<p>Calabi–Yau manifold. (Image attribution: Lunch on Wikipedia)</p>

This workshop is on recent developments in representation theory, quantum field theory, and motivic homotopy theory. Contemporary representation theory has been increasingly influenced by the mathematical development of supersymmetric gauge theories and mirror symmetry. Motivic homotopy theory—a version of homotopy theory for algebraic varieties and solutions to polynomial equations—has seen widespread modern applications. This three-day workshop aims to introduce researchers to these fields and their open problems, and to highlight recent progress. The workshop will include research lectures as well as activities to promote interaction and connections between the participants. This workshop is open to all mathematicians.

Under the light of suspended spheres, the QFT wizard dangles groups and their duals over a boiling cauldron of motives. (Drawn by Rok Gregoric)

The goal of this introductory workshop is to showcase some of the recent developments in motives and quantum field theory, with a focus on giving a high-level, but "outsider-friendly" introduction to both subjects. It will feature lectures introducing the modern formalism of sheaf theories, supersymmetric gauge theories, geometric representation theory and motivic spectra.

Image by Elliot Kienzle

This workshop will bring together theoretical physicists, representation theorists, algebraic geometers and symplectic geometers interested the connections between quantum field theory and geometric representation theory. The main topics to be discussed are mathematical aspects of 2d, 3d and 4d supersymmetric field theories, such as: topological twists and the resulting Higgs and Coulomb branches, relations to quantization and categorification, representations of vertex operator algebras, connections to enumerative geometry and quantum K-theory and elliptic cohomology, relations to knot homology and, finally, connections to the (relative) geometric Langlands program.

The workshop will bring together researchers at the forefront of ongoing work in motivic homotopy theory.  Topics will include the application of motivic techniques to: geometry of affine algebraic varieties and algebraic vector bundles; computations in classical algebraic topology such as homotopy groups of spheres; and enumerative geometry.  The workshop will also consider the internal foundational development of motivic homotopy theory itself.

The Simons Laufer Mathematical Sciences Institute (SLMath) will host the 2026 Modern Math Workshop (MMW) as part of the SACNAS Annual Conference. The Modern Math Workshop encourages undergraduates to pursue careers in the mathematical sciences, and builds research and networking opportunities among undergraduates, graduate students and recent PhDs in the mathematical sciences.

The formalization of mathematics is undergoing a quiet revolution. Computational proof assistants like Lean now make it possible to build verified, machine-checkable libraries of mathematics at scale, and the frontier of partial differential equations (PDEs) stands as one of the most profound and underexplored territories in this landscape. At the same time, large language models and AI for mathematics (AI4Math) are beginning to reshape how researchers discover, conjecture, and verify mathematical results.

This workshop brings together three communities, PDE analysts, Lean/Mathlib developers, and AI/AI4Math researchers from both academia and industry, to chart a shared path forward. Our central aim is to produce a clear, community-endorsed roadmap (1-year, 3-year, and 5-year horizons) for the formalization and AI-assisted analysis of PDEs, spanning undergraduate and graduate textbook material through to active research frontiers. We will identify the PDE topics most ripe for formalization in Mathlib, the AI tools best positioned to accelerate that work, and the collaborative structures needed to sustain long-term progress.

A contour used in [Calegari–Dimitrov–Tang, 2024] to prove L(2, χ−3) ∈/ Q.

In 1978, Apéry proved that ζ(3) was irrational. Almost 50 years later, we still don’t know if ζ(5) is irrational. Why not? The goal of this workshop is to bring together experts on explicit rational approximations and motivic differential equations to consider new approaches to these questions.

The perfectoid infinite level modular curve

This workshop will focus on geometric Sen theory, its connections to the p-adic Simpson correspondence, and applications to modularity. In addition to three mini-courses, the workshop will feature research talks on related recent developments in p-adic Hodge theory and p-adic geometry.  

