle Los Biologos 245 - Urb. San Cesar - Primera Etapa La Molina, Lima 12, Peru

Invited speakers:

• Fernando Cukierman (Universidad de Buenos Aires, Argentina),
• Lucas Das Dores (University of Liverpool, United Kingdom),
• Richard Gonzales (Pontificia Universidad Catolica del Peru, Peru),
• Sergey Gorchinskiy (Steklov Mathematical Institute of RAS, Russia),
• Vladimir Guletskii (University of Liverpool, United Kingdom),

For additional information, please contact: escuela7@imca.edu.pe

We would like to study the definition of sites linked to schemes in finite chararcteristic. In the recent years such a study has merged several techniques: from anaytic spaces to perfectoids to the developing of various p-adic cohomological theories: syntomic, rigid, overcongent ones. We aim to gather in Padova some of the best known experts in the field. It would be also an opportunity to celebrate the legacy of the work of BERNARD LE STUM on all these subjects.

First iteration of a recurring one-day number theory conference, especially targeting faculty and students at nearby universities.

The planned topics include recent advances in ramification of â„“-adic sheaves, study of irregular holonomic D-modules in higher dimensions, irregular Hodge theory, exponential motives, companions, finiteness results for local systems, etc. There are well-known analogies between wild ramification in characteristic p and irregular singularities of meromorphic differential equations, and one of our aims is to bring experts in these and related areas together.

he workshop Women in Geometry and Topology is an endeavour organized by the GEOMVAP research group at UPC and financed under the AGAUR project 2017SGR932.

The group GEOMVAP focuses in Geometry and Topology in the broad sense and its applications to several topics suchs as Celestial Mechanics, Control Theory, Mathematical Physics, Phylogenetics and Robotics. GEOMVAP promotes, in particular, Responsible, Research and Innovation within the framework of Horizon 2020. Among the RRI initiatives we strive for gender equality, public engagement, science communication and the visibility of women in Science and Society.

The Workshop Women in Geometry and Topology will feature several plenary talks by top female mathematicians and some contributed talks (contributed by speakers of any gender identity).

There will be two public lectures by Marta Macho (Scientific Culture Chair UPV/EHU, RSME Medal 2015, Emakunde Equality Prize 2016) and Carme Torras (Narcís Monturiol Medal 2000) addressed to the general public (you don't need to be a mathematician to follow them you just need to be curious!). A panel (open to the public) will also be organized in order to discuss the situation of women in mathematics, the gender gap and strategies for breaking the glass-ceiling inside and outside academia. Other complementary activities will be announced in due time.

The publication of "extended abstracts" from the congress is expected in the prestigious Springer-Birkhäuser Research Perspectives CRM Barcelona collection, within the Trends in Mathematics series.

MESIGMA is a symposium that brings together researchers from several fields of mathematics and computer science.

As the scientific framework for MESIGMA is widely set, we consider it to be an incentive environment for finding the “shared territory” of diverse scientific disciplines. Therefore, MESIGMA is intended to be a little, pleasant socializing with conversations about mathematics and binding different scientific ideas.

Weekend regional conference in number theory and arithmetic geometry, featuring two mini-courses and additional research lectures.

The aim of this workshop is to provide an overview of recent developments in theory of p-adic L-functions associated to automorphic representations, covering both the construction of p-adic L-functions, and their relations to Euler systems in Galois cohomology via regulator maps. The workshop will consist of three mini-courses, aimed at younger researchers, and more specialised individual lectures.

There will be three mini-courses, each consisting of four lectures, on the following topics:

Construction of p-adic L-functions for automorphic forms on GL(2n), using the automorphic modular symbols introduced in work of Dimitrov Construction of p-adic L-functions for GSp(4), using Pilloni’s higher Hida theory evalulation of global cohomology classes under the syntomic regulator, using the methods of Darmon—Rotger. The other talks will explore connections of these topics with other related areas of current research, such as Iwasawa theory, the theory of Hecke varieties and the theory of L-invariants.

Modularity. Until relatively recently, the celebrated Taylor--Wiles method for establishing the automorphy of Galois representations carried several significant limitations. First, the method applied only to Galois representations expected to come from cohomological automorphic forms of regular weight. For classical modular forms this excludes the case of weight 1 forms. Second, the locally symmetric space in whose cohomology the automorphic form is expected to arise was required to be an algebraic variety (a Shimura variety). This excludes for instance the case of elliptic curves over imaginary quadratic fields, where the locally symmetric space is 3-dimensional, and so cannot even admit a complex structure. Finally, in the absence of results towards Serre's conjecture on the modularity of mod p Galois representations, the Taylor--Wiles method generally only establishes the potential automorphy of Galois representations, i.e., automorphy after a finite base change.

