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UW Madison mathematics Colloquium is ONLINE on Fridays at 4:00 pm.

Fall 2020

September 25, 2020, Joseph Landsberg (Texas A&M)

(Hosted by Gurevitch)

From theoretic computer science to algebraic geometry: how the complexity of matrix multiplication led me to the Hilbert scheme of points.

In 1968 Strassen discovered the way we multiply nxn matrices (row/column) is not the most efficient algorithm possible. Subsequent work has led to the astounding conjecture that as the size n of the matrices grows, it becomes almost as easy to multiply matrices as it is to add them. I will give a history of this problem and explain why it is natural to study it using algebraic geometry and representation theory. I will conclude by discussing recent exciting developments that explain the second phrase in the title.

October 9, 2020, Carolina Araujo (IMPA)

(Hosted by Ellenberg)

Symmetries in Algebraic Geometry and Cremona transformations

In this talk I will discuss symmetries of complex algebraic varieties. When studying a projective variety $X$, one usually wants to understand its symmetries. Conversely, the structure of the group of automorphisms of $X$ encodes relevant geometric properties of $X$. After describing some examples of automorphism groups of projective varieties, I will discuss why the notion of automorphism is too rigid in the scope of birational geometry. We are then led to consider another class of symmetries of $X$, its birational self-maps. Birational self-maps of the projective space $\mathbb{P}^n$ are called Cremona transformations. Describing the structure of the group of Cremona transformations of the plane is a classical problem that goes back to the 19th century. In higher dimensions, not so much is known, and a natural problem is to construct interesting subgroups of the Cremona group. I will end by discussing a recent work with Alessio Corti and Alex Massarenti, where we investigate subgroups of the Cremona group consisting of symmetries preserving some special meromorphic volume forms.

October 23, 2020, Jeremy Quastel (University of Toronto)

(Hosted by Gorin)

Towards KPZ Universality

The 1-d KPZ universality class contains random interface growth models as well as random polymer free energies and driven diffusive systems. The KPZ fixed point has now been determined, through the exact solution of a special model in the class, TASEP, and is expected to describe the asymptotic fluctuations for all models in the class. It is an integrable Markov process, with transition probabilities described by a system of integrable PDE’s. Very recently, new techniques have become available to prove the convergence of the KPZ equation itself, as well as some non-integrable extensions of TASEP, to the KPZ fixed point. This talk will be a gentle introduction to these developments with no prior knowledge assumed. The results are, variously, joint works with Daniel Remenik, Konstantin Matetski, and Sourav Sarkar.

November 6, 2020, Yiannis Sakellaridis (Johns Hopkins University)

(Hosted by Gurevitch)

Harmonic analysis, intersection cohomology, and L-functions.

The goal of this lecture will be to describe a link between geometric-topological objects (certain intersection complexes on singular loop spaces), and objects of arithmetic interest (L-functions). The link between the two is by a Fourier/spectral transform. I will begin by giving an overview of Iwasawa–Tate theory, which expresses the Riemann zeta function as the Mellin transform of a certain theta series, and will conclude by describing joint work with Jonathan Wang (MIT), which expresses other L-functions as spectral transforms of functions obtained from intersection complexes on singular arc spaces. No prior familiarity with notions such as L-functions or intersection cohomology will be assumed.

November 20, 2020, Ngoc Mai Tran (University of Texas)

(Hosted by Rodriguez)

Does your problem have a tropical solution?

Tropical mathematics is mathematics done in the min-plus (or max-plus) algebra. The power of tropical mathematics comes from two key ideas: (a) tropical objects are limits of classical ones, and (b) the geometry of tropical objects is polyhedral. In this talk I'll demonstrate how these two ideas are used to solve a variety of problems in different domains the last 10 years, from deep neural networks, semigroups theory, auction theory and extreme value statistics.

December 4, 2020, Federico Ardila (San Francisco)

(Hosted by Ellenberg)

Measuring polytopes through their algebraic structure.

Generalized permutahedra are a beautiful family of polytopes with a rich combinatorial structure, and strong connections to optimization and algebraic geometry. We prove they are the universal family of polyhedra with a certain Hopf-algebraic structure. This Hopf-algebraic structure is compatible with McMullen’s foundational work on the polytope algebra.

Our construction provides a unifying framework to organize and study many combinatorial families; for example:

1. It uniformly answers open questions and recovers known results about graphs, posets, matroids, hypergraphs, simplicial complexes, and others.

2. It shows that permutahedra and associahedra “know" how to compute the multiplicative and compositional inverses of power series.

3. It explains the mysterious fact that many combinatorial invariants of matroids, posets, and graphs can also be thought of as measures on polytopes, satisfying the inclusion-exclusion relations.

This is joint work with Marcelo Aguiar (2017) and Mario Sanchez (2020).