<p>Mapping tensors to their moment polytyopes to their asymptotic spectrum.</p>

Tensors play a central role in various areas of mathematics, physics and computer science: from constructing fast matrix multiplication algorithms in algebraic complexity theory to the study of entanglement in quantum information. While matrices (order-two tensors) are very well understood, tensors have a much more intricate structure, and much about them is still unknown, despite much interest. In this lecture series we will discuss foundations and recent advances in the theory of tensors and their applications across computer science and physics.

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.

MSRI-UP 2027 will focus on statistical models applied to real-world data using the free statistical software R. Students should have completed the calculus sequence, and prior coursework in statistics and programming experience in R or a similar language are desired. Beyond these prerequisites, students should be willing to learn quickly, work in a team, and communicate results clearly. The first two weeks of the program will consist of crash-course training in regression, inference, and R. Research projects will be chosen from three areas: epidemiological statistics, education statistics, and labor economics. The summer projects will be directed by Dr. Minho Kim (Colorado College).

<p>A real-world Flying Wing. Lai proved Hamilton&rsquo;s conjecture on the existence of Flying Wing Ricci solitons.</p>

Participants will learn the basic theory of Ricci flow and Ricci solitons—they will learn the fundamental equations and inequalities, how to do basic tensor calculations, how to control the potential function, how to obtain global curvature bounds, how to do bootstrap arguments, how to control the geometry at infinity, how to construct new Ricci solitons, the current cutting edge existence theory for Ricci solitons, including the existence and classification of flying wing solitons. Many arguments involve the maximum principle, integration by parts, and/or bootstrap arguments. In short, the student will learn the art of the a priori estimate. This is one of the main methods for understanding the geometry of Ricci solitons.

The goal of our school is to make connections between different flavors of geometry: euclidean geometry, hyperbolic geometry, translation surfaces and graph theory. We will focus on growth, asymptotics and extremal problems.

The course will introduce graduate students to the many modern methods to construct minimal surfaces from a variational or min-max principle as well as applications to problems in geometry and topology. Topics include Almgren-Pitts min max theory, Allen-Cahn construction and PDE methods, the proof of the Willmore conjecture, the resolution of Yau’s conjecture on the infinitude of minimal hypersurfaces in general Riemannnian manifolds and corresponding Weyl law and equidistribution results.

Scheduled at ICARM in Pittsburgh, Computer-Aided Reasoning for Algebraic Geometry is a special summer school activity for 40–50 graduate students and junior researchers. This workshop positions algebraic geometry, formalization, and artificial intelligence as the three vertices of a connected framework, bridging the gaps between these distinct skill sets. Participants will explore topics beyond standard introductory courses through a rigorous schedule of blackboard-style foundations lectures, Lean formalization sessions, and hands-on machine learning tutorials. The program features a heavy emphasis on collaborative afternoon group projects led by expert mentors, allowing students to apply state-of-the-art autoformalization tools and ML pipelines directly to open problems in algebraic geometry.

The aim of this summer school is to introduce students to landmark results in mathematical ecology while emphasizing the modern techniques and ideas developed to address these emerging challenges. The school will begin with a review of classical questions and results, together with a discussion of new problems arising from both novel data sources and changing environmental conditions. Particular emphasis will be placed on identifying ecological questions that are amenable to mathematical formulation.

The core of the school will focus on modern dynamical systems approaches. A sequence of lectures will introduce nonautonomous systems, including their attractors, stability theory, and bifurcation structure, with a strong emphasis on transient dynamics and tipping phenomena. This will be complemented by an introduction to ergodic theory and random dynamical systems, providing tools to model and analyse stochastic influences and variability. The final part of the school will return to open problems in ecology, highlighting how these modern mathematical frameworks can be used to address unresolved questions and to formulate new ones.