In a major breakthrough, Calegari--Geraghty have introduced a derived version of the Taylor--Wiles method which has the potential to remove the first two of these restrictions. To realize the potential of the Calegari--Geraghty method requires overcoming a number of significant challenges in the theory of automorphic forms and the arithmetic of Shimura varieties. For instance one needs to know the existence of Galois representations attached to torsion classes in the cohomology of locally symmetric spaces, as well as strong forms of local-global compatibility for those representations. Scholze [SchTorsion] (and independently Boxer [Boxer] in some special cases) has addressed the former, and work of Cariani--Scholze [CS] on the vanishing of torsion in the cohomology of non-compact Shimura varieties has made progress towards the latter. These advances already have remarkable applications, such as the proof of potential modularity of elliptic curves over imaginary quadratic fields, as well as the Sato--Tate conjecture for such curves [tenauthor].

In addition to examining these many important developments, the workshop will contemplate possible future improvements to the Calegari--Geraghty method, such as may come from incorporating the derived deformation theory of Galatius--Venkatesh [GV]. We will also explore the prospects for proving actual (rather than potential) modularity of elliptic curves over some CM fields. Another expected topic is work in progress by Boxer--Calegari--Gee--Pilloni on the potential automorphy of abelian surfaces, using the Calegari--Geraghty method, as well as Pilloni's ``higher Hida theory'' for coherent cohomology of Shimura varieties [Pilloni].

Moduli of Galois representations. In ongoing work, Emerton and Gee are constructing moduli stacks which parameterize p-adic Galois representations arising from p-adic local fields. In the classical deformation theory of Galois representations, one considers formal families of deformations of a fixed mod p Galois representation; in contrast, the Emerton--Gee stacks admit non-constant families of mod p Galois representations, raising the possibility of arguing by interpolating between them. Furthermore, thanks to the global geometry of these spaces one has more algebro-geometric tools at one's disposal to study them.

The Emerton--Gee moduli stacks are built out of moduli spaces of integral p-adic Hodge theory data. Several incarnations of p-adic Hodge theory play a role in constructing and understanding these spaces, including Breuil-Kisin modules, Wach modules, and Tong Liu's (ϕ,Gˆ)-modules. Understanding how these different theories interact should a play an important role in the further development of this field. There remains many open questions about these stacks. What are the components of the special fiber? Are they normal? Cohen--Macaulay? What kind of singularities do they have? What is the structure of the line bundles/coherent sheaves on these spaces? Answers to these questions would have broad implications for modularity and the p-adic Langlands program.

The geometry of the Emerton--Gee stacks is closely linked to the Breuil--M\'ezard conjecture, which first arose in the context of attempt to generalize the Taylor--Wiles method. This conjecture measures the complexity of local Galois deformation rings (i.e., the versal deformation rings at closed points of Emerton--Gee stacks) in terms of the modular representation theory of GLn;\ understanding the geometry of local deformation spaces is essential for proving modularity lifting theorems. The Breuil--M\'ezard conjecture is in turn closely connected to the so-called weight part of Serre's conjecture, which can be viewed as a step towards the conjectural p-adic local Langlands correspondence.

For example, Caraiani--Emerton--Gee--Savitt [CEGS] are able to use known results about the geometric Breuil--M\'ezard conjecture and the weight part of Serre's conjecture for GL2 to analyze the irreducible components of certain Emerton--Gee stacks and relate them to the modular representation theory of GL2. The moduli stack perspective has also already played a role in the proof of the weight part of Serre's conjecture in generic situations in higher dimensions [LLLM1, LLLM2] and in on-going work of Emerton--Gee on the existence of crystalline lifts of mod p representations.

Despite considerable progress (e.g.\ [Herzig, GHS]), there still is no unconditional statement of the weight part of Serre's conjecture beyond the case of GL2. The Emerton-Gee moduli stack may be helpful for understanding this conjecture, as illustrated by the work of [CEGS]. One objective of the workshop will be to formulate an unconditional weight part of Serre's conjecture in terms of the Emerton-Gee stack, and to understand how such a conjecture relates to modular representation theory and to the Breuil-M\'ezard conjecture.

Finally, there are already tantalizing hints, for instance the work of [EGS] proving Breuil's local-global compatibility conjecture for types in the p-adic Langlands program, that the Emerton--Gee moduli stacks will play an important role in future developments on the modularity of Galois representations. However, this avenue is as yet largely unexplored. Another goal of this workshop is to bring together leading experts involved in these two strands of research in order to explore the possible synergies between them.

