To volunteer to give a talk, or for anything else regarding the seminar, contact Claudiu Raicu or Eric Riedl.

Abstracts can be found below. The seminar will meet on Tuesdays, 2:30–3:30pm, either on Zoom or in 258 Hurley, ** unless otherwise noted.** Related events are also listed below.

Date | Speaker | Title |
---|---|---|

Tuesday, Sep. 6 | Steven Karp (Notre Dame) | Wronskians, total positivity, and real Schubert calculus |

Tuesday, Sep. 13 | Joe Waldron (Michigan State) | Purely inseparable Galois theory |

Tuesday, Sep. 20 | Mark Skandera (Lehigh University) | Type-BC character evaluations and P-tableaux |

Tuesday, Sep. 27 | Joe Cummings (Notre Dame) | A Fano compactification of the free group character variety |

Tuesday, Oct. 4 | Joshua Mundinger (Chicago) | Quantization of restricted Lagrangian subvarieties in positive characteristic |

Tuesday, Oct. 11, 2:30–3:30pm | Bernd Ulrich (Purdue) | TBA |

Tuesday, Oct. 11, 3:30–4:30pm 127 Hayes-Healy Note special time/location |
Hunter Simper (Purdue) | TBA |

Tuesday, Oct. 18 | No seminar (Fall break) | |

Tuesday, Oct. 25 | Cheng Meng (Purdue) | TBA |

Tuesday, Nov. 1 | ||

Tuesday, Nov. 8 | Brandon Alberts (Eastern Michigan) | TBA |

Tuesday, Nov. 15 | Nitin Chidambaram (Bonn) | TBA |

Tuesday, Nov. 22 | Mihai Fulger (Connecticut) | TBA |

Tuesday, Nov. 29 | Martha Precup (Washington University in St. Louis) | TBA |

Tuesday, Dec. 6 | Lena Ji (Michigan) | TBA |

**Speaker**- Steven Karp (Notre Dame)
**Title**- Wronskians, total positivity, and real Schubert calculus
**Abstract**- The totally positive flag variety is the subset of the complete flag variety Fl(n) where all Plücker coordinates are positive. By viewing a complete flag as a sequence of subspaces of polynomials of degree at most n-1, we can associate a sequence of Wronskian polynomials to it. I will present a new characterization of the totally positive flag variety in terms of Wronskians, and explain how it sheds light on conjectures in the real Schubert calculus of Grassmannians. In particular, a conjecture of Eremenko (2015) is equivalent to the following conjecture: if V is a finite-dimensional subspace of polynomials such that all complex zeros of the Wronskian of V are real and negative, then all Plücker coordinates of V are positive. This conjecture is a totally positive strengthening of a result of Mukhin, Tarasov, and Varchenko (2009), and can be reformulated as saying that all complex solutions to a certain family of Schubert problems in the Grassmannian are real and totally positive.

**Speaker**- Joe Waldron (Michigan State)
**Title**- Purely inseparable Galois theory
**Abstract**- Given a field K of characteristic p, a classical result of Jacobson provides a Galois correspondence between finite purely inseparable subfields of K of exponent one (i.e. those which contain K^p), and sub-restricted Lie algebras of Der(K). I will discuss joint work with Lukas Brantner in which we extend this Galois correspondence to subfields of arbitrary exponent using methods from derived algebraic geometry.

**Speaker**- Mark Skandera (Lehigh University)
**Title**- Type-BC character evaluations and P-tableaux
**Abstract**- For w in S_n corresponding to a smooth type-A Schubert variety X(w), there exists a poset P = P(w) which provides a combinatorial interpretation of the Kazhdan-Lusztig basis element C_w(1) in Z[S_n]. Evaluations of characters at C_w(1) may be computed by counting special P-tableaux. We state a type-BC analog of these results. Namely, for w in the hyperoctahedral group B_n such that the type-B and type-C Schubert varieties are simultaneously smooth, there exists a type-BC poset P = P(w) which provides a combinatorial interpretation of the Kazhdan-Lusztig basis element C_w(1) in Z[B_n]. Again, evaluations of characters at the Kazhdan-Lusztig basis element C_w(1) of Z[B_n] may be computed by counting special P-tableaux.

**Speaker**- Joe Cummings (Notre Dame)
**Title**- A Fano compactification of the free group character variety
**Abstract**- We show that a certain compactification of the free group character variety $\mathcal{X}(F_g, Sl_2(\mathbb{C}))$ is Fano. This compactification has been studied previously by Manon, and separately by Biswas, Lawton, and Ramras. Part of the proof involves the construction of a large family of integral reflexive polytopes arising from trivalent graphs of genus $g$. This project is joint with Chris Manon.

**Speaker**- Joshua Mundinger (Chicago)
**Title**- Quantization of restricted Lagrangian subvarieties in positive characteristic
**Abstract**- In order to answer basic questions in modular and geometric representation theory, Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties in positive characteristic. These are certain noncommutative algebras over a field of positive characteristic which have a large center. I will discuss recent work describing how to construct an important class of modules over such algebras.