<p>A K3 surface</p>

The study of complex projective surfaces is yet to attain the level of realization and widespread prevalence as that of curves. The classification of surfaces is far from complete, and many basic problems remain open. This is particularly pronounced in geometric topology, where understanding of algebraic curves (i.e., Riemann surfaces) is essential knowledge but, for just one example, applications of surface theory to studying the geometric topology of four-dimensional manifolds are very limited at the present moment.

The purpose of this summer school is to close the background gap and connect young researchers with different backgrounds that are, perhaps unbeknownst to themselves, well-poised to fruitfully collaborate on important problems at the interface of geometric topology and surface geometry.

<p>A simplification of the Hopf link in the quantum GL(N)-category</p>

The purpose of this summer school will be to introduce students to the world of tensor categories. We will approach this topic in a variety of ways. Beyond simply exploring the formal definitions, many examples will be introduced and studied, and the natural occurrences of tensor categories “in the wild” will be explored. One such example will be the Temperley-Lieb-Jones category, and the corresponding “quantum” knot invariants.

The PIMS–CRM Summer Schools in Probability are a highlight of Canadian probability. Launched by PIMS in 2004, they take the form of two main 4-week courses by top researchers in probability theory and related fields, along with a number of keynote addresses and shorter mini-courses. The school is geared primarily at doctoral students and postdoctoral scholars, who are also given the opportunity to present their own research, should they desire. The schools have played a major role in the development of an exceptionally strong community of young probabilists in Canada, North America and overseas. This will be the 14th time this school has run. 

Blocks, but not as you know them

The summer school will provide a comprehensive introduction to the structure and classification of blocks. Participants will explore the algebraic, homological, combinatorial, categorical, geometric, and computational perspectives that shape the area. Beginning with basic concepts such as defect groups, decomposition numbers, and Cartan matrices, the lectures will develop Morita and derived equivalences and higher representation-theoretic methods, before turning to finite reductive groups and geometric constructions arising from Deligne–Lusztig varieties. Hands-on computational sessions will complement this theoretical material, illustrating how character tables, decomposition matrices, and block invariants can be explored using Julia, and in particular its Chevie package for computations involving finite reductive groups.

The Whitney Umbrella. Courtesy of Herwig Hauser, Fac. Math., University of Vienna, Austria

The school will introduce students to the fundamental problem of resolution of singularities of varieties and schemes. This is a subject that is relevant for all areas of algebraic geometry and commutative algebra. Students will learn algorithms to resolve singularities, and learn how to apply them in specific situations and understand how these techniques can be used in their research.

This workshop presents introductory talks and new trends in tropical geometry and algebraic combinatorics, including interactions between tropical geometry and enumerative, logarithmic, nonarchimedean, and real algebraic geometry, mirror symmetry and symplectic geometry, moduli spaces in tropical geometry, the geometry of matroids, integrable systems and dynamical combinatorics, combinatorial representation theory, geometry of polynomials, combinatorial varieties and connections to symmetric function theory and cluster algebras.

Fano plane

The workshop will include three minicourses on key topics in Algebraic Combinatorics & Tropical Geometry: combinatorial representation theory, combinatorial algebraic varieties and their connections to various polynomial bases, combinatorial and algebro-geometric facets of matroids, enumerative tropical geometry, real aspects of tropical geometry and connections with A1-homotopy theory.
The minicourses will be intended for a broad audience, with the goal of providing background and motivation in these topics.

<p>This figure encodes the combinatorics of a triangulation of the permutahedron.&nbsp;&nbsp;Figure appears in Defant--Sherman-Bennett--<wbr />Williams&nbsp;<a data-saferedirecturl="https://www.google.com/url?q=https://arxiv.org/pdf/2509.11497&amp;source=gmail&amp;ust=1769544428157000&amp;usg=AOvVaw1E-v0MZ1v_EyqlWLiqlPNw" href="https://arxiv.org/pdf/2509.11497" target="_blank">https://arxiv.org/<wbr />pdf/2509.11497</a>.</p>

This workshop will bring together researchers at the cutting edge of algebraic combinatorics.   The topics will include integrable systems, dynamical combinatorics, combinatorial representation theory with connections to theoretical computer science, the geometry of polynomials, cluster algebras, combinatorial algebraic geometry, and symmetric functions.  

