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# Events for 09/27/2019 from all calendars

## Probability Seminar

## Mathematical Physics and Harmonic Analysis Seminar

## Algebra and Combinatorics Seminar

## Geometry Seminar

**Time:** 11:30AM - 12:30PM

**Location:** BLOC 628

**Speaker:** Patricia Alonso Ruiz, TAMU

**Title:** *Order statistics in stochastic geometry*

**Abstract:** This talk will discuss and give an overview of questions concerning the asymptotic distribution of order statistics for functionals of spatial Poisson point processes. We will highlight some techniques from stochastic geometry applied in this setting.

**Time:** 1:50PM - 2:50PM

**Location:** BLOC 628

**Speaker:** Blake Keeler, UNC

**Title:** *Random Waves and the Spectral Function on Manifolds without Conjugate Points*

**Abstract:** In this talk, we discuss off-diagonal Weyl asymptotics on a compact manifold M, with the goal of understanding the statistical properties of monochromatic random waves. These waves can be thought of as randomized "approximate eigenfunctions," and their statistics are completely determined by an associated covariance kernel which coincides exactly with a rescaled version of the spectral function of the Laplace-Beltrami operator. We will prove that in the geometric setting of manifolds without conjugate points, one can obtain a logarithmic improvement in the two-point Weyl law for this spectral function, provided one restricts to a shrinking neighborhood of the diagonal in M x M. This then implies that the covariance kernel of a monochromatic random wave locally converges to a universal limit at a logarithmic rate as we take the frequency parameter to infinity. This result generalizes the work of Berard, who obtained the logarithmic improvement in the on-diagonal case for manifolds with nonpositive curvature.

**Time:** 3:00PM - 3:50PM

**Location:** BLOC 506A **

**Speaker:** Amy Huang, Texas A&M University

**Title:** *Syzygies of determinantal thickenings and gl(m|n) representations*

**Abstract:** The coordinate ring $S = \mathbb{C}[x_{i,j}]$ of space of $m \times n$ matrices carries an action of the group $\mathrm{GL}_m \times \mathrm{GL}_n$ via row and column operations on the matrix entries. If we consider any $\mathrm{GL}_m \times \mathrm{GL}_n$-invariant ideal $I$ in $S$, the syzygy modules $\mathrm{Tor}_i(I,\mathbb{C})$ will carry a natural action of $\mathrm{GL}_m \times \mathrm{GL}_n$. Via BGG correspondence, they also carry an action of $\bigwedge^{\bullet} (\mathbb{C}^m \otimes \mathbb{C}^n)$. It is a recent result by Raicu and Weyman that we can combine these actions together and make them modules over the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. We will explain how this works and how it enables us to compute all Betti numbers of any $\mathrm{GL}_m \times \mathrm{GL}_n$-invariant ideal $I$. The latter part will involve combinatorics of Dyck paths

**Time:** 4:00PM - 5:00PM

**Location:** BLOC 628

**Speaker:** Souvik Goswami, Texas A&M University

**Title:** *Height pairing on Bloch's higher cycles and mixed Hodge structures.*

**Abstract:** In a previous work with JosÃ© Ignacio Burgos, we have studied the higher arithmetic Chow groups. As a by product, an Archimedean height pairing between higher cycles has been defined. Classically, Hain has shown that the Archimedean component of the height pairing between ordinary cycles can be interpreted as the class of a biextension in the category of mixed Hodge structures. In the current work we study the mixed Hodge structure defined by a pair of higher cycles intersecting properly and show that, in a special case, the Archimedean height pairing is one of the periods attached to such mixed Hodge structure. This is joint work in progress with Greg Pearlstein and JosÃ© Ignacio Burgos.

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