# Geometry Seminar

## Fall 2023

Date: | September 8, 2023 |

Time: | 4:00pm |

Location: | BLOC 302 |

Speaker: | Georgy Scholten, Sorbonne Université |

Title: | Global Optimization of Analytic Functions over Compact Domains |

Abstract: | In this talk, we introduce a new method for minimizing analytic Morse functions over compact domains through the use of polynomial approximations. This is, in essence, an effective application of the Stone-Weierstrass Theorem, as we seek to extend a local method to a global setting, through the construction of polynomial approximants satisfying an arbitrary set precision in L-infty norm. The critical points of the polynomial approximant are computed exactly, using methods from computer algebra. Our Main Theorem states probabilistic conditions for capturing all local minima of the objective function $f$ over the compact domain. We present a probabilistic method, iterative on the degree, to construct the lowest degree possible least-squares polynomial approximants of f which attains a desired precision over the domain. We then compute the critical points of the approximant and initialize local minimization methods on the objective function f at these points, in order to recover the totality of the local minima of f over the domain. |

Date: | September 15, 2023 |

Time: | 4:00pm |

Location: | BLOC 302 |

Speaker: | Chong-Kyu Han, Seoul National University |

Title: | First integrals, stability of dynamics, and affine control with prescribed invariant subsets |

Abstract: | Starting with the classical theory of first integrals I will talk about the notion of weak first integral and how to find implicit solutions of quasi-linear systems of first order PDEs. In determined cases this method has applications to determining the stability and affine control of the population dynamics of Volterra-Kolmogorov type. |

Date: | September 18, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Stephen Karp, Notre Dame |

Title: | Positivity in real Schubert calculus |

Abstract: | Cchubert calculus involves studying intersection problems among linear subspaces of C^n. A classical example of a Schubert problem is to find all 2-dimensional subspaces of C^4 which intersect 4 given 2-dimensional subspaces nontrivially (it turns out there are 2 of them). In the 1990s, B. and M. Shapiro conjectured that a certain family of Schubert problems has the remarkable property that all of its complex solutions are real. This conjecture inspired a lot of work in the area, including its proof by Mukhin-Tarasov-Varchenko in the 2000s, based on a correspondence between solutions of such a Schubert problem and eigenspaces of a family of commuting operators. I will present a strengthening of this correspondence which explicitly solves the Schubert problem inside the group algebra of the symmetric group. This implies a positive version of the Shapiro-Shapiro conjecture, which thereby resolves some conjectures of Sottile, Eremenko, Mukhin-Tarasov, and myself. This is joint work with Kevin Purbhoo. |

Date: | September 22, 2023 |

Time: | 4:00pm |

Location: | BLOC 302 |

Speaker: | Timothy Duff, U. Washington Seattle |

Title: | Geometry of two, three, or four cameras |

Abstract: | I will introduce a line of work that aims to characterize the set of all valid algebraic constraints that relate any number of perspective cameras, 3D points, and their 2D projections. More formally, this framework involves the study of certain multigraded vanishing ideals. This leads to several new results, as well as new proofs of old results about the well-studied multiview ideal. For example, a "folklore theorem" from geometric computer vision states: "all algebraic constraints on the 2D projections of 3D points can be obtained from those involving 2, 3, or 4 cameras." (joint w/ S. Agarwal, E. Connelly, J. Loucks-Tavitas, R. Thomas) I will also discuss a complementary line of work focused on practical estimation methods. Incremental 3D reconstruction systems usually focus on estimating the relative orientation of two cameras. This in turn requires solving systems of algebraic equations with (very) special structure. I will describe recent progress extending the domain of such solvers to problems involving three or four cameras, including non-perspective cameras with lens distortion. The key players are numerical homotopy continuation methods, and the Galois/monodromy groups that capture their inherent complexity. (joint w/ P. Hruby, K. Kohn, V. Korotynskiy, V. Larsson, A. Leykin, L. Oeding, T. Pajdla, M. Pollefeys, M. Regan) |

Date: | September 29, 2023 |

Time: | 4:00pm |

Location: | BLOC 302 |

Speaker: | Klemen Sivic, University of Ljubljana |

Title: | Applications of Borel fixed point theorem to linear algebra |

Abstract: | The Borel fixed point theorem says that an action of a solvable algebraic group on a projective variety has a fixed point. In the talk we will show how this theorem can help us to determine the upper bound on dimension of certain matrix spaces. In particular, we will consider spaces of matrices with bounded number of eigenvalues and spaces of matrices with commutators of bounded rank. |

