Geometry and Topology

Speaker: Liang Ze Wong (Institute of High Performance Computing)

"Cubes with connections from algebraic weak factorization systems"

Time: 19:00 - 20:00

Room: Zoom Meeting ID: 958 6908 4555

The humble interval is a gateway to both homotopy theory and higher-dimensional geometry. The interval allows us to define homotopies between continuous functions, while taking products of the interval with itself gives rise to the square, cube, tesseract and beyond. Abstracting away from topological spaces, one may speak of interval objects, cylinder objects or even cylinder functors on other categories, and use these to define homotopies there. However, the connection to higher-dimensional cubes is lost in the process. In this talk, I will show that in categories with algebraic weak factorization systems (AWFS) -- which provide a conducive setting for abstract homotopy theory -- we can recover cylinder functors that share both the homotopical and "cubical" aspects of the interval. More precisely, for any object $X$ in a category $\mathcal{C}$ equipped with coproducts and an AWFS, there is a functor from the category of cubes-with-connections to $\mathcal{C}$ that sends the 0-cube to $X$, the 1-cube to the cylinder on $X$, and so on. As a corollary, any such category is enriched in cubical sets with connections. (Joint work with Chris Kapulkin.)

Algebra Seminar

Speaker: Pranav Chakravarthy (Western)

"Homotopy type of equivariant symplectomorphisms of rational ruled surfaces."

Time: 13:30 - 14:30

Room: Zoom: 998 5635 1219

Darboux's theorem states that all symplectic manifolds locally look alike. Consequently, there are no local invariants in symplectic geometry, and one must look for global invariants to probe symplectic manifolds. Such invariants can be obtained by investigating the homotopy type of mapping spaces related to the symplectic structure. In this talk, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $CP^2$ blown up once under the presence of hamiltonian group actions of either $S^1$ or finite cyclic groups.

Geometry and Combinatorics

Speaker: Anthony Bahri (Rider University)

"On the cohomology of polyhedral products - additive results"

Time: 14:30 - 15:30

Room: online

The talk is a report on the recent completion of a project dating from 2010. Our goal is an explicit description of the additive and multiplicative structure of the cohomology of a polyhedral product. The result, for field coefficients and general CW pairs, comes with a transparency sufficient to allow for computation. The problem has an extensive history which I shall try to outline briefly. Some of the earliest results in this direction, for the case of moment-angle complexes, appeared in the work of M. Franz, I. Baskakov, V. Buchstaber and T. Panov and also S. Lopez de Medrano. The focus in this lecture will be on the additive results. This is joint work with M. Bendersky, F.R. Cohen and S. Gitler.

Geometry and Topology

Speaker: Reid Barton (University of Pittsburgh)

"Model categories for o-minimal homotopy theory"

Time: 15:30 - 16:30

Room: Zoom Meeting ID: 958 6908 4555

O-minimality is a branch of model theory with roots in real algebraic geometry which provides a family of settings for "tame topology": flexible enough to include most functions used in homotopy theory but without pathologies such as space-filling curves.

I will introduce a model category of spaces based on the definable sets of any o-minimal structure. These model categories resemble the Serre--Quillen model structure on topological spaces but inherit technical advantages from their construction. At the same time, they provide a context in which to better understand the weak polytopes of Knebusch (generalized to the o-minimal setting by Piekosz). This talk is based on joint work with Johan Commelin.

Algebra Seminar

Speaker: Felix Baril Boudreau (Western)

"L-functions of Elliptic Curves over Function Fields"

Time: 13:30 - 14:30

Room: Zoom: 998 5635 1219

Given a global function field of characteristic p > 0 and an elliptic curve over it, one can study its L-function. The L-function of an elliptic curve is an interesting object at the interface of complex analysis and algebra and lead to the famous Birch and Swinnerton-Dyer conjecture.

Moreover, explicitly computing an L-function is a non-trivial task. Naive methods involve point-counting and are inefficient. Therefore, alternatives should be welcome.

In this talk, we will first introduce the necessary background on elliptic curves and L-functions. Then, we will discuss a possible approach to effectively tackle the problem of computing L-functions of elliptic curves over global function fields.

