Geometry and Combinatorics

Speaker: Botong Wang (University of Wisconsin (Madison))

"The Hodge theory of hyperplane arrangements and matroids"

Time: 15:30

Room: Zoom

Given a hyperplane arrangement, we associate two projective varieties: the wonderful compactification and the matroid Schubert variety. Adiprasito, Huh and Katz used the Hodge-Riemann relations of the wonderful compactification (and their combinatorial generalizations) to prove that the coefficients of their characteristic polynomials form a log-concave sequence. In a joint work with Huh, we proved Dowling and Wilson's Top-heavy conjecture for realizable matroids by applying the hard Lefschetz theorem to the matroid Schubert varieties. In a more recent work with Braden, Huh, Matherne and Proudfoot, we proved the Top-heavy conjecture to arbitrary matroids. In this talk, I will go over some of the key ideas about the proof of the Top-heavy conjectures. If time permits, I will also mention some on-going works with my students Colin Crowley and Connor Simpson towards generalizations to type A Coxeter matroids.

Geometry and Topology

Speaker: Viktoriya Ozornova (Ruhr-Universitaet Bochum)

"Pasting diagrams in $(\infty,2)$-categories"

Time: 11:30

Room: Zoom Meeting ID: 958 6908 4555

In the world of $(\infty,1)$-categories, it is well-known that the composition of specified morphisms is well-defined up to a contractible choice. The situation for $(\infty,2)$-categories is more subtle, as there are many potential ways of composing 2-cells. In a joint work in progress with Hackney, Riehl, Rovelli, we prove a uniqueness statement for the composition of so-called pasting diagrams, which I will explain during the talk.

Algebra Seminar

Speaker: Tung T. Nguyen (University of Chicago)

"Power sums and special values of L-functions"

Time: 14:30

Room: Zoom

The zeta functions are a pillar of number theory. Zeta functions have been objects of great interest for number theorists due to their beauty, mystery, and power. I will discuss this study in the simplest case: the Hurwitz zeta functions. Recently, some surprising direct connections between the special values of Hurwitz zeta functions and power sums were found. In my talk, I will introduce these discoveries. In particular, we will see that special values of Hurwitz zeta functions have some nice integral representations. This is joint work with Jan Minac and Nguyen Duy Tan.

Geometry and Combinatorics

Speaker: Laurentiu Maxim (University of Wisconsin, Madison)

"Hodge theory on Alexander invariants"

Time: 15:30

Room: Zoom

I will give an overview of recent developments in the study of Hodge-theoretic aspects of Alexander-type invariants associated with smooth complex algebraic varieties. Our results are motivated by (and can be regarded as global analogues of) similar statements for the Milnor fiber cohomology of complex hypersurface singularity germs. (Joint work with E. Elduque, C. Geske, M. Herradon-Cueto and B. Wang).

Geometry and Topology

Speaker: Ronnie Chen (University of Illinois Urbana-Champaign)

"Gabriel-Ulmer duality for continuous categories"

Time: 15:30

Room: Zoom Meeting ID: 958 6908 4555

The classical Gabriel-Ulmer duality asserts a dual adjoint equivalence between finitely complete categories, and a full sub-2-category of the complete, filtered-cocomplete categories which are known as locally finitely presentable (LFP). The definition of LFP category involves ``exactness conditions'' asserting compatibility between limits and filtered colimits, together with a different sort of condition which amounts to admitting enough structure-preserving functors to Set; removing this last condition yields the continuous locally presentable (CLP) categories in the sense of Johnstone-Joyal. We prove an analog of Gabriel-Ulmer duality for all CLP categories, by replacing the dualizing category Set with the category CPUMet of complete partial ultrametric spaces. As with Gabriel-Ulmer duality, this result has a logical interpretation, as a strong conceptual completeness theorem for the ``lex fragment'' of a continuous first-order logic for partial ultrametric structures.

Algebra Seminar

Speaker: Harris Daniels (Amherst College)

"Entanglements of Division Fields of Elliptic Curves"

Time: 14:30

Room: Zoom

Central objects in the study of elliptic curves are the fields of definition of the points of order n. Relatively little is known about the way in which a fixed elliptic curve's division fields can intersect. In this talk, we lay the foundations for a systematic study of the entanglements of division fields of elliptic curves from a group theoretic perspective. In particular, we classify the way in which two prime-level division fields can intersect for infinitely many elliptic curves. This is joint work with Jackson S. Morrow.

Geometry and Topology

Speaker: Alexander Campbell (Macquarie University)

"A model-independent construction of the Gray monoidal structure for $(\infty,2)$-categories"

Time: 19:00

Room: Zoom Meeting ID: 958 6908 4555

In this talk I will describe joint work with Yuki Maehara in which we give a model-independent (i.e. a purely $\infty$-categorical) construction of the (non-symmetric) Gray monoidal structure on the $\infty$-category of $(\infty,2)$-categories. Our construction is a generalisation to the $\infty$-categorical setting of a construction of the Gray monoidal structure for 2-categories due to Ross Street, which uses the techniques from Brian Day's PhD thesis for extending a monoidal structure along a dense functor. The proof of our construction uses, among other things, the results from Yuki's PhD thesis on the Gray tensor product for 2-quasi-categories. I will also mention a few of the open problems concerning the Gray monoidal structure for $(\infty,2)$-categories, and explain how our results can be used to simplify (though not yet solve) one of these problems.

