Seminars Sorted by Series

A locally testable code (LTC) is an error-correcting code equipped with a tester T that, given an input string x, queries only a small number of positions and rejects x with probability proportional to its distance from the code. Classic examples of...

High dimensional expansion comes in two flavors: spectral, which relates to random walks;  and cosystolic, which relates to chains of linear maps. The later is a more mysterious notion, which turns out related to a variety of applications such as...

A valency argument is an elegant and well-known technique for proving impossibility results in distributed computing. It is an example of an extension-based proof, which is modelled as an interaction between a prover and a protocol. Even though...

The $r$-colour Ramsey number $R_r(k)$ is the minimum $n \in \mathbb{N}$ such that every $r$-colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove, for each fixed $r \ge 2$, that 

$$R_r(k)...

We will go in-depth into the proof that for all functions $t(n) \geq n$, every multitape Turing machine running in time $t$ can be simulated in space only $O(\sqrt{t \log t})$. We will attempt to be completely self-contained. To that end, we will...

I will talk about the well known relation between cocycles and covers.  Then, I will discuss the relation of cocycle expansion to approximate covers, and finally how these show up in low-soundness PCP agreement tests.

In this talk, we will discuss new algorithms for solving instances of random and semirandom planted constraint satisfaction problems (CSPs). Random CSP are generated by first choosing a solution x and then sampling constraints so that x satisfies...

In this talk, I will begin with a quick primer to parameterized complexity, present some key insights from recent hardness of approximation results in the area, and end with a proof sketch of the following result: Assuming the Exponential Time...

Around 1992, Aldous made the following bold conjecture. Let A be any set of transpositions in the symmetric group Sym(N). Then the spectral gap of the Cayley graph Cay(Sym(N),A) is identical to that of a relatively tiny N-vertex graph defined by A...

Around 1992, Aldous made the following bold conjecture. Let A be any set of transpositions in the symmetric group Sym(N). Then the spectral gap of the Cayley graph Cay(Sym(N),A) is identical to that of a relatively tiny N-vertex graph defined by A...

Given linear equations over a finite field and under infinity norm, the Short-Integer-Solution (SIS^inf) problem asks to find a solution where each entry is a small number.

In 2021, Chen, Liu, and Zhandry presented an efficient quantum algorithm for...

Can we encode data in a way that it is recoverable even when 1) most data becomes corrupted, and 2) we can only read a sub-constant fraction of the database? This is the central question of local list decoding, a powerful tool from coding theory...

In this talk we will construct (approximate) locally list decodable codes from high dimensional expanders (HDXs).

Last week we saw that locally list decoding is a powerful tool from coding theory. We also got a subspace-based construction of...

Inspired by the Erdos--Renyi model for random graphs, Linial and Meshulam devised in 2006 a model for random 2-dimensional simplicial complexes. The goal of this talk (and the next) is to present some nice results about the behavior of these random...

Last week we defined the Linial--Meshulam model for random 2 dimensional simplicial complexes and discussed two notions of connectivity for it: Vanishing of its 1st cohomology with F2 coefficients, and vanishing of its fundamental group. This time...

The study of Turán numbers of graphs and hypergraphs is a rich problem in extremal combinatorics. The Turán problem asks, given a fixed forbidden (hyper)graph F, what is the maximum number of edges in an F-free (hyper)graph in terms of the number of...

This is the introductory talk for a semester-long learning seminar on the notion of solid modules and the resulting 6-functor formalism for coherent cohomology.

Steven will cover the material from the first two lectures by Peter Scholze (available electronically at http://people.mpim-bonn.mpg.de/scholze/Condensed.pdf). In particular, he will introduce the notion of condensed set and condensed abelian group...

Quasi-coherent sheaves cannot support a six-functor formalism in the same way that e.g. \ell-adic sheaves do. Clausen and Scholze have shown that this can be fixed by extending the category of quasi-coherent sheaves to `solid sheaves’. In this talk...

This talk is an introduction to analytic rings. Some examples and non-examples will be covered. Some nice properties of the module categories will be proved.

We will discuss the notion of solid abelian groups.

We will globalize the notion of solid modules to an arbitrary scheme/discrete adic space.

We will discuss the 6-functor formalism for solid quasi-coherent sheaves.

The topic of the learning seminar this semester is "Analytic K-theory''. More precisely, the goal is to use condensed mathematics to obtain a notion of quasi-coherent sheaves/complexes in rigid analytic geometry, develop Efimov's approach to K...

Define the Grothendieck group of a commutative ring. Then define K1 and K2 and 
state Matsumoto’s theorem. Explain the definitions of higher K-theory in terms of the 
+-construction and in terms of the∞-group completion of groupoid of finite...

Introduce the notion of stable compactly generated $\infty$-category and discuss its properties and some examples. In particular, prove that the $\infty$-category of small idempotent complete stable $\infty$-categories is equivalent to the $\infty$...

Introduce the notion of Verdier sequences of presentable stable $\infty$-categories and then discuss the behavior of dualizable $\infty$-categories in Verdier sequence; in particular, discuss Thomason's trick. Introduce the notion of localizing...

Recall the notion of condensed analytic ring, then introduce its animated variant and discuss its properties. Explain the construction of induced analytic structures on animated condensed rings. Introduce the notion of steady maps of analytic rings...

Introduce the notion of nuclear object in a symmetric monoidal stable ($\infty$-)category and discuss its properties. In particular, prove that if the monoidal unit is compact, then an object is dualizable if and only if it is compact and nuclear...

For a general complete Huber pair $(A,A^+)$, define the analytic ring $(A,A^+)_\blacksquare$. Prove that the associations $\mathrm{Spa}\, (A,A^+)\mapsto \mathcal{D}((A,A^+)_\blacksquare)$, $\mathrm{Spa}\, (A,A^+)\mapsto \mathcal{D}^\omega((A,A^+)_...

Define the category of solid quasi-coherent sheaves $\mathcal{D}_\blacksquare(X)$ on a rigid-analytic varity $X$. Show that this assignment comes with a natural $6$-functor formalism. Finally, prove the Grothendieck--Serre duality in the rigid...

Prove that for a qcqs analytic adic space $X$, the category of nuclear sheaves on $X$ is dualizable in two steps. First, establish the result locally over affinoid Tate adic spaces using the notion of weak proregularity and then prove a general...

Review the notion of cd-structure for a Grothendieck topology and explain how it helps check the sheaf conditions. Introduce and discuss the Nisnevich topology for schemes and analytic adic spaces; in particular, discuss their equivalent...

Introduce the notion of hypersheaf and discuss its basic properties and equivalent characterizations. In particular, discuss the notions of homotopy and cohomological dimensions and their relation to hyperdescent. Sketch the proof of the fact for...

Explain the formulation of the Grothendieck–Riemann–Roch theorem for analytic adic spaces: go through [And23, pp. 32-38] and define all relevant objects and maps. Before explaining the construction of the Chern class map, define the sheaf KU∧p on...