Local models for Galois deformation spaces. Although the two flavors of moduli spaces (Shimura varieties, Galois deformation spaces) that we have contemplated in this proposal are rather different, Kisin [Kis09a] observed that there is a surprising and fundamental relation between them:\ namely, their singularities are both modeled by relatively simpler moduli spaces called local models of Shimura varieties. These local models have been studied extensively in the context arithmetic of Shimura varieties, so that much is known about their geometry. Kisin's observation led to improved modularity lifting theorems, which in turn played a key role in the eventual proof of Serre's original conjecture for GL2/Q.

Beyond dimension two, in order to study regular weight Galois deformation spaces, there is an additional condition which comes from a subtle analogue of Griffiths transverality in p-adic Hodge theory. In [LLLM1,LLLM2], Le--Le Hung--Levin--Morra give explicit presentations for certain potentially crystalline deformation rings with Hodge--Tate weights (0,1,2) by studying this Griffiths transversality condition, and as an application prove cases of the weight part of Serre's conjecture and other related conjectures in dimension three. In higher dimension, the connection with local models is weaker and does not capture the Griffiths transversality condition. Ongoing work of Le--Le Hung--Levin--Morra constructs local models for Galois deformation spaces in generic situations and will shed light on the structure of generic parts of the Emerton-Gee moduli stack. Further, there are mysterious connections between these local models and objects in geometric representation theory which have not yet been explored.

There are a number of parallels between the mod p and p-adic stories. A striking example of this is Breuil--Hellmann--Schraen's recent proof of a Breuil--M\'ezard type conjecture for locally analytic representations, which furthermore leads to a proof of the locally analytic socle conjecture of Breuil [BHS]. They study the geometry of a p-adic family of Galois representations called the trianguline variety. In another parallel to the mod p picture, they create a link between the geometry of these p-adic families to objects in geometric representation theory.

By sharing these new developments broadly with other experts in the field, the workshop aims to spur further development of connections between moduli of Galois representations and the geometry of (generalized) local models, and of parallels between the p-adic and mod p settings; and to contemplate what the implications might be for the geometry of Emerton--Gee stacks.

We organize an Oberwolfach Seminar on Topological Cyclic Homology and Arithmetic. The purpose of the seminar is to introduce the higher algebra refinements of determinant and trace, namely, algebraic K-theory and topological cyclic homology, along with their budding applications in arithmetic geometry and number theory. In particular, we will use these ingredients to build Clausen's Artin map from K-theory of locally compact topological R-modules to the dual of his Selmer K-theory of R, and explain that for R a finite, local, or global field, this implies Artin reciprocity. If you wish to participate, please follow the instructions described here to register at ag@mfo.de by August 11, 2019.

The symbiotic relationship between the illustration of mathematics and mathematical research is now flowering in algebra and number theory. This workshop aims to both showcase and develop these connections, including the development of new visualization tools for algebra and number theory. Topics are wide-ranging, and include Apollonian circle packings and the illustration of the arithmetic of hyperbolic manifolds more generally, the visual exploration of the statistics of integer sequences, and the illustrative geometry of such objects as Gaussian periods and Fourier coefficients of modular forms. Other topics may include expander graphs, abelian sandpiles, and Diophantine approximation on varieties. We will also focus on diagrammatic algebras and categories such as Khovanov-Lauda-Rouquier algebras, Soergel bimodule categories, spider categories, and foam categories. The ability to visualize complicated relations diagrammatically has led to important advances in representation theory and knot theory in recent years.

NTS-LA is a biannual regional number theory theory conference located in Los Angeles. While each meeting with feature two plenary talks by faculty and one plenary talk from a graduate student from outside of Southern California, the majority of talks will consist of 20 minute contributed talks. The purpose of these meetings is to establish a community of people interested in number theory in Southern California, to allow faculty at institutions that do not have funds for regular seminars to attend high-quality research talks, and to provide a friendly environment for students and faculty to present their research.

The goals of this conference include facilitating interaction between modular forms and arithmetic geometry researchers, providing graduate students with an opportunity to present their work, and strengthening networks for mathematicians from underrepresented groups including but not limited to women.

On the evening of the second day of the conference (November 2), we plan to have a reception with a public lecture by Professor Raman Parimala, followed by a showing of the CWM film “Journeys of Women in Mathematics.” Talks and registration will be held at Emory University's Mathematics and Science Center (rooms E208, W201, and the Atrium).