The discriminant locus in the base of a Lagrangian torus fibration on the quintic threefold.

This workshop will focus on enumerative aspects of tropical geometry and tropical moduli spaces. The meeting will bring together researchers from several different areas in real and complex geometry, symplectic geometry, mirror symmetry and non-Archimedean geometry.

<p>A wall-crossing in a moduli problem</p>

One of the central problems in algebraic geometry is to classify so-called algebraic varieties: geometric shapes cut out by polynomial equations. Algebraic varieties are parametrized by certain moduli spaces (roughly: parameter spaces whose points correspond to these different varieties). The geometry of these moduli spaces encodes the ways of continuously deforming these shapes. Furthermore, classification questions for algebraic varieties often boil down to understanding the geometry of these moduli spaces. In the past few years, powerful new tools have been developed in moduli theory, especially for higher dimensional varieties – those which are of complex dimension at least two. The goal of this summer school is to provide an introduction to many of these recently emerging breakthroughs to enable graduate students to begin working in this area. The program will be motivated and often guided by examples and is intended to be accessible to a wide variety of students

<p>Homotopy measures how spheres can be tangled in spaces; the logo shows a sphere tangled in a grove of California redwoods<br />The background painting is &ldquo;Giant Redwood Trees of California&rdquo; by Albert Bierstadt in 1874</p>

The goal of this school will be to introduce students to several powerful cohomological tools that were brought to commutative algebra by Avramov in the 80s and 90s: Lie algebra methods from homotopy theory, and support theoretic methods from the representation theory of finite groups. These tools have have seen a huge array of applications that continue to grow, with several major developments in recent years, opening new connections to algebraic topology, noncommutative algebraic geometry, and representation theory.

This Summer Graduate School aims to introducing graduate students to some aspects of contemporary modeling and analysis of dynamical systems in their interactions with machine learning and artificial intelligence (AI) applications.

This two week summer school, jointly organized by SLMath with OIST, will offer the following two mini-courses:

1. Measure-theoretical analysis, divergence-measure fields, and nonlinear PDEs of divergence form
   This course will present some recent developments in the theory of divergence-measure fields via measure-theoretic analysis and its applications to the analysis of nonlinear PDEs of conservative form – nonlinear conservation laws.
2. Perron’s method and Wiener-type criteria in the potential theory of elliptic and parabolic PDEs
   This course will present some recent developments precisely characterizing the regularity of the point at ∞ for second order elliptic and parabolic PDEs and broadly extending the role of the Wiener test in classical analysis.

<p>Artificial image generation by flow-based generative models starting from noise</p>

The overarching goal of this summer school is to expose students to the latest developments in the mathematics of generative models. Our ultimate goal is to teach them how to conduct research in this exciting area in machine learning and use their knowledge to make contributions to applied mathematics using these new tools.

Models of random growth and of the separation of phases occurring when one substance is suspended in another often evince universal features, in which scaling exponents are shared among a broad class of such models. A foothold for understanding such features is often offered by studying a few special models that are exactly solvable, which means that exact formulas of algebraic or integrable origin are available. Showing that a broader range of models also have the features is a task that may rely on a range of robust probabilistic or geometric tools. The summer school will offer an introduction to random growth and phase separation, with an emphasis on tools that offer the prospect of proving universality for a wider class of models.

The aim of this summer school is to provide an introduction to theoretical ideas that have been developed with the objective of understanding machine learning methods and their domain of applicability. The focus will be on proof technique and general mathematical tools. The lecturers are two worldwide experts in the area and the material is regularly taught in Mathematics and Statistics Departments of the top world Universities.