Algebra Seminar

Speaker: Nitin Chidambaram (MPI Bonn)

"Kernels for GIT wall crossings"

Time: 14:30

Room: Zoom

In this talk, I'll present a recent proposal of a Fourier-Mukai kernel for VGIT problems associated to flops due to Ballard-Diemer-Favero. When the GIT problem is that of $\mathbb G_m $ acting on a smooth scheme, they check that these kernels produce the `grade-restriction windows' that have been well studied in the literature. In the non-smooth case, we show that this proposal produces windows by working on "smooth" dg-resolutions. If time permits, we will also discuss some work in progress concerning generalizations to general groups.

Geometry and Combinatorics

Speaker: Robin van der Veer (Leuven)

"MLE, tropical geometry and slopes of Bernstein-Sato ideals"

Time: 15:30

Room: Zoom

Let $X$ be a smooth subvariety of a complex torus. For general data vectors the MLE problem on $X$ has exactly $|\chi(X)|$ solutions. We investigate what happens to these solutions when the data vector approaches a non-general value. Assuming that $X$ is schön we relate this behaviour to special rays in the tropical variety of $X$. We also explain how the non-general data vectors are related a Bernstein-Sato ideal associated to X. Based on joint work with Anna-Laura Sattelberger.

Geometry and Topology

Speaker: Ezra Getzler (Northwestern University)

"Complete Segal spaces in the theory of derived stacks"

Time: 15:30

Room: Zoom Meeting ID: 958 6908 4555

Derived stacks play the same role in the theory of moduli that projective resolutions play in the study of modules. In the work of Toen and Vezzosi, Lurie, and Pridham, derived stacks are realized as fibrant objects in a combinatorial model category. We prefer to think of derived stacks as the objects of a category of fibrant objects, the right fibrations of simplicial derived schemes with a fixed base $B_*$.

An important model for $\infty$-categories is Rezk's theory of complete Segal spaces. Their definition actually makes sense for simplicial objects in any category of fibrant objects. In joint work with Kai Behrend, we show that the complete Segal spaces in a category of fibrant objects are the objects of a category of fibrant objects if the category of fibrant objects satisfying an additional axiom:

If $f \colon X \to Y$ is a trivial fibration and $g \colon Y \to Z$ is a morphism such that $gf$ is a fibration, then $g$ is a fibration.

Many (if not all) categories of fibrant objects satisfy this axiom: for example, it holds for Kan complexes.

There is a functor from quasicategories to complete Segal spaces, which shows that these two theories are closely related. Boavida de Brito and Rasekh extend this to a functor from cartesian fibrations to complete Segal spaces in the category of right fibrations. Thus, the above result may be interpreted as a concrete realization of cartesian fibrations in derived geometry.

Algebra Seminar

Speaker: Brandon Doherty (Western)

"Cubical models of (infinity,1)-categories"

Time: 14:30

Room: Zoom

We describe a new model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We discuss the proof that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor, and a new theory of cones in cubical sets which is used in this proof. We also introduce the homotopy category and mapping spaces of a fibrant cubical set, and characterize weak equivalences between fibrant cubical sets in terms of these concepts. This talk is based on joint work with Chris Kapulkin, Zachery Lindsey, and Christian Sattler, arXiv:2005.04853.

Geometry and Combinatorics

Speaker: (Western)

"no seminar this week"

Time: 15:30

Room:

Geometry and Topology

Speaker: Ralph Kaufmann (Purdue University)

"Natural appearances of cubical structures"

Time: 15:30

Room: Zoom Meeting ID: 958 6908 4555

Cubical structures appear naturally in several situations, sometimes clandestinely. In particular, we will consider such structures related to Feynman categories and discuss their origins. In this context, they appear for instance in coproduct structures, in a W-construction and relative W-constructions given via push-forwards, which give natural cubical complexes that are homotopy equivalent to moduli spaces.

Colloquium

Speaker: Graham Denham (Western)

"Lorentzian polynomials 1"

Time: 15:30

Room: Via Zoom

A complex polynomial in n variables satisfies the (Hurwitz) half-plane property if its value is nonzero when its inputs all have positive real part. This classical definition is the start of an interesting story that factors through the closely related theory of stable polynomials, due to Borcea and Brändén [Duke J. Math 2008] and leads to the notion of Lorentzian polynomials, recently introduced by Brändén and Huh [Annals of Math 2020]. Lorentzian polynomials also have an elementary definition, though subtle properties and close links to (statistical) negative dependence, matroid theory, and discrete convexity.

In this two-part Basic Notions seminar, Graham plans to spend a few minutes saying why he thinks this is interesting. Then together we will watch a recording of an introductory lecture that June Huh gave at the IAS in 2019. We will break in the middle, since the lecture is 90 minutes long. We will resume in the second week and conclude with some informal discussion.