The recent results of Green and Tao on the existence of arbitrarily long arithmetic progressions of prime numbers have showed the strength of the interactions between combinatorics, number theory and dynamical systems. Other advances, like the results of Bourgain, Green, Tao, Sarnak and Ziegler on the randomness principle for the Möbius function, the resolution of the Gelfond conjectures concerning the sum of digits of prime and square numbers, as well as those of Golston, Pintz and Yildirim and then Zhang and Maynard on small gaps between primes, the recent results of Pintz on the existence of arbitrarily long arithmetic progressions of generalized twin prime numbers show the vitality of this domain of research. The difficulty of the transition from the representation of an integer in a number system to its multiplicative representation (as a product of prime factors) is at the source of many important open problems in mathematics and computer science. The conference will be devoted to the study of independence between the multiplicative properties of integers and various ”deterministic” functions, i. e. functions produced by a dynamical system of zero entropy or defined using a simple algorithm. This area is developing very fast at international level and the conference will be an opportunity to help to develop techniques that were recently introduced to study of relations between prime numbers, polynomial sequences and finite automata, the study of the pseudorandom properties of certain arithmetic sequences and the search of prime numbers in deterministic sequences. This goal is related to several recent works by Bourgain, Green, Sarnak, Tao and others concerning the orthogonality of the Möbius function with deterministic sequences and obtaining prime number theorems for these sequences.

Visit the conference website for more info.

Zeta functions are ubiquitous objects in Number Theory and Arithmetic Geometry. They are analytic, algebraic, or combinatorial in nature. Families of zeta functions (or more generally of L-functions) naturally appear in a broad variety of active research fields e.g. au- tomorphic forms and Artin representations, Drinfeld modules, arithmetic dynamics, abelian varieties over global fields, inequities in the distribution of sequences indexed by prime num- bers or more generally by places of global fields...

The main purpose of the “Zeta functions” conference is to gather experts of the theoretical and computational branches of number theory and arithmetic geometry together with students and young researchers to have them interact and explore further the richness of the information encoded by zeta and L-functions. Our conference proposal aims at synthesizing complementary points of view coming from distant fields: the analytic approach in the classical theory of zeta and L-functions, the theory of Artin L-functions in connection with the Langlands program, zeta and L-functions coming from arithmetic geometry in the spirit of the Weil conjectures, zeta functions arising in dynamics...

One of the original aspects of the project lies in the interaction between theoretical considerations and numerical and algorithmic features for diverse families of zeta and L-functions. Rather than a meeting meant for experts in a particular topic we will put the emphasis on the exchange of ideas between people coming from related fields in Number Theory and on inviting young researchers and students to further pursue the study of these interactions that have already proven fruitful and that we believe are still very promising.

The first Algebra, Codes and Cryptography conference will be held in Dakar, Senegal on Thursday to Saturday, December 5-7, 2019. The conference aims to provide a forum for researchers from all over the world to present results and exchange ideas on topics in Non-Associative Algebra, Non-commutative Algebra, Cryptology, Coding Theory and Information Security.

The mathematical theory and practice of both cryptography and coding underpins the provision of effective security and reliability for data communication, processing and storage. This seventeenth International Conference in an established and successful IMA series on the theme of "Cryptography and Coding" solicits original research papers on all technical aspects of cryptography and coding are solicited for submission. Submissions are welcome on any cryptographic or coding-theoretic topic including, but not limited to:

• Foundational theory and mathematics;
• The design, proposal, and analysis of cryptographic or coding primitives and protocols
• Secure implementation and optimisation in hardware or software; and
• Applied aspects of cryptography and coding. The proceedings will be published in Springer's Lecture Notes in Computer Science series, and will be available at the conference.

The mathematical theory and practice of both cryptography and coding underpins the provision of effective security and reliability for data communication, processing and storage. This seventeenth International Conference in an established and successful IMA series on the theme of “Cryptography and Coding” solicits both original research papers and presentations on all technical aspects of cryptography and coding.

The programme will focus on the areas of Algebraic K-theory, Algebraic Cycles and Motivic Homotopy Theory. These are fields at the heart of studying algebraic varieties from a cohomological point of view, which have applications to several other fields like Arithmetic Geometry, Hodge theory and Mathematical Physics.

It was in the 1960s that Grothendieck first observed that the various cohomology theories for algebraic varieties shared common properties, which led him to explain the underlying kinship of such cohomology theories in terms of a universal motivic cohomology theory of algebraic varieties. The theory of Algebraic Cycles, Higher Algebraic K-theory, and Motivic Homotopy Theory are modern versions of Grothendieck's legacy. In recent years it has seen some spectacular developments, on which we want to build further.

The programme will also specifically explore the connections between the following areas:

Algebraic K-theory, Motivic Cohomology, and Motivic Homotopy Theory;
Hodge theory, Periods, Regulators, and Arithmetic Geometry;
Mathematical Physics.

For this, we shall bring together mathematicians working on different aspects of this broad area for extended periods of time, promoting exchange of ideas and stimulating further progress.

During the programme there will be four workshops. At the very beginning, there will be a workshop aimed at giving a younger generation of mathematicians an overview of and introduction to this interesting, but broad area. Later there will be a workshop for each of the three areas listed above, aimed at the latest developments and applications of that area.

