8/29/2022 – Hari Rau-Murthy: The tensor product of an algebra with a space
You might know how to take the tensor product of two algebras. But the tensor product of an algebra with a space!!! The result is a ring. I will introduce \(A \otimes X\) where \(A\) is an algebra and \(X\) is a space. The tensor product of \(A\) with a finite set of \(n\) elements is the \(n\)-fold tensor product of \(A\). One can use simplicial sets to prolong this construction to spaces.
I will introduce Hoschild Homology, \(\mathrm{HH}(A)\), and I will give a short proof of the theorem that \(\mathrm{HH}(A) = A \otimes S^1\)due to McClure in 1997. Time permitting, I’ll tell you about different filtrations on the ring \(A \otimes X\) that come from Goodwillie calculus and the Adams spectral sequence. This filtrations give ways to compute \(A \otimes X\) when \(X\) is smooth.
9/5/2022 – Nikolai Konovalov: Rational homotopy and homology
We will discuss the homotopy category of spaces with some its rational invariants.
Notes: Konovalov
9/12/2022 – Daniel Soskin: Determinantal inequalities for totally positive matrices
Totally positive matrices are matrices in which each minor is positive. An important role in representation theory of quantum groups is played by dual canonical basis. Lusztic has shown that specialization of elements of the dual canonical basis at \(q=1\) are totally non-negative polynomials. To this end there is an interest in bounded functions on the locus of totally positive matrices. I will present results on multiplicative determinantal inequalities (joint work with M.Gekhtman). Also, I will present a majorizing monotonicity of symmetrized Fischer’s products which are a natural generalization of Hadamard-Fischer inequalities. Majorizing monotonicity of symmetrized Fischer’s products were already known for hermitian positive semi-definite matrices which brings additional motivation to verify if they hold for totally positive matrices as well (joint work with M.Skandera). The main tools we employed are network parametrization, Temperley-Lieb and monomial trace immanants.
Notes: Soskin
9/19/2022 – Matt Scalamandre: Morse theory
Morse theory is a tool for translating information about the derivatives of a nice real-valued function on a manifold into topological information about that manifold. We will use this tool to provide simplicial and handle decompositions of smooth manifolds, and discuss other topological applications (e.g. to cobordisms, exotic spheres, \(3\)-manifolds, …).
9/26/2022 – Lorenzo Riva: Morse theory II: Differential Boogaloo
We will exploit the Morse theoretic tools developed last week, along with a healthy amount of linear algebra and a bunch of pictures, to outline a proof of the h-cobordism theorem, one of whose powerful consequences is a solution of the Poincaré conjecture in high dimensions.
Notes: Riva
10/3/2022 – Jacob Zoromski: Infinite Minimal Free Resolutions
First, we will introduce the notion of minimal free resolutions over a polynomial ring \(S\) and discuss their significance. Hilbert’s Syzygy Theorem says that such resolutions always have finite length. Then, we will examine minimal free resolutions over a quotient \(R = S/I\) which (almost) always have infinite length. There are many open problems in the study of such resolutions, and we will explain a few of them.
10/10/2022 – Richard Birkett: Skew Products on the Berkovich Projective Line
For over a century, complex dynamicists have studied what happens when you iterate a rational function \(f\) in \(\mathbb{C}(z)\) on \(\mathbb{P}^1(\mathbb{C})\). This has yielded amazing results and beautiful fractals like the Mandelbrot set.
What happens when we replace \(\mathbb{C}\) with another field \(K\)? This is peculiar when the norm/metric on \(K\) is non-Archimedean, one where disks can never partially overlap. For example consider \(K = \mathbb{C}_p\), the \(p\)-adic numbers; a function field \(\mathbb{C}(t)\) is another.
Remarkably, it turns out that we can trade the rational map \(f :K \to K\) for a piecewise-linear map of a real tree, the Berkovich projective line. I will describe the tree geometrically, and a new type of map called a skew product which allows for applications to complex dynamics in two dimensions.
10/24/2022 – Panel on non-academic jobs
This seminar will be a panel discussion featuring four Notre Dame PhD’s in mathematics who are currently working in a job outside of academia, or in academia but not as a math professor.
