Abstract. Motivic integration was introduced by Kontsevich in 1995 as a tool to prove that birational Calabi–Yau manifolds have the same Hodge numbers (generalizing previous results using p-adic integration). Since then, it's found numerous applications in algebraic and arithmetic geometry, particularly towards the study of cohomological invariants and singularities. In this course, we'll introduce arc schemes and the Grothendieck ring of varieties before developing the basic theory of motivic integration; we'll then apply the theory to prove Kontsevich's result on Hodge numbers. Additional time permitting, we'll give applications to the study of Igusa zeta functions and Hodge spectra. Throughout, we will focus on examples and motivation rather than technical proofs; the only prerequisite is basic algebraic geometry.

Abstract. I will describe foundational aspects of homotopy theory and derived algebra, from the perspective of ∞-categories. First, I'll cover some basics of homotopy theory and that of ∞-categories as a context in which to do homotopy theory, including the ∞-category of spaces as the most important example. Next, I'll discuss ∞-(co)limits, and the corresponding notion of homotopy (co)limits for spaces. Another formal framework in which to do homotopy theory is that of model categories, and I will describe the relation between these two frameworks. Next, I'll describe the idea behind derived/homotopy coherent/higher algebra, and how this leads us to stable ∞-categories, as well as to the stable ∞-category of spectra, the most important example of such categories. Finally, I'll discuss how to do more of the derived algebra via operads in spaces and via ∞-operads. All throughout, there will be an emphasis on detail and precision.

Abstract. This mini-course is a gentle introduction to a modern field of mathematics via examples. Starting in the seventies with the work of Stanley and Hochster, new results in commutative algebra have been proved by studying associated combinatorial objects. Ever since, new synergies between algebraic and combinatorial objects have been established. Each day I will introduce one of the main four families of examples: monomial ideals, binomial ideals, determinantal ideals, and linear ideals. The course will emphasize applications of the results from the literature to specific computations, so that we can understand via concrete examples what we can find out about these algebraic objects using the combinatorial perspective. The course will be a mix of lecture and problem solving as each day we work through a worksheet.

Abstract. In 2015, Adiprasito, Huh, and Katz resolved a longstanding conjecture of Rota and Welsh stating that the coefficients of the characteristic polynomial of a matroid (for example, the chromatic polynomial of a graph) form a log-concave sequence. For matroids representable over a field, Huh and Katz had already proven this by associating an algebraic variety to a matroid and using inequalities from Hodge theory. Inspired by this, the general proof defines a "matroid Chow ring" and proves precise analogues of the Hodge-theoretic results without using algebraic geometry, creating an exciting new vantage point from which to study matroids.

This minicourse will outline Adiprasito, Huh, and Katz's argument, across 5 talks:

• The requisite background on matroids.
• The inequalities from Hodge theory which motivate analogous results for the matroid Chow ring.
• The "wonderful compactification" construction, which connects the Chow ring of a representable matroid to an actual Chow ring.
• How the matroid Chow ring allows us to show log-concavity for the characteristic polynomial.
• Sketching the proof of the Hodge-theoretic inequalities for the matroid Chow ring.

Abstract. This minicourse recaptures a part of “A master course on algebraic stacks” taught by Bertrand Toën in University of Toulouse in 2005 (hence the minicourse title). More specifically, we will take the perspective a homotopy theorist to try to understand stacks, or more specifically, stacks in groupoids.
Originally conceived by Grothendieck for construction of moduli “spaces” in descent theory, via the algebraic notions of “pseudofunctors” and “fibered categories”, the notion of stacks is a (higher) categorical device which takes into account of the automorphisms of the objects being parametrized. However, as many other higher categorical notions, the algebraic definition of a stack can sometimes be less motivating and always cumbersome to write down.
In this minicouse, we will see precisely, via the homotopical viewpoint, how a stack (resp. stack in groupoids) is nothing but a 2-sheaf (resp. (2,1)-sheaf), and how this viewpoint connects to Grothendieck’s algebraic definitions. We will discuss basic examples of stacks, such as stacks of sheaves, of geometric/algebraic spaces, and of quasi-coherent modules. We will also briefly demystify some of the key ideas like “stackification”, quotient stacks, classifying stacks, and algebraic (Artin, or Deligne–Mumford) stacks. Finally, in the last day of the minicourse, we will outline the construction of (∞,1)-stacks based on simiplicial (pre)sheaves, following a part of the lectures of Toën–Vezzosi on homotopical algebraic geometry.