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 2026, MSRI-UP will focus on Numerical Methods for Differential Equations The research program will be led by Dr. Johnny Guzman of Brown University.

The school will consist of three interrelated courses, aimed to introduce the main concepts in Geometric Measure Theory.

(1) Sets and measure in the Euclidean space, Guido De Philippis (Courant Institute of Mathematical Sciences)
(2) Theory of currents, Annalisa Massaccesi (University of Padua)
(3) Allard regularity theory, Camillo De Lellis (Institute of Advanced Study)

<p>The five largest clusters in critical site percolation on a large three-dimensional box</p>

Percolation and spin models such as the Ising model have a history that goes back over 100 years. The subject has taken a central role in probability theory over the last few decades, in particular through interactions with various other areas of mathematics. These include graph theory, theoretical
computer science, statistical physics, quantum field theory, complex analysis, partial differential equations, and geometric group theory. Through examples, the summer school aims to illustrate some of the successful techniques and ideas in the subject area.

One of the hottest topics in analytic number theory involves the use of statistics and probability in understanding different aspects of algebraic and analytic number theory, through various new lenses. This is reflected in some of the most exciting number theory research of the last few years (for example, of Bhargava, of Ellenberg and Venkatesh, of Alexander Smith, of Sawin and Wood, of Adam Harper, of Koukoulopoulos and Maynard, of Helfgott and Radziwill, of Pilatte,....). As a consequence the CRM will host a thematic semester Mar 2-July 3, 2026 on these topics involving some of the world leaders in the subject. Since this new area can roughly be split in two into Algebraic and Analytic, we will focus for two months on each, with the SMS school placed in the middle. The 2026 SMS will introduce junior mathematicians to important trends in number theory. 

Building on the vision Bob Moses set out in the 1993 Algebra Initiative Colloquium, the 2025 Critical Issues in Mathematics Education (CIME) Workshop celebrated the legacy of Moses’ call for a mathematics literacy that supports citizenship in the 21st century. A mathematics towards citizenship has proved elusive as a goal for mathematicians, mathematics educators, researchers, students, teachers, and leaders—even as efforts among different individuals, organizations, and communities labor for greater access to high-quality mathematics learning experiences. The 2026 CIME Workshop, Mathematics Literacy for 21st Century Citizenship - Part II, endeavors to design opportunities for learning, collaborating, and struggling towards a mathematics for lived citizenship, as a theme and organizing principle for communities across the Nation.

For Moses, the notion of lived citizenship is not simply a matter of gaining legal status but of demanding that the legal status be meaningful. Thus mathematics literacy for lived or participatory citizenship might be thought of to include contributing to your community, not just being part of a community, and beyond belonging, to actively participate in making your community better, in making life better for others.

The conception of mathematics for lived citizenship is an important foundation for the goals for the 2026 CIME Workshop, in order to envision the development of coordinated actions in the short and long term within the mathematics and mathematics education communities. To ground and catalyze these efforts, we invite participants to consider a mathematics for lived citizenship through three foundational ideas:

• Claiming mathematics -- establishing agency in and ownership of math.
• Building Agency -- claiming and leveraging mathematics knowledge to overcome present day, generational, community, and national challenges.
• Collaboration and collective effort -- using mathematics literacy to make systemic change.

Three questions will guide our work:

• How does the unfolding history of mathematics impact lived citizenship?
• How do we understand the relationship between mathematical flourishing and the evolution of vibrant democracy in the United States? 
• In light of the above two questions, how do we reimagine our roles, responsibilities, and actions in the short and long term?

Limit set of an Anosov representation

This workshop will focus on recent advances on geometric and dynamical approaches to the study of discrete subgroups of higher rank Lie groups and their deformation spaces. The goal will be to present results and exchange ideas from different areas of mathematics, and we hope to create bonds between several different mathematical communities.