The study of integer lattices serves as a bridge between number theory and geometry and has for centuries received the attention of illustrious mathematicians including Lagrange, Gauss, Dirichlet, Hermite and Minkowski. In computer science, lattices made a grand appearance in 1982 with the celebrated work of Lenstra, Lenstra and Lovász, who developed the celebrated LLL algorithm to find short vectors in integer lattices. The role of lattices in cryptography has been equally, if not more, revolutionary and dramatic, playing first a destructive role as a potent tool for breaking cryptosystems, and later as a new way to realize powerful and game-changing notions such as fully homomorphic encryption. These exciting developments over the last two decades have taken us on a journey through such diverse areas as quantum computation, learning theory, Fourier analysis and algebraic number theory.

We stand today at a turning point in the study of lattices. The promise of practical lattice-based cryptosystems together with their apparent quantum-resistance is generating a tremendous amount of interest in deploying these schemes at internet scale. However, before lattice cryptography goes live, we need major advances in understanding the hardness of lattice problems that underlie the security of these cryptosystems. Significant, ground-breaking progress on these questions requires a concerted effort by researchers from many different areas: (algebraic) number theory, (quantum) algorithms, optimization, cryptography and coding theory.

The goal of the proposed special semester is to bring together experts in these areas in order to attack some of the main outstanding open questions, and to discover new connections between lattices, computer science, and mathematics. The need to thoroughly understand the computational landscape and cryptographic capabilities of lattice problems is greater now than ever, given the possibility that secure communication on the internet and secure collaboration on the cloud might soon be powered by lattices.

This workshop is the 17th in a series of weekend workshops that bring together the mathematical community sharing interests in algebraic combinatorics, commutative algebra and combinatorial algebraic geometry.

Invited Speakers

Chris Francisco (Oklahoma State University) Ezra Miller (Duke University) Stephanie van Willigenburg (University of British Columbia) Josephine Yu (Georgia Institute of Technology) (tentative)

Call for papers:

If you would like to give a talk at this conference, please send titles and abstracts to Sara Faridi (faridi@dal.ca) by November 15, 2019.

The concept of a group is central to essentially all of modern mathematics. In Number theory and geometry, where groups take central stage in various shapes such as symmetry groups, Galois groups, fundamental groups, reflection groups and permutation groups, the conceptual unification that it provides is most strikingly illustrated. The School will help the students acquiring a good background on the Langlands program, which, after all, is about relations between symmetries in geometry, analysis and number theory. In this school, we present groups and the natural objects they act on in a variety of arithmetic and geometric contexts. Special emphasis will be given to concrete examples, and practical and computational aspects of groups and their actions will be stressed. The topics to be treated include finite fields, coding theory, covering spaces, representation theory, modular forms and Galois theory.

The Banff International Research Station will host the "Equivariant Stable Homotopy Theory and p-adic Hodge Theory" workshop in Banff from March 1 to March 06, 2020.

Algebraic topology has had a long and fruitful collaboration with algebraic geometry, with each providing techniques and problems to the other. This workshop is aimed at an exciting, evolving incarnation of this story: applications of equivariant stable homotopy to number theory. Recent work on the foundations of equivariant stable homotopy theory (starting with the Hill--Hopkins--Ravenel work on the Kervaire invariant one problem) and Lurie's development of the foundations of ''derived algebraic geometry'' now allows systematic exploration and organization of ''equivariant derived algebraic geometry''. This allows us to do ordinary algebraic geometry in commutative ring spectra.

New foundations in this area have been spectacularly applied to phenomena seen in the trace methods approach to computing algebraic K -theory. For instance, although the theory of equivariant commutative ring spectra was described decades ago, few of the subtleties in the theory were understood or explored. The modern approaches to computing algebraic K-groups step through equivariant commutative ring spectra via the natural S1-action on topological Hochschild homology. Ongoing and transformative work by Bhatt--Morrow-Scholze in p-adic Hodge theory uses cyclotomic spectra and therefore subtle equivariant information. This workshop, at the vanguard of work in this area, seeks to bring together experts in algebraic topology, (derived) algebraic geometry, and number theory to explore these exciting new connections.

Organizing Committee: Caucher Birkar (University of Cambridge), Christopher Hacon (the University of Utah), Chenyang Xu (M.I.T.) with help from Jingjun Han (Johns Hopkins University).

Principal Japanese Organizers: Keiji Oguiso (University of Tokyo), Shunsuke Takagi (University of Tokyo).

This one year long program at Johns Hopkins University will feature 3 graduate-level courses, one conference, three Kempf lectures, three Monroe H. Martin lectures, several colloquiums and weekly seminars.

