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Upcoming Programs

  1. The Analysis and Geometry of Random Spaces

    Organizers: LEAD Mario Bonk (University of California, Los Angeles), Joan Lind (University of Tennessee), Steffen Rohde (University of Washington), Eero Saksman (University of Helsinki), Fredrik Viklund (Royal Institute of Technology), Jang-Mei Wu (University of Illinois at Urbana-Champaign)
    Graphisc

    This program is devoted to the investigation of universal analytic and geometric objects that arise from natural probabilistic constructions, often motivated by models in mathematical physics. Prominent examples for recent developments are the Schramm-Loewner evolution, the continuum random tree, Bernoulli percolation on the integers,  random surfaces produced by Liouville Quantum Gravity, and Jordan curves and dendrites obtained from random conformal weldings and laminations. The lack of regularity of these random structures often results in a failure of classical methods of analysis. One goal of this program is to enrich the analytic toolbox to better handle these rough structures.

    Updated on Nov 20, 2019 02:12 PM PST
  2. Complex Dynamics: from special families to natural generalizations in one and several variables

    Organizers: LEAD Sarah Koch (University of Michigan), Jasmin Raissy (Institut de Mathématiques de Bordeaux), Dierk Schleicher (Université d'Aix-Marseille (AMU)), Mitsuhiro Shishikura (Kyoto University), Dylan Thurston (Indiana University)
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    The mating of these two dendritic Julia sets is equal to the Julia set of a rational map of degree 2; that Julia set is equal to the entire Riemann sphere. Picture by Arnaud Chéritat

    Holomorphic dynamics is a vibrant field of mathematics that has seen profound progress over the past 40 years. It has numerous interconnections to other fields of mathematics and beyond. 

    Our semester will focus on three selected classes of dynamical systems: rational maps (postcritically finite and beyond); transcendental maps; and maps in several complex variables. We will put particular emphasis on the interactions between each these, and on connections with adjacent areas of mathematics. 

    Updated on Nov 20, 2019 02:12 PM PST
  3. Floer Homotopy Theory

    Organizers: Mohammed Abouzaid (Columbia University), Andrew Blumberg (Columbia University), Kristen Hendricks (Rutgers University), Robert Lipshitz (University of Oregon), LEAD Ciprian Manolescu (Stanford University), Nathalie Wahl (University of Copenhagen)
    335 image
    Illustrated by Nathalie Wahl

    The development of Floer theory in its early years can be seen as a parallel to the emergence of algebraic topology in the first half of the 20th century, going from counting invariants to homology groups, and beyond that to the construction of algebraic structures on these homology groups and their underlying chain complexes.  In continuing work that started in the latter part of the 20th century, algebraic topologists and homotopy theorists have developed deep methods for refining these constructions, motivated in large part by the application of understanding the classification of manifolds. The goal of this program is to relate these developments to Floer theory with the dual aims of (i) making progress in understanding symplectic and low-dimensional topology, and (ii) providing a new set of geometrically motivated questions in homotopy theory. 

    Updated on Oct 02, 2020 03:01 PM PDT
  4. Analytic and Geometric Aspects of Gauge Theory

    Organizers: Laura Fredrickson (University of Oregon), Rafe Mazzeo (Stanford University), Tomasz Mrowka (Massachusetts Institute of Technology), Laura Schaposnik (University of Illinois at Chicago), LEAD Thomas Walpuski (Humboldt-Universität)
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    The mathematics and physics around gauge theory have, since their first interaction in the mid 1970’s, prompted tremendous developments in both mathematics and physics.  Deep and fundamental tools in partial differential equations have been developed to provide rigorous foundations for the mathematical study of gauge theories.  This led to ongoing revolutions in the understanding of manifolds of dimensions 3 and 4 and presaged the development of symplectic topology.  Ideas from quantum field theory have provided deep insights into new directions and conjectures on the structure of gauge theories and suggested many potential applications.  The focus of this program will be those parts of gauge theory which hold promise for new applications to geometry and topology and require development of new analytic tools for their study.