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The Langlands program, pictured as a sea monster (Photo credit: V. Lafforgue)

This workshop will focus on the recent proof of the global unramified geometric Langlands conjecture as well as potential future developments in this field.

Foliations around a pseudo-Anosov singularity (Image credit: Chi Cheuk Tsang)

This workshop will bring together ideas from diverse areas of mathematics that meet in the setting of geometry and topology in low dimensions.  This includes the study of flows, foliations, and fibrations of three-manifolds and the related study of geometry (e.g. hyperbolic or conformal structures) of the manifolds and of the leaves or fibers, and their mapping class groups.  This is a rich and interconnected area and many adjacent topics will also be featured.

The joint introductory workshop for the programs in Geometry and Dynamics for Discrete Subgroups of Higher Rank Lie Groups and Topological and Geometric Structures in Low Dimensions will feature lectures introducing subjects of interest to both programs, including Teichmuller Theory, geometry in higher rank, foliations and flows, Anosov groups and thermodynamic formalism,  mapping class groups, counting and equidistribution, and related topics. Minicourses will be targeted at early career researchers as well as specialists looking to find connections between the different subjects.

This workshop features a series of invited talks by experts across the fields of low-dimensional topology, homogeneous dynamics, and the geometry of symmetric spaces. Spanning all career stages, the event aims to foster a collaborative and supportive environment, particularly for early-career mathematicians, encouraging engagement, learning, and exploration in a welcoming academic setting.

Tribute to Hamilton's graffiti of the Quaternion Division Algebra, County Dublin, Ireland. Photo: Professor Peter Gallagher, Director Dublin Institute for Advanced Studies Dunsink Observatory (courtesy DIAS)

Infinite-dimensional division algebras are essential in noncommutative algebra and noncommutative algebraic geometry, yet they have remained cryptic and largely unclassified. This workshop will address three key classical open problems concerning them: the Kurosh Problem, the Free Subalgebra Problem and Artin's Conjecture. We will review decades of progress on these wide-open problems and emphasize novel techniques and emerging theories and concepts that show promise in facilitating breakthroughs.

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The motion of a random string.

The workshop aims to bring together researchers working on different facets of stochastic PDEs. The field of stochastic PDEs has seen many new techniques recently appear to tackle different problems, including renormalization, large scale and long-time behaviours, stochastic fluid dynamics, and homogenization. The goal of the workshop is to facilitate discussions and allow different communities to engage with one another one.

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Top: Neutrino interactions and neutrino-atom interactions. Bottom: Collision of two "waves"

This workshop will explore the latest advances in kinetic theory and stochastic particle dynamics in mean field regimes, covering both classical themes and emerging areas. Topics will include the derivation of kinetic type equations from particle and plasma systems, state-of-the-art numerical methods, studies of multiscale phenomena, and the applications of kinetic equations in physics, chemistry, computer sciences appearing in  life sciences, social sciences, and machine learning. This workshop will offer an exciting opportunity to connect researchers from all stages and sub-areas and spark new ideas.

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Computational proof assistants let us build global, digital libraries of mathematics where every theorem is fully verified by a computer. In this workshop, you'll get to explore this exciting new technology and the ideas behind it, with a special focus on formalizing partial differential equations (PDEs). The Lean mathematical library, Mathlib, is a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant. The main goal of this workshop is to help you start tangible projects that will contribute to the PDE section of the Mathlib library.

Parameter scan for deploying external electric field to control two-stream instability for Vlasov-Poisson.

The goal of the workshop is to introduce non-experts to two active research areas: kinetic theory and stochastic partial differential equations. Kinetic theory studies the properties of interacting particle systems modeling various processes in non-equilibrium statistical mechanics. Stochastic partial differential equations describe dynamics subjected to random noises. The methods from the two areas complement each other in studies of the phenomena arising in physics, economics, life sciences, etc.

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The Pathways workshop will bring together leading experts working at the intersection of kinetic theory and stochastic partial differential equations (SPDEs).