Tentative schedule for the conference: March 16--22, 2020.

Arithmetic groups provide a fruitful link between various areas, such as geometry, topology, representation theory and number theory. Methods from geometry and topology hinge on the fact that arithmetic groups are lattices in Lie groups, whereas the theory of automorphic forms establishes a connection to representation theory and number theory. This interplay is especially intriguing in the setting of hyperbolic 3-manifolds. Indeed many conjectures in 3-manifold theory tend to be much more accessible for hyperbolic 3-manifolds whose fundamental groups are arithmetic, and conversely such manifolds provide the simplest set-up in which some of the most exciting new phenomena in the Langlands program can be studied. This conference will bring together researchers with various backgrounds around links between number theory and 3-manifolds. Central topics of the conference are the cohomology of arithmetic groups, the relation between torsion and L²-torsion, profinite invariants of 3-manifolds, and number theoretic ramifications.

Conference on moduli spaces.

The elucidation of the way in which the additive and multiplicative structure of the integers are intertwined with one another is one of the most important and central themes in number theory. In August 2012, Shinichi Mochizuki (the proposer and chief organizer of the present RIMS Research Project) released preprints of a series of papers concerning "Inter-universal Teichmüller Theory", a theory that constitutes an important advance with regard to elucidating this intertwining. Moreover, the proof of the "ABC Conjecture", which follows as a consequence of the theory, attracted worldwide attention.

The present RIMS Research Project seeks to bring together various researchers not only from the "inter-universal Teichmüller theory community", but also researchers interested in various forms of mathematics related to inter-universal Teichmüller theory, and to provide all such researchers an opportunity to engage in lively discussions concerning the various developments.

Periods occur in various branches of mathematics and as the title of our conference indicates, their study intertwines arithmetic, Diophantine analysis, differential equations, and algebraic geometry. Many interesting results have been proved in recent years and many challenging problems on periods are still open. The aim of our conference is to bring together specialists who cover all these different points of view and their ramifications, with special attention towards possible applications to broader areas of the techniques developed in the study of periods and their realizations.

Yves André has contributed in many ways to this ongoing adventure and this conference will not only be the opportunity to listen to a broad range of recent developments in mathematics around the topic of periods, but also to celebrate his 60th birthday.

The purpose of the workshop is to bring together leading, as well as early career researchers on arithmetic statistics and on algorithmic aspects of algebra and number theory, with the aim of fostering collaborations within and between these communities, and to offer early career researchers the opportunity to get a broad overview of the most recent achievements and of the most pressing problems in these fields. Another purpose is to celebrate the mathematics of Hendrik W. Lenstra Jr. on the occasion of his 71st birthday.

Recent trends, such as the NIST initiative to standardize post-quantum cryptography, point to large-scale adoption of lattice-based cryptography in the near future. There has consequently been a great deal of attention devoted to making various aspects of lattice-based cryptography practical.

This workshop will focus on questions related to the transition of lattice-based cryptography from theory to practice including the hardness of lattice problems arising from algebraic number theory, and algorithmic solutions to practical issues such as time and space-efficiency, side-channel resistance, and ease of hardware implementations.

The workshop will bring together theoretical and applied cryptographers and computational number-theorists, and will also encourage interaction amongst different communities within and outside cryptography.

Over the last 33 years, the Annual Workshop on Automorphic Forms and Related Topics has remained a small and friendly conference. Those attending range from students to new PhD's to established researchers. For young researchers, the conference has provided support and encouragement. For accomplished researchers, it has provided the opportunity to mentor as well as a forum for exchanging ideas.

The workshop has become internationally recognized for both its high-quality research talks and its supportive atmosphere for junior researchers. Participants present cutting-edge research in all areas related to automorphic forms. These include mock modular forms, Maass wave forms, elliptic curves, Siegel and Jacobi modular forms, special values of L-functions, random matrices, quadratic forms, applications of modular forms, and many other topics.

In addition to research talks, the workshop has, in the past years, featured panel discussion sessions on the topics of grant writing, mentoring and research partnerships, REUs and outreach, and opportunities for international collaborations. Based on the success of these sessions, we plan to have similar panel sessions this year as well.

This year, the 2020 Automorphic Forms Workshop will be held in Moab, Utah at the Moab Arts and Recreation Center. Moab, in southern Utah, is near Arches and Canyonlands National Parks and other scenic landmarks. The Workshop will be organized and hosted by Brigham Young University.

This workshop is one of the workshops of a special RIMS year "Expanding Horizons of Inter-universal Teichmüller Theory". The workshop will review fundamental developments in several branches of anabelian geometry, as well as report on recent developments. The list of speakers includes major contributors to anabelian geometry and birational anabelian geometry. Anabelian geometry, together with higher class field theory and the Langlands correspondences, is one of three generalisations of class field theory.