    Updated on Oct 28, 2020 09:12 AM PDT
  5. Algebraic Cycles, L-Values, and Euler Systems

    Organizers: Henri Darmon (McGill University), Ellen Eischen (University of Oregon), LEAD Benjamin Howard (Boston College), David Loeffler (University of Warwick), Christopher Skinner (Princeton University), Sarah Zerbes (University College London), Wei Zhang (Massachusetts Institute of Technology)
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    Some Gaussian periods for the 255,255-th cyclotomic extension. Image credit: E. Eischen, based on earlier work by W. Duke, S. R. Garcia, T. Hyde, and R. Lutz

    The fundamental conjecture of Birch and Swinnerton-Dyer relating the Mordell–Weil ranks of elliptic curves to their L-functions is one of the most important and motivating problems in number theory. It resides at the heart of a collection of important conjectures (due especially to Deligne, Beilinson, Bloch and Kato) that connect values of L-functions and their leading terms to cycles and Galois cohomology groups. 

    The study of special algebraic cycles on Shimura varieties has led to progress in our understanding of these conjectures. The arithmetic intersection numbers and the p-adic regulators of special cycles are directly related to the values and derivatives of L-functions, as shown in the pioneering theorem of Gross-Zagier and its p-adic avatars for Heegner points on modular curves. The cohomology classes of special cycles (and related constructions such as Eisenstein classes) form the foundation of the theory of Euler systems, providing one of the most powerful methods known to prove vanishing or finiteness results for Selmer groups of Galois representations. 

    The goal of this semester is to bring together researchers working on different aspects of this young but fast-developing subject, and to make progress on understanding the mysterious relations between L-functions, Euler systems, and algebraic cycles.

    Updated on Apr 12, 2021 10:17 AM PDT
  6. Diophantine Geometry

    Organizers: Jennifer Balakrishnan (Boston University), Mirela Ciperiani (University of Texas, Austin), Philipp Habegger (University of Basel), Wei Ho (University of Michigan), LEAD Hector Pasten (Pontificia Universidad Católica de Chile), Yunqing Tang (Princeton University), Shou-Wu Zhang (Princeton University)
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    A rational point on a curve of genus 3

    While the study of rational solutions of diophantine equations initiated thousands of years ago, our knowledge on this subject has dramatically improved in recent years. Especially, we have witnessed spectacular progress in aspects such as height formulas and height bounds for algebraic points, automorphic methods, unlikely intersection problems, and non-abelian and p-adic approaches to algebraic degeneracy of rational points. All these groundbreaking advances in the study of rational and algebraic points in varieties will be the central theme of the semester program “Diophantine Geometry” at MSRI. The main purpose of this program is to bring together experts as well as enthusiastic young researchers to learn from each other, to initiate and continue collaborations, to update on recent breakthroughs, and to further advance the field by making progress on fundamental open problems and by developing further connections with other branches of mathematics. We trust that younger mathematicians will greatly contribute to the success of the program with their new ideas. It is our hope that this program will provide a unique opportunity for women and underrepresented groups to make outstanding contributions to the field, and we strongly encourage their participation.

    Updated on Feb 25, 2021 04:59 PM PST
  7. Commutative Algebra

    Organizers: Aldo Conca (Università di Genova), Dale Cutkosky (University of Missouri), LEAD Claudia Polini (University of Notre Dame), Claudiu Raicu (University of Notre Dame), Steven Sam (University of California, San Diego), Kevin Tucker (University of Illinois at Chicago), Claire Voisin (Collège de France; Institut de Mathématiques de Jussieu)
    9 points theorem
    Image for theorem about 9 point on cubic curve, the special case of Cayley–Bacharach theorem.

    Commutative algebra is, in its essence, the study of algebraic objects, such as rings and modules over them, arising from polynomials and integral numbers.     It has numerous connections to other fields of mathematics including algebraic geometry, algebraic number theory, algebraic topology and algebraic combinatorics. Commutative Algebra has witnessed a number of spectacular developments in recent years, including the resolution of long-standing problems, with new techniques and perspectives leading to an extraordinary transformation in the field. The main focus of the program will be on these developments. These include the recent solution of Hochster's direct summand conjecture in mixed characteristic that employs the theory of perfectoid spaces, a new approach to the Buchsbaum--Eisenbud--Horrocks conjecture on the Betti numbers of modules of finite length, recent progress on the study of Castelnuovo--Mumford regularity, the proof of Stillman's conjecture and ongoing work on its effectiveness, a novel strategy to Green's conjecture on the syzygies of canonical curves based on the study of Koszul modules and their generalizations, new developments in the study of various types of multiplicities, theoretical and computational aspects of Gröbner bases, and the implicitization problem for Rees algebras and its applications.

    Updated on Oct 19, 2021 11:00 AM PDT