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Bhargav Bhatt: p-adic motives I

7th April, 2023

In the 1960s, Grothendieck dreamt that algebraic varieties can be linearized in a universal way, leading to his philosophy of motives. Subsequent ideas of many mathematicians (especially Beilinson and Deligne) led to a beautiful conjectural framework surrounding the notion of a motive. In the last decade, thanks to the discovery of perfectoid geometry and subsequent developments, some aspects of this framework have also been realized unconditionally in the context of p-adic motives on p-adic varieties. In these lectures, I will survey some of this landscape, with an emphasis on the concrete applications that have guided the theoretical developments.

Artem Chernikov: Recognizing Groups in Erdős Geometry and Model Theory

28th March, 2023

Erdős-style geometry is concerned with combinatorial questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be typically viewed as asking for the possible number of intersections of a given (semi-)algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups.  Techniques from model theory - Hrushovski's group configuration and its variants - are very useful in recognizing these groups, and allow to obtain higher arity and dimension generalizations of the Elekes-Szabó theorem. In fact, all of this is not just about polynomials and works in the larger setting of definable sets in o-minimal structures.

Zijun Zhou: Virtual Coulomb branch and quantum K-theory

21st March, 2023

In this talk, I will discuss a virtual variant of the quantized Coulomb branch constructed by Braverman-Finkelberg-Nakajima, where the convolution product is modified by a virtual intersection. The resulting virtual Coulomb branch acts on the moduli space of
quasimaps into the holomorphic symplectic quotient T *N///G. When G is abelian, over the torus fixed points, this representation is a Verma module. The vertex function, a K-theoretic enumerative invariant introduced by A. Okounkov, can be expressed as a Whittaker function of the algebra. The construction also provides a description of the quantum q-difference module. As an application, this gives a proof of the invariance of the quantum q-difference module under the variation of GIT.

Rhiannon Savage: Algebra and Geometry in Monoidal Quasi-abelian Categories

7th March, 2023

A quasi-abelian category is an additive category with all kernels and cokernels, along with some additional conditions allowing us to extend notions from homological algebra to them. A key example is the category of complete bornological spaces which is derived equivalent to the category of inductive limits of Banach spaces. In this talk, we will introduce the key concepts in the theory of quasi-abelian categories and we will discuss their potential applications. In particular, we will see how we can extend ideas from Koszul duality to quasi-abelian categories, as well as their use more generally as a setting for a new theory of derived analytic geometry proposed by my supervisor Kobi Kremnizer and his collaborators.

Karim Alexander Adiprasito: Lefschetz Without Positivity: An Overview

6th February, 2023

motivated class is of full rank.

We drop the keyword projective, and replace it with the word "generic". I will describe how, for instance, a complete, quasismooth toric and NON-projective variety still satisfies the Lefschetz property, provided the torus action is sufficiently generic. I will survey the theory and the techniques (including an intriguing relation to Parseval-Rayleigh identities in positive characteristic), as well as some of the motivations and applications to quantitative topology, combinatorics, discrete geometry and more. I will end with some cool conjectures.

Toni Mikael Annala: Stable Homotopy without Homotopy

30th January, 2023

Many cohomology theories in algebraic geometry, such as crystalline and syntomic cohomology, are not homotopy invariant. This is a shame, because it means that the stable motivic homotopy theory of Morel-Voevodsky cannot be employed in studying the deeper aspects of such theories, such as cohomology operations that act on the cohomology groups. In this talk, I will discuss ongoing efforts, joint with Ryomei Iwasa and Marc Hoyois, to set up a workable theory of non-homotopy invariant stable motivic homotopy theory, with the goal of providing effective tools of studying cohomology theories in algebraic geometry by geometric means.

Igor Klep: Factorization of non-commutative polynomials and Nullstellensätze for the free algebra

10th November, 2022

The singularity set of a non-commutative polynomial f=f(x1, . . . ,xd) is the graded set Z(f)=(Zn(f))n, where Zn(f)={ X ∈ Mnd : det f(X) = 0 }. Two main results will be presented. First, irreducible factors of f are shown to be in a natural bijective correspondence with irreducible components of Zn(f) for every sufficiently large n. In particular, f is irreducible if and only if Zn(f) is eventually irreducible. Second, we give Nullstellensätze for non-commutative polynomials. For instance, given two non-commutative polynomials f1, f2, we have Z(f1) ⊆ Z(f2) if and only if each irreducible factor of f1 is (up to stable associativity) an irreducible factor of f2. Along the way an algorithm for factorization of non-commutative polynomials will be presented.

Grigory Garkusha: Homological Algebra for Enriched Grothendieck Categories

8th November, 2022

Enriched Grothendieck categories naturally occur in algebraic geometry, where associated abelian categories rarely have projectives but have plenty of information encoded by enriched category theory. In this talk general properties of derived categories for Grothendieck categories of enriched functors and various recollements of such categories will be presented. Applications are given for Voevodsky's triangulated categories of motives.

Carolina Araujo: Higher Fano manifolds

13th October, 2022

Fano manifolds are complex projective manifolds having positive first Chern class. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit holomorphic sections. In recent years, there has been great effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo Brauer obstruction). In this talk, I will discuss a possible notion of higher Fano manifolds in terms of positivity of higher Chern characters, and describe special geometric features of these manifolds.