The Casa Matemática Oaxaca (CMO) will host the "Advances in Mixed Characteristic Commutative Algebra and Geometric Connections" workshop in Oaxaca, from June 7 to June 12, 2020.

One of the big ideas in modern mathematics is that integers (like 1, 2, 3, 4, 5, ...) in many formal ways behave similarly to polynomial equations (like y = x^2, which defines the parabola). Frequently, and perhaps surprisingly, many questions in mathematics are easier to study for polynomials than for integers. Hence intuition and results for polynomials can tell us about the integers. Commutative algebra lives at the intersection of both perspectives, and one fundamental object of study is polynomials with integer coefficients, this is called the mixed characteristic case. Recently, Yves Andre proved a long standing open conjecture in commutative algebra in this mixed characteristic setting, relying on constructions of Scholze (and then Bhatt gave a simplified proof of the same conjecture).

This workshop aims to foster and discuss these and other recent tools, to study some remaining open problems in mixed characteristic. The workshop will bring together a diverse group of researchers from different fields, such as commutative algebra, algebraic geometry, and number theory.

CTNT 2020 will take place during the week of June 8th-14th, 2020 (summer school June 8-12, and research conference 12-14), at University of Connecticut.

Combinatorial anabelian geometry concerns the reconstruction of scheme- or ring-theoretic objects from more primitive combinatorial constituent data. In this sense, it is closely philosophically related to inter-universal Teichmüller theory.

The purpose of the present workshop is to expose fundamental, introductory aspects of combinatorial anabelian geometry, as well as more recent developments related to the Grothendieck-Teichmüller group and the absolute Galois groups of number fields and mixed-characteristic local fields.

The workshop will also treat results concerning the "resolution of nonsingularities" of hyperbolic curves over mixed-characteristic local fields, such results are closely related to combinatorial anabelian geometry over mixed-characteristic local fields.

Andrew Ranicki and his theory of algebraic surgery played a central role in linking manifold theory, algebraic K-theory, and its close cousin L-theory. These areas have seen great developments and advances in the last decade from distinct research communities. This workshop will bring together mathematicians working on the topology of high-dimensional manifolds and their automorphisms with those working on the algebraic K-theory (and its cousins hermitian K-theory and L-theory) of rings and ring spectra, in order to share recent progress in these areas and kindle a fresh interaction between them.

See conference website

The goal of this summer school is to introduce participants to modern tools used in the study of fixed point theory in algebraic topology and homotopy theory. The workshop will be centered around mini-courses whose goal will be to introduce and apply tools such as categorical approaches to duality, spectra and trace methods in algebraic K-theory to the study of classical fixed point theory.

The intended audience for this summer school should be familiar with the material in Hatcher (except the appendices). This reflects our goal that the school be accessible to second and third year students with an interest in algebraic topology from any PhD granting institution. The school will be structured around a few mini-courses which will run in an active-learning style.

Scientific Leaders

Jonathan Campbell, Vanderbilt University

Inbar Klang, École Polytechnique Fédérale de Lausanne

Kate Ponto, University of Kentucky

Cary Malkiewich, Binghamton University

John Lind, California State Chico

Sarah Yeakel, University of Maryland

Inna Zakharevich, Cornell University

This will be a workshop in arithmetic and algebraic geometry, similar to the previous iteration (https://stacks.github.io/2017/). The intended participant is a graduate student, or a postdoc, or even a senior researcher. You will work on a single topic in a small group together with a mentor for a week with the aim of producing a text that will be considered for inclusion in the Stacks Project. Part of this process will be seeing how one builds new theory from the foundations. There will also be one or two talks per day covering advanced topics in arithmetic or algebraic geometry.

The Stacks project workshop will have some optional activities you won't see at other workshops. Adding references to and finding mistakes in the Stacks Project (and fixing them) as well as activities related to LaTeX use, Git, and GitHub. Overall these will be aimed at helping you contribute efficiently to the Stacks Project.

This program is focused on the two-way interaction of logical ideas and techniques, such as definability from model theory and decidability from computability theory, with fundamental problems in number theory. These include analogues of Hilbert's tenth problem, isolating properties of fields of algebraic numbers which relate to undecidability, decision problems around linear recurrence and algebraic differential equations, the relation of transcendence results and conjectures to decidability and decision problems, and some problems in anabelian geometry and field arithmetic. We are interested in this specific interface across a range of problems and so intend to build a semester which is both more topically focused and more mathematically broad than a typical MSRI program.