Date: | October 6, 2023 |

Time: | 4:00pm |

Location: | BLOC 302 |

Speaker: | Stephen McKean, Harvard |

Title: | Real bitangents to plane quartics |

Abstract: | Using tools from the enriched enumerative geometry program, Larson and Vogt gave a signed count for the number of real bitangents to a plane quartic. This signed count depends on the choice of an auxiliary “orienting” line. Larson and Vogt proved that the signed count is always non-negative and conjectured an upper bound of 8. Mario Kummer and I proved this conjecture using some real algebraic geometry and a few basic facts about quadrilaterals. In this talk, I will explain the background leading up to Larson and Vogt’s conjecture, as well as my joint work with Kummer. |

Date: | October 16, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Michail Savvas, UT Austin |

Title: | Stabilizer reduction and partial desingularization in derived geometry |

Abstract: | Suppose that a group acts on a variety. When can the variety and the action be resolved so that all stabilizers are finite and the space of orbits desingularized, at least partially? Kirwan gave an answer to this question in the 1980s through an explicit blowup algorithm for smooth varieties with group actions in the context of Geometric Invariant Theory (GIT). In this talk, we will explain how to generalize Kirwan's algorithm to Artin stacks and their moduli spaces in derived algebraic geometry, which, in particular, include classical, potentially singular, quotient stacks that arise from group actions in GIT. Based on joint works with (subsets of) Eric Ahlqvist, Jeroen Hekking, Michele Pernice and David Rydh. |

Date: | October 30, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | M. Perlman, U. Minn. |

Title: | Local cohomology on spaces with finitely many orbits |

Abstract: | We consider a smooth complex variety endowed with the action of a linear algebraic group, such as a toric variety, space of matrices, or flag variety. Given an orbit, the local cohomology modules with support in its closure encode a great deal of information about its singularities and topology. We will discuss how techniques from representation theory, D-modules, and quivers may be used to compute these local cohomology modules and their related invariants. Our focus will be the spaces of (hyper)matrices and their orbit closures. |

Date: | November 6, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Thomas Brazelton, Harvard University |

Title: | Equivariant enumerative geometry |

Abstract: | Classical enumerative geometry asks geometric questions of the form "how many?" and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of "equivariant enumerative geometry", which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the S4 orbits of the 27 lines on any smooth symmetric cubic surface. |

Date: | November 17, 2023 |

Time: | 4:00pm |

Location: | BLOC 302 |

Speaker: | Keller VandeBogert, Notre Dame |

Title: | Stable Sheaf Cohomology on Flag Varieties |

Abstract: | The Borel-Weil-Bott (BWB) theorem is a fundamental result that gives a (relatively simple) method of computing the cohomology of line bundles on flag varieties over a field of characteristic 0. The analogue of BWB in positive characteristic is a wide-open problem despite many important results over the decades, and it remains out of reach even from a computational perspective. In this talk, I'll speak on joint work with Claudiu Raicu that shows that, despite the chaos, there is a notion of stability for the cohomology of line bundles on flags in arbitrary characteristic. Moreover, there are many cases where we can compute this stable sheaf cohomology explicitly, and these computations yield sharp, characteristic-free vanishing results for finite-length Koszul modules. |

Date: | December 4, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Jake Levinson, Université de Montréal |

Title: | Minimal degree fibrations in curves and asymptotic degrees of irrationality |

Abstract: | A basic question about an algebraic variety X is how similar it is to projective space. One measure of similarity is the minimum degree of a rational map from X to projective space, called the degree of irrationality. This number, and the corresponding minimal-degree maps, are in general challenging to compute, but capture special features of the geometry of X. I will discuss some recent joint work with David Stapleton and Brooke Ullery on asymptotic bounds for degrees of irrationality for divisors X on projective varieties Y. Here, the minimal-degree rational maps $X \dashrightarrow \mathbb{P}^n$ turn out to "know" about Y and factor through rational maps $Y \dashrightarrow \mathbb{P}^n$ fibered in curves. This leads to the useful notion of "minimal fibering degree in curves". |