Seminars Sorted by Series

Speaker #1 (Tran): On a projective variety, Simpson showed that there is a homeomorphism between the moduli space of semisimple flat bundles and that of polystable Higgs bundles with vanishing Chern classes. Recently, Bakker, Brunebarbe and...

The goal of these lectures is to present the fundamentals of Simpson’s correspondence, generalizing classical Hodge theory, between complex local systems and semistable Higgs bundles with vanishing Chern classes on smooth projective varieties.

Many geometric spaces carry natural collections of special submanifolds that encode their internal symmetries. Examples include abelian varieties and their sub-abelian varieties, locally symmetric spaces with their totally geodesic subspaces, period...

Many geometric spaces carry natural collections of special submanifolds that encode their internal symmetries. Examples include abelian varieties and their sub-abelian varieties, locally symmetric spaces with their totally geodesic subspaces, period...

This is the organizational meeting for a learning seminar during the fall term on topics related to non-abelian p-adic Hodge theory. 

With every bounded prism Bhatt and Scholze associated a cohomology theory of formal p-adic schemes. The prismatic cohomology comes equipped with the Nygaard filtration and the Frobenius endomorphism. The Bhatt-Scholze construction has been advanced...

Multiplier ideals in characteristic zero and test ideals in positive characteristic are fundamental objects in the study of commutative algebra and birational geometry in equal characteristic.  We introduced a mixed characteristic version of the...

This talk will introduce background, achievements and challenges in the quest to find non-Archimedean versions of the celebrated Corlette-Simpson correspondence, which on Kähler manifolds relates representations of the fundamental group to certain...

I will discuss the following conjecture: an irreducible $\bar{Q}$$_{\ell}$-local system L on a smooth complex algebraic variety S arises in cohomology of a family of varieties over S if and only if L can be extended to an etale local system over...

In this talk, we introduce the analytic de Rham stack for rigid varieties over $Q_p$ (and more general analytic stacks). This object is an analytic incarnation of the (algebraic) de Rham stack of Simpson, and encodes a theory of analytic D-modules...

Cohomology of classifying space/stack of a group G is the home which resides all characteristic classes of G-bundles/torsors. In this talk, we will try to explain some results on Hodge/de Rham cohomology of BG where G is a $p$-power order...

Etale cohomology of $F_p$-local systems does not behave nicely on general smooth p-adic rigid-analytic spaces; e.g., the $F_p$-cohomology of the 1-dimensional closed unit ball is infinite. 

However, it turns out that the situation is much better if...

The notion of $p$-adic shtukas are introduced by Scholze in his Berkeley lectures on $p$-adic geometry. They are closely related to $p$-divisible groups when their ``legs" are bounded by some minuscule cocharacter. But compared to $p$-divisible...

In this talk, I will first describe how classical Dieudonne module of finite flat group schemes and $p$-divisible groups can be recovered from crystalline cohomology of classifying stacks. Then, I will explain how in mixed characteristics, using...

I will present a new theory of motivic cohomology for general (qcqs) schemes. It is related to non-connective algebraic K-theory via an Atiyah-Hirzebruch spectral sequence. In particular, it is non-$A^1$-invariant in general, but it recovers...

I will present two settings where $q$-De Rham and prismatic vector bundles can be described in terms of modules over an appropriate ring of $q$-twisted differential operators and also the relation with former results. 

This is a joint work with...

Motivated by the desire to express in terms of de Rham data the pro-étale cohomology with non-trivial $\mathbb{Q}_p$-coefficients of rigid spaces $X$, defined over $\mathbb{Q}_p$ or $\mathbb{C}_p$, I will explain how to define D-modules on the...

Topological Hochschild homology is an important invariant, closely related to algebraic K-theory, and can be seen as a noncommutative analogue of de Rham chains.

In this talk, I will describe various computations of the ring/monoid of endomorphisms...

In this talk, I will present a computation of the image of the Hodge-Tate logarithm map (defined by Heuer) in the case of smooth Stein varieties. When the variety is the affine space, Heuer has proved that this image is equal to the group of closed...

A few years ago, Bhatt-Morrow-Scholze introduced an invariant of $p$-adic formal schemes called syntomic cohomology, which has a close relationship to (étale-localized) algebraic $K$-theory. In a recent paper, Antieau-Mathew-Morrow-Nikolaus showed...

In the first talk, I will give a broad survey of classical inequalities that arise in enumerative and algebraic combinatorics.  I will discuss how these inequalities lead to questions about combinatorial interpretations, and how these questions...

In the second talk, I will concentrate on polynomial inequalities and whether the defect (the difference of two sides) has a combinatorial interpretation.  For example, does the inequality  $x^2+y^2 \geq 2xy$  have a combinatorial proof and what...

Chapter 14 of the classic text "Computational Complexity" by Arora and Barak is titled "Circuit lower bounds: complexity theory's Waterloo". I will discuss the lower bound problem in the context of algebraic complexity where there are barriers...

A class of tensors, called "concise (m,m,m)-tensors  of minimal border rank", play an important role in proving upper bounds for the complexity of matrix multiplication. For that reason Problem 15.2 of "Algebraic Complexity Theory" by Bürgisser...

The theme of the lecture is the notion of points over F1, the field with one element. Several heuristic computations led to certain expectations on the set of F1-points: for example the Euler characteristic of a smooth projective complex variety X...

This will be an expository talk on the structure and classification of equivariant vector bundles on toric varieties. I will emphasize Klyachko's classification results from the 1980s and 1990s and discuss more recent re-formulations of this...

The Kronecker coefficients of the Symmetric group $S_n$ are the multiplicities of an irreducible $S_n$ representation in the tensor product of two other irreducibles. They were introduced in 1938 by Murnaghan and generalize the beloved Littlewood...

In the third talk, I will concentrate on inequalities for linear extensionsof finite posets.  I will start with several inequalities which do have a combinatorial proof.  I will then turn to Stanley's inequality and outline the proof why its defect...

A classical construction associates a Poincare duality algebra to a homogeneous polynomial on a vector space. This construction was used to give a presentation for cohomology rings of complete smooth toric varieties by Khovanskii and Pukhlikov and...

A sequence of nonnegative real numbers $a_1, a_2, \ldots, a_n$, is log-concave if $a_i^2 \geq a_{i-1}a_{i+1}$ for all $i$ ranging from 2 to $n-1$. Examples of log-concave inequalities range from inequalities that are readily provable, such as the...

I will motivate the study of the Schubert variety of a pair of linear spaces via Kempf collapsing of vector bundles. I'll describe equations defining this variety and how this yields a simplicial complex determined by a pair of matroids which...

Lecture Series Framework:  A unifying framework for F1-geometry, tropical schemes and matroid theory. In this series of 3 lectures, I will present a recent approach towards F1-geometry and its links to tropical geometry, matroid theory, Lorentzian...