The aim of the workshop is to discover how the problems in number theory and algebraic geometry arising from the Hilbert’s tenth problem for rationals interact with the ideas and techniques in mathematical logic, such as definability from model theory and decidability and degree-theoretic complexity from computability theory. This interaction includes various analogues of Hilbert’s tenth problem and related questions, focusing on the connections of algebraic, number-theoretic, model-theoretic, and computability-theoretic properties of structures and objects in algebraic number theory, anabelian geometry, field arithmetic, and differential algebra.

Our workshop will focus research efforts on the interaction of number-theoretic questions with questions of decidability, definability, and computability, bringing together researchers approaching these questions from various sides to work on the core issues. This Introductory Workshop will serve as the introductory event of the MSRI semester program and is designed to introduce the basic structures and ideas of the different communities, and to highlight problems of active current interest.

This workshop will be directed primarily toward beginning learners of inter-universal Teichmüller theory, who are, nonetheless, familiar with basic aspects of arithmetic geometry. The purpose of the workshop is to further dissemination activities concerning inter-universal Teichmüller theory by concentrating on the exposition of the fundamental ideas underlying theory, as well as of certain basic technical results and notions that are used in the theory.

The approach to exposing such fundamental ideas will differ substantially from that of previous workshops on inter-universal Teichmüller theory in that it will focus on discussing various fundamental issues, as well as misunderstandings and questions, that arise in the course of studying the theory. Such discussions will build on the extensive experiences and know-how, with regard to exposing the theory, of researchers who have already acquired a thorough understanding of the theory.

The Banff International Research Station will host the "Arithmetic Aspects of Algebraic Groups" workshop in Banff from September 6 to September 11, 2020.

The investigation of arithmetic groups has been an active and important area of mathematical research ever since it arose in the work of Gauss, Klein, Poincare, and other famous mathematicians of the 18th and 19th centuries. New points of view have recently led to progress on classical problems, opened new directions of inquiry, and revealed unexpected connections with other areas of mathematics. The workshop will bring together experts in the area, researchers in related fields, and young mathematicians who wish to learn about the most recent advances and the most promising directions for the future of the field.

This workshop is one of four workshops of special RIMS year "Expanding Horizons of Inter-universal Teichmüller Theory".

This workshop will differ from previous workshops on inter-universal Teichmüller theory in that it will be directed primarily toward advanced learners of the theory, as well as researchers who have already acquired a thorough understanding of the theory.

The talks of the workshop will focus on issues related to the exposition or formulation of the theory from a more advanced point of view, as well as on recent new research developments related to the theory.

The GTA 2020 conference brings together world class researchers in mathematics. Its main objectives are to discuss recent advances in the fields of algebraic geometry, number theory and their applications, as well as to foster international collaborations on related topics.

The Banff International Research Station will host the "WIN5: Women in Numbers 5" workshop in Banff from November 15 to November 20, 2020.

Despite recent progress in gender equality in STEM fields, women continue to be underrepresented in the research landscape of many areas of mathematics, including number theory. The Women in Numbers (WIN) network was created in 2008 for the purpose of increasing the number of active female researchers in number theory. For this purpose, WIN sponsors regular conferences, taking place approximately every three years, where female scholars gather to collaborate on cutting-edge research in the field and produce publishable scientific results. The WIN workshops provide an ongoing forum for involving each new generation of junior faculty and graduate students in state-of-the-art research in number theory. They have to come be highly regarded among the broader number theory community due to the quality of research produced by these collaborations.

WIN5 is the fifth in this series of events, bringing together female number theorists at various career stages for research collaboration and mentorship. As always, the scientific program will centre on onsite collaboration on open research problems in number theory, conducted in small groups comprised of senior and junior scholars as well as graduate students. Groups will publish their initial finding in a peer-reviewed conference proceedings volume, and research partnerships formed at the WIN5 workshop are expected to last well beyond the duration of the event. WIN projects have the potential to grow into fruitful long-term research alliances that have a transforming influence on participants' careers and a significant positive impact on the research landscape in number theory. Past WIN workshop project groups have matured into highly effective research teams producing ongoing scholarly work of exceptional scientific quality.

The Casa Matemática Oaxaca (CMO) will host the "Langlands Program: Number Theory and Representation Theory" workshop in Oaxaca, from November 29 to December 04, 2020.

Langlands functoriality conjectures predict a vast generalization of the classical reciprocity laws of Class Field Theory, providing crossroads between Number Theory and Representation Theory. The conjectures are both local and global and pertain a connected reductive group and its Langlands dual group.

We aim to introduce young mathematicians in M\'exico and Latin-America to topics of current research in the Langlands Program. We will also promote the participation women and of graduate students from a diverse background in a workshop where experts in the field from across the world will gather to expand upon the frontiers of current research. In addition to research talks, there will be three courses that will also be accessible to mathematicians working in closely related fields.