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<p>Laminations arise naturally in hyperbolic geometry and (pseudo-) Anosov flows [Image by Jeffrey Brock]</p>

This school will serve as an introduction to the SLMath semester “Topological and Geometric Structures in Low-Dimensions”.  The school consists of two mini-courses: one on Teichmüller Theory and Hyperbolic 3-Manifolds and the other on Anosov Flows on Geometric 3-Manifolds.  Both topics lie at the interface of low-dimensional geometric topology (specifically, surfaces, foliations, and 3-manifolds) and low-dimensional dynamics.  The first course will be targeted towards students who have completed the standard first year graduate courses in geometry, topology, and analysis while the second course will geared towards more advanced students who are closer to beginning research. However, we expect that all students will benefit from both courses.

<p>Flats and hyperbolic planes in a higher rank symmetric space</p> Drawn by Steve Trettel.

Lie groups are central objects in modern mathematics; they arise as the automorphism groups of many homogeneous spaces, such as flag manifolds and Riemannian symmetric spaces. Often, one can construct manifolds locally modelled on these homogeneous spaces by taking quotients of their subsets by discrete subgroups of their automorphism groups. Studying such discrete subgroups of Lie groups is an active and growing area of mathematical research. The objective of this summer school is to introduce young researchers to a class of discrete subgroups of Lie groups, called Anosov subgroups.

<p>The traditional scientific method cycle, with Francis Bacon and Rene Descartes, its concievers in the center, alongside formal and statistical AI machinary, as a propsective evolution of the method.&nbsp;<br />&nbsp;</p>

The summer school aims to expose participants to formal methods that can facilitate principled scientific discovery. The school will cover some of the basic automated statistical inference (in the form of machine learning techniques) and reasoning methods that are commonly used in scientific discovery, as well as novel techniques developed to tackle open questions and issues. This summer school will address novel computational methods for scientific discovery and focus on fusing axiomatic knowledge and experimental data to enable principled derivations of models of natural phenomena along with certificates of the consistency of these models with background knowledge specified as axioms.

One of the core elements of applied mathematics is mathematical modeling consisting of nonlinear equations such as ODEs, and PDEs. A fundamental difficulty which arises is that most nonlinear models cannot be solved in closed form. Computer assisted proofs are at the forefront of modern mathematics and have led to many important recent mathematical advances. They provide a way of melding analytical techniques with numerical methods, in order to provide rigorous statements for mathematical models that could not be treated by either method alone. In this summer school, students will review standard computational and analytical techniques, learn to combine these techniques with more specialized methods of interval arithmetic, and apply these methods to establish rigorous results in otherwise intractable problems

The school will consist of two courses: Homological Mirror Symmetry and Algebraic Models for Spaces. These courses will be planned and taught by organisers with the help of teaching assistants for the problem sessions. The school will be aimed at a wide range of graduate students, from students with a Bachelor degree to beginning PhD students. The lectures and problem sessions will be complemented by a poster session in week one and a total of four introductory research talks on Friday afternoons. 

<p>The discriminant ∆ detects singular varieties. The picture shows three different scenarios: solutions of quadratic polynomials, cubic plane curves and cubic surfaces.</p>

This summer school will offer a hands-on introduction to discriminants, with a view towards modern applications. Starting from the basics of computational algebraic geometry and toric geometry, the school will gently introduce participants to the foundations of discriminants. A particular emphasis will be put on computing discriminants of polynomial systems using computer algebra software. Then, we will dive into three applications of discriminants: algebraic statistics, geometric modeling, and particle physics. Here, discriminants contribute to the study of maximum likelihood estimation, to finding practical parametrizations of geometric objects, and to computations of scattering amplitudes. We will explain recently discovered unexpected connections between these three applications. In addition to lectures, the summer school will have daily collaborative exercise sessions which will be guided by the teaching assistants and will include software demonstrations.