Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, October 9, 2018.

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Tuesday, October 9, 2018

11:00 am in 345 Altgeld Hall,Tuesday, October 9, 2018

Chromatic homotopy is algebraic when $p > n^2+n+1$

Piotr Pstrągowski (Northwestern Math)

Abstract: It is generally accepted that the structure of the $E(n)$-local homotopy theory, where $E(n)$ is the p-local Johnson-Wilson spectrum at height $n$, becomes increasingly algebraic when $p$ is large with respect to $n$. To give two examples of this phenomena, when the prime is large, the $E(n)$-based Adams spectral sequence collapses on the second page for degree reasons, giving an algebraic description of the homotopy of the $E(n)$-local sphere. Moreover, it is a result of Hovey and Sadofsky that the only invertible spectra in this range are the spheres, showing that the $E(n)$-local Hopkins' Picard group is isomorphic to the integers at large primes, in stark contrast to what happens when $p$ is small. In this talk, we show that when $p > n^2+n+1$, the homotopy category of $E(n)$-local spectra is equivalent to the homotopy category of differential $E(n)_*E(n)$-comodules, giving a precise sense in which chromatic homotopy is "algebraic" at large primes. This extends the work of Bousfield at $n = 1$ to all heights, and affirms a conjecture of Franke.

1:00 pm in 347 Altgeld Hall,Tuesday, October 9, 2018

The Ricci iteration on homogeneous spaces

Artem Pulemotov (University of Queensland)

Abstract: The Ricci iteration is a discrete analogue of the Ricci flow. Introduced in 2007, it has been studied extensively on Kähler manifolds, providing a new approach to uniformisation. In the talk, we will define the Ricci iteration on compact homogeneous spaces and discuss a number of existence, convergence and relative compactness results. This is largely based on joint work with Timothy Buttsworth (Queensland), Yanir Rubinstein (Maryland) and Wolfgang Ziller (Penn).

1:00 pm in 345 Altgeld Hall,Tuesday, October 9, 2018

From univalence to the fundamental group of the circle

Egbert Rijke (UIUC Math)

Abstract: We show how the univalence axiom can be used to construct the fundamental cover of the circle, which is then used to show that the loop space of the circle is equivalent to the integers.

2:00 pm in 243 Altgeld Hall,Tuesday, October 9, 2018

Independent sets in algebraic hypergraphs

Anush Tserunyan (Illinois Math)

Abstract: A modern trend in extremal combinatorics is extending classical results from the dense setting (e.g. Szemerédi's theorem) to the sparse random setting. More precisely, one shows that a property of a given "dense" structure is inherited by a randomly chosen "sparse" substructure. A recent breakthrough tool for proving such statements is the Balogh–Morris–Samotij and Saxton–Thomason hypergraph containers method, which bounds the number of independent sets in homogeneously dense finite hypergraphs, thus implying that a random sparse subset is not independent. Another trend in combinatorics is proving combinatorial properties for algebraic, or more generally, model theoretically definable structures. Jointly with A. Bernshteyn and M. Delcourt, we combine these trends, establishing a containers-type theorem for hypergraphs definable over an algebraically closed field: if such a hypergraph is "dense", then Zariski-generic low-dimensional sets of vertices induce a relatively "dense" subhypergraph (in particular, they are not independent).

3:00 pm in 243 Altgeld Hall,Tuesday, October 9, 2018

On the Hitchin fibration for algebraic surfaces

Tsao-Hsien Chen (University of Chicago)

Abstract: In his work on non-abelian Hodge theory, Simpson constructs the Hitchin map from the moduli spaces of Higgs bundles over an arbitrary smooth algebraic variety X to an affine space, generalizing Hitchin's construction in the case of when X is a Riemann surface. Very little is known about the geometry of Simpson's Hitchin map except in the case when X is one-dimensional. In the talk I will report on some recent developments on the structure of Hitchin map for higher dimensional varieties with emphasis on the case of algebraic surfaces. Joint work with B.C. Ngo.