Abstract. Class field theory is the study of abelian Galois extensions. A great example would be cyclotomic extensions of Q. In fact, the Kronecker-Weber theorem states that all finite abelian extensions of Q are contained in some cyclotomic extensions. So we can construct any abelian extensions of Q very explicitly. Another fantastic point of the theory is that we can obtain explicitly abelian extensions where only a certain set of primes ramify. Apart from Q, we only know how to construct all abelian extensions explicitly for imaginary quadratic field. This is by using theory of complex multiplication. The story is a mystery for other number fields.
As a lot have said, things are easier in function fields. Indeed, we have the explicit class field theory for not just the rational function field F_q(t), but in fact all global function field (that is, finite extensions of F_q(t)). This is developed by Hayes in 1980s, using the theory of Drinfeld module. In this course, our goal is to understand the construction. We will begin with introducing the theory of Drinfeld modules, and move towards the construction of 'cyclotomic fields' afterwards. We will be following David Goss' book Basic Structure of Function Field Arithmetic, in particular Chapter 3, 4 and 7.

Prerequisites: Galois theory, basic understanding of ramification theory of primes. No prerequisite is required for function field or algebraic geometry.

Abstract. Algebraic K-Theory has its roots in Grothendieck's proof of the celebrated Grothendieck-Riemann-Roch (GRR), a generalisation of earlier analytical results of Hirzebruch. In his proof, Grothendieck, among other things , developed the algebraic K^0-group of coherent sheaves on a scheme. Following his success, Atiyah and Hirzebruch developed topological K-theory, a generalised Eilenberg Steenrod cohomology theory, by applying the K^0-functor to topological vector bundles. Using results of Bott on the periodicity of certain homotopy groups, they were able to extend the topological K^0-functor to a sequence of functors K^i's which satisfied the Eilenberg Steenrod axioms. Coupled with the Atiyah-Hirzebruch spectral sequence (AHSS), this theory is extremely powerful and has far reaching applications to Index Theory, Stable Homotopy theory, etc.
However, despite attempts by Bass, Milnor etc, defining the higher K- group of Schemes remained a difficult problem until the work of Quillen in the late 60's and early 70's when he gave two constructions for higher K-groups - the 'Plus' construction for rings and the 'Q' construction for exact categories both of which are equivalent when appropriate. These constructions used hitherto unknown and novel techniques from homotopy theory such as building a homotopy theory for categories, and presented the K groups as the homotopy groups of a certain space.
Since then Algebraic K-Theory has absolved itself of its difficulty by exhibiting deep relationships with fields such as l-adic cohomology, Motivic Cohomology, Intersection Theory, Stable Homotopy Theory and so on. On the other hand computing the K-group of integers remains an open problem.
In this course we start by reviewing the GRR, topological K-theory and the AHSS as way of motivation. Then we move on to discuss the + construction and the Q construction. We then apply some of the main technical theorems of Quillen Q-construction to establish the relationship of K-Groups with the Chow Ring, and compute K-groups of various projective bundles on smooth schemes etc. Time remaining we may discuss Waldhausen K-Theory.
While the prerequisites are basic algebraic topology (singular cohomology, higher homotopy groups, basics of spectral sequences), basic algebraic geometry (passing familiarity with schemes and sheaf cohomology) and basic category theory (representable functors, adjunctions etc); we will try to be as self contained as possible and all are welcome to attend!