Geometry and Topology

Speaker: Joachim Kock (Universitat Autonoma de Barcelona)

"Decomposition spaces, incidence algebras and Mobius inversion"

Time: 11:30 - 12:30

Room: Zoom

I'll start briefly with the classical theory of incidence algebras for posets (Rota 1963) and Leroux's generalisation to certain categories called Mobius categories (1975). A key element in this theory is Mobius inversion, a counting device exploiting how combinatorial objects can be decomposed. Then I will survey recent work with Imma Galvez and Andy Tonks developing a far-reaching generalisation to something called decomposition spaces (or 2-Segal spaces [Dyckerhoff-Kapranov]). There are three directions of generalisation involved: firstly, the theory is made objective, meaning that it works with the combinatorial objects themselves, rather than with vector spaces spanned by them. This can be seen as a systematic way of turning algebraic proofs into bijective proofs. The role of vector spaces is played by slice categories. Secondly, the theory incorporates homotopy theory by passing from categories to infinity-categories in the form of Segal spaces. (This is relevant even for classical combinatorics to deal with symmetries.) Finally, the Segal condition is replaced by something weaker (decomposition spaces): where the Segal condition expresses composition, the new condition expresses decomposition. This allows to cover a wide range of combinatorial Hopf algebras that cannot directly be the incidence algebra of any poset or Mobius category, such as the Butcher-Connes-Kreimer Hopf algebra of trees, or Schmitt's chromatic Hopf algebra of graphs. It also turns out to have interesting connections to representation theory, covering all kinds of Hall algebras: the Waldhausen S-construction of an abelian category is an example of a decomposition space. I will finish with the general Mobius inversion principle for decomposition spaces. Throughout I will stress the general ideas behind, avoiding technicalities.

Geometry and Combinatorics

Speaker: Seonjeong Park (KAIST)

"On generic torus orbit closures in Richardson varieties"

Time: 09:00 - 10:00

Room: online

The flag variety $\mathcal{F}\ell_n$ is a smooth projective variety consisting of chains $(\{0\}\subset V_1\subset\cdots\subset V_n=\mathbb{C}^n)$ of subspaces of $\mathbb{C}^n$ with $\dim_{\mathbb{C}} V_i=i$. Then the standard action of $\mathbb{T}=(\mathbb{C}^\ast)^n$ on $\mathbb{C}^n$ induces a natural action of $\mathbb{T}$ on $\mathcal{F}\ell_n$. For $v$ and $w$ in the symmetric group $\mathfrak{S}_n$ with $v\leq w$ in Bruhat order, the Richardson variety $X^v_w$ is defined to be the intersection of the Schubert variety $X_w$ and the opposite Schubert variety $w_0X_{w_0v}$, and it is an irreducible $\mathbb{T}$-invariant subvariety of $\mathcal{F}\ell_n$. A point $x$ in $X^v_w$ is said to be generic if $(\overline{\mathbb{T}x})^\mathbb{T}=(X^v_w)^\mathbb{T}$. In this talk, we are interested in the $\mathbb{T}$-orbit closures in the flag variety which can appear as a generic $\mathbb{T}$-orbit closure in a Richardson variety. We discuss topology and combinatorics of such $\mathbb{T}$-orbit closures. This talk is based on joint work with Eunjeong Lee and Mikiya Masuda.

Algebra Seminar

Speaker: Avi Steiner (Western)

"Vanishing criteria for tautological systems"

Time: 13:30 - 14:30

Room: Zoom: 998 5635 1219

Tautological systems are vast generalizations of $A$-hypergeometric systems to the case of an arbitrary reductive algebraic group. Much of the interest in such systems has come from their application to period integrals of Calabi-Yau hypersurfaces. As with $A$-hypergeometric systems, part of the input data is a parameter $\beta$. I will discuss joint work with P. Gorlach, T. Reichelt, C. Sevenheck, and U. Walther discussing criteria which bounds the number of parameters $\beta$ which give a non-trivial tautological system.

Geometry and Combinatorics

Speaker: Bill Trok (University of Kentucky)

"Very Unexpected Hypersurfaces"

Time: 14:30 - 15:30

Room: Zoom

Often in Algebraic Geometry linear systems will have an "expected dimension" which should hold most of the time, the problem then becomes to study when this expectation fails. In this talk we discuss a version of this problem. In particular we introduce the concept of a "very unexpected hypersurface" passing through a fixed set of points Z. These occur when Z imposes less than the "expected" number of conditions on the ideal sheaf of a generic linear subspace. We show in certain cases these can be characterized via combinatorial data and geometric data from the Hyperplane Arrangement dual to Z. We close by discussing relationships between this problem and certain motivating problems in Combinatorics, Matroid Theory and Hyperplane Arrangements.