Random Matrix Theory Seminar

Speaker: Luuk Verhoeven (Western)

"Gauge theory from classical to fuzzy"

Time: 14:40

Room: MC 106

We'll take a look at physical models coming from gauge theory. Starting from (parts of) the standard model as usual, followed by the NCG example of almost commutative manifolds and onto the goal of a gauge theory over a ``fuzzy'' manifold as in Perez-Sanchez.

Transformation Groups Seminar

Speaker: Kumar Shukla (Western)

"Applications of Localization Formula"

Time: 10:30

Room: MC 204

We will go over some applications of the Atiyah-Bott localization formula. We will evaluate the integral of the volume form of $S^2$ as a sum over the fixed point set of the rotation action of $S^1$ on $S^2$. A more non-trivial application will be in the enumerative problem of counting number of lines in a cubic surface in $\mathbb{CP}^3$. We will explain why this problem is well-posed and then compute the number using Atiyah-Bott formula.

Random Matrix Theory Seminar

Speaker: Luuk Verhoeven (Western)

"Gauge theory from classical to fuzzy II"

Time: 14:40

Room: MC 106

We'll continue to look at physical models coming from gauge theory. Starting from (parts of) the standard model as usual, followed by the NCG example of almost commutative manifolds and onto the goal of a gauge theory over a fuzzy'' manifold as in Perez-Sanchez.

Analysis Learning Seminar

Speaker: Blake Boudreaux (Western)

"Nef line bundles from a curvature perspective"

Time: 15:30

Room: MC 107

This is our second talk about "nef" line bundles; this time it will be from the perspective of curvature. We will begin with a brief discussion of currents. This will lead into a discussion of curvature of line bundles, and a new definition of a nef line bundle. We will conclude by showing this new definition is equivalent to the old one.

Transformation Groups Seminar

Speaker: Kumar Shukla (Western)

"Counting Lines in a Cubic Surface using Localization Formula"

Time: 10:30

Room: MC 204

By a cubic surface in $\mathbb{CP}^3$, we mean the zero set of a homogenous degree 3 polynomial in 4 variables. Cayley computed the number of lines on a 'generic' cubic surface to be 27. This number can be computed as the integral of the Euler class of a certain bundle over the Grassmannian $G(2, 4)$ of lines in $\mathbb{CP}^3$. To evaluate this integral, we observe that there is a certain action of 4-torus on $G(2, 4)$, and then we apply the Atiyah-Bott localization formula.

Transformation Groups Seminar

Speaker: Rafael Gomes (Western)

"Localization formula for equivariant cohomology"

Time: 14:30

Room: MC 107

In this talk, we will see how we can obtain a formula to compute the equivariant cohomology of a $G$-manifold $M$ for $G$ a group in terms of the equivariant cohomology of the fixed points $F$ of the $G$-actions. This formula depends on the equivariant Euler class of the normal bundle of the fixed points and both the push forward and pull back of the inclusion map $F\to M$ and simplifies computations quite a lot.

Random Matrix Theory Seminar

Speaker: Luuk Verhoeven (Western)

"Gauge theory from classical to fuzzy III"

Time: 14:40

Room: MC 106

We'll continue to look at physical models coming from gauge theory. Starting from (parts of) the standard model as usual, followed by the NCG example of almost commutative manifolds and onto the goal of a gauge theory over a fuzzy'' manifold as in Perez-Sanchez.

Random Matrix Theory Seminar

Speaker: Katrina Lawrence (Western)

"The Applications of Random Matrix Theory in Machine Learning and Brain Mapping"

Time: 14:30

Room: MC 107

Brain mapping analyzes the wavelengths of brain signals and outputs them in a map, which is then analyzed by a radiologist. Introducing Machine Learning (ML) into the brain mapping process reduces the variable of human error in reading such maps and increases efficiency. A key area of interest is determining the correlation between the functional areas of the brain on a voxel (3-dimensional pixel) wise basis. This leads to determining how a brain is functioning and can be used to detect diseases, disabilities, and sicknesses. As such, random noise presents a challenge in consistently determining the actual signals from the scan. This paper discusses how an algorithm created by RMT can be used as a tool for machine learning, as it detects the correlation of the functional areas of the brain. Random matrices are simulated to represent the voxel signal intensity strength for each time interval where a stimulus is presented in an fMRI scan. Using the Marchenko-Pastur law for Wishart Matrices, a result of Random Matrix Theory (RMT), it was found that no matter what type of noise was added to the random matrices, the observed eigenvalue distribution of the Wishart Matrices would converge to the theoretical distribution. This means that RMT is robust and has a high test-re-test reliability. These results further indicate that a strong correlation exists between the eigenvalues, and hence the functional regions of the brain. Any eigenvalue that differs significantly from those predicted from RMT may indicate the discovery of a new discrete brain network.

Algebra Seminar

Speaker: Séverin Philip (Institut Fourier Université Grenoble Alpes)

"Postponed to April 29 because of Good Friday"

Time: 14:30

Room: ZOOM

Geometry and Combinatorics

Speaker: Max Wakefield (US Naval Academy)

"A non-associative incidence near-ring with a generalized Möbius function"

Time: 15:30

Room: zoom

There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized Möbius function. Under the product this generalized Möbius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the Möbius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical Möbius functions. Using this generalized Möbius function we define analogues of the characteristic polynomial and Möbius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized Möbius polynomial has -1 as root if the matroid is modular. Using results from Ardila and Sanchez we also prove that this generalized characterisitc polynomial is a matroid valuation.

Algebra Seminar

Speaker: Senate Meeting - No Seminar (Western)

"no seminar"

Time: 14:30

Room: MC 108

Algebra Seminar

Speaker: Séverin Philip (Institut Fourier Université Grenoble Alpes)

"On the degree of semi-stable reduction for abelian varieties"

Time: 14:30

Room: ZOOM

For an abelian variety $A$ over a number field $K$ a seminal theorem of Grothendieck asserts the existence of a finite extension $L/K$ such that $A$ acquires semi-stable reduction over $L$. My work is centered around studying this phenomenon of acquiring 'semi-stability' where arithmetic and geometry meet. I will showcase this with elliptic curves. The main object of study will be the minimal degree $d(A)$ of the extensions $L/K$ that appear this way and I will give almost optimal bounds depending only on the dimension of $A$ for its maximum (for varying $A$ and $K$ of a given dimension). These bounds will come from a study of the so-called finite monodromy groups of $A$, groups that I will introduce and show that they give a theoretical computation of $d(A)$. This computation and the work of Silverberg and Zarhin will give our upper bound. The lower bound will come from the study of a moduli space for abelian varieties and an explicit construction using the theory CM of abelian varieties.