Abstract. This one-week mini-course is an introduction to elasticity theory --- the study of deformable bodies --- and geometry with a particular emphasis on the recent geometric rigidity theorem of Friesecke, James, and Muller. After introducing the basic concepts of strains and displacements and reviewing John’s counterexample to Linfty-Linfty rigidity, we prove L2-L2 rigidity following FJM. Time permitting, we explain the use of this rigidity theorem to derive Kirchhoff’s plate theory as a Gamma-limit and other recent developments along these lines. The course is aimed at graduate students having some basic familiarity with Sobolev spaces, though we will spend some time reviewing the basics in the first lecture. No knowledge of elasticity theory will be assumed.

Abstract. The Atiyah–Singer Index Theorem is an immensely powerful theorem relating the analytical index (the difference of the dimension of the kernel and cokernel of the operator) to the topological index (the integral of the chern character and the Todd Class appropriately defined). In this course we will present the K-Theoretic proof of the same by roughly following Gregory Landweber's article 'K-Theory and Elliptic Operators', which itself is a distilled account of the original papers of Atiyah and Singer - 'The Index of Elliptic Operators I and III'. We would like to note that the proof will be presented from a topological perspective and analytical facts will be blackboxed. However we will introduce the requisite K-Theory and therefore the prerequisites for the course are just some familiarity with Algebraic Topology and Topological Vector bundles. Time remaining we may prove equivariant versions of the same.

Abstract. We will discuss some topics about Bruhat-Tits building that are normally used in the study of p-adic groups and their representation theory. Rough schedule that I am now having in mind as follows: Monday, a brief discussion of p-adic numbers and some properties, then define the p-adic group we concerned throughout the minicourse and examine its structure. Tuesday, define parahoric subgroups, and how they reduce many problems in representation theory to the case of a finite group. Thursday, define the Bruhat–Tits building. Friday, construct an important class depth zero supercuspidal representations.

Prerequisites. Some familiarity on root systems and Weyl groups would be helpful.

Abstract. In this minicourse we will define the Fargues–Fontaine curve. We will discuss geometric structures such as vector bundles on this curve. We will discuss geometric interpretations of comparison theorems in p-adic cohomology theories in terms of the curve. Also, we will mention geometric interpretations of problems in p-adic galois representation theory in terms of this curve. Prerequisites on perfectoid rings will be discussed briefly when needed.

Abstract. Abstract: A celebrated theorem of Atiyah and Segal provides a comparison map from the ring of complex representations Rep(G) of a finite group, to the 0th complex K-theory KU^0 (BG) of a certain space BG. This comparison map is not an isomorphism, but rather it presents KU^0(BG) as the completion of Rep(G) with respect to a certain augmentation ideal. In this mini course, we will investigate a way in which one can “decomplete” KU^0(BG) in order for it to have all the information about the representation theory of G.

Our approach will be algebro-geometric, and we hope to provide a geometric explanation for the failure of this comparison map to be an isomorphism. This explanation will ultimately rely in the geometry of p-divisible groups, we will see that there is a natural p-divisible group associated to Rep(G), and that KU^0(G) only sees a connected component of this p-divisible group.

Time permitting, we will see what kind of advantages this tempered version of KU has, as it comes equipped with a great deal of favorable categorical properties, and discuss potential applications to the representation theory of the infinite symmetric group.

Abstract.

Abstract. Translation surfaces are down-to-earth geometric objects that you can easily draw, and are right now receiving a lot of attention from dynamicists, algebraic geometers, and low-dimensional topologists. In this course we hope to:

• show how basic questions about billiards on polygonal tables can be answered cleanly via the study of translation surfaces
• show how the symmetry groups of translation surfaces constrain the dynamics of the straight-line flow on these surfaces
• give a crash-course introduction to some basic ergodic theory
• explain how studying moduli spaces of translation surfaces gives us theorems about the surfaces themselves
• discuss the famous “Magic Wand” theorem and give examples of why it deserves that name