The panelists are:
Ben Jones ( Notre Dame PhD, 2007, Senior Applied Scientist at Amazon Web Services, Portland)
Ben Lewis (Notre Dame PhD, 2017, Data Scientist at Shopify)
Jeremy Mann (Notre Dame PhD, 2020, Senior Data Scientist at Concert AI, Oakland, CA)
Taylor Ball (Notre Dame PhD, 2021)
The purpose of this panel discussion is to give current Notre Dame students and other interested parties a better idea of career options outside of academia after getting a math PhD. Each panelist will discuss their own career path and useful skills to develop for getting a teaching position or a job in industry. This is the second in a series of annual panel discussions and we hope that the panelists and others can serve as useful resources for Notre Dame math students interested in working outside of academia.
10/31/2022 – Bridget Schreiner: Representation stability for ordered configuration spaces
Let \(\mathrm{Conf}(n, M) = \{(x_1, \dotsc, x_n) \in M^n: x_i \neq x_j\}\) denote the space of ordered configurations of n points in a manifold \(M\). The nth symmetric group acts on this space by permuting the coordinates. When M is a non-compact manifold without boundary, we have maps \(\mathrm{Conf}(n, M) \to \mathrm{Conf}(n+1, M)\) by “adding a point near infinity”, inducing maps on homology \(H_q \mathrm{Conf}(n, M) \to H_q \mathrm{Conf}(n+1, M)\). These maps do not satisfy homological stability, that is, they are not isomorphisms for \(n >> q\). However, accounting for the action of the symmetric group, we have a different type of stability called representation stability. This talk will introduce the notion of representation stability and give a proof of stability for configuration spaces in the case that \(M = \mathbb{R}^d\) If time permits, we will also discuss notions of higher stability.
11/14/2022 – Brian Reyes: An introduction to weak solutions
In this talk we will see what are weak solutions with some examples starting from the idea of the Fundamental Theorem of Calculus all the way leading up to Sobolev spaces. We will talk about the importance of the Dirac delta function and how the ability to exchange integration with differentiation is crucial in analysis. Integration by parts together with the idea of test functions is one that nowadays is standard but is actually not that old since it was developed in the last century. Finally we talk about how weak solutions relate to real solutions via regularity with a general overview of the usage in PDE.
11/21/2022 – Randy Van Why (Princeton): A glance at low dimensional symplectic topology
Symplectic geometry is a peculiar geometry from an outsiders perspective. The field has its roots in physics but the theory has expanded far beyond the scope of the conceptual frameworks from the genesis of the field. Currently, there is no wide consensus on what symplectic geometry is, what it is good for, or what it possibly has to say about geometry/topology in the large.
In this talk, I hope to introduce everyone to my favorite facet of this enigmatic field: symplectic geometric topology. I will use complex affine varieties as a motivator for discussing some constructions in symplectic 4-manifolds and contact 3-manifolds. If time permits, I will discuss some little known connections between modern symplectic geometry and classical geometric topology.
11/28/2022 – Ethan Reed: Using Geometric Methods to Construct Generalized Eagon-Northcott Complexes
In Commutative Algebra, Generalized Eagon-Northcott Complexes are a family of complexes that generalize the Koszul Complex. Many properties of these complexes can be proved algebraically, but they can alternatively be proven using algebraic geometric methods. In this talk, we will describe the relevant tools of algebraic geometry and sketch how to use them to construct Generalized Eagon-Northcott Complexes. More specifically, we will use a relative hypercohomology spectral sequence from twists of a Koszul resolution of sheaves on a projectivized bundle to derive the existence of these complexes.
12/5/2022 – Lizda Nazdira Moncada Morales: Graphs, Hypergraphs, and their Edge Ideals
Classification of minimal free resolutions of monomial ideals is one of the central problems in combinatorial commutative algebra. By polarization, studying the minimal free resolution of a graded monomial ideal is equivalent to studying the minimal free resolutions of a particular class of squarefree monomial ideals: edge ideals. The homological properties of an edge ideal of a (hyper)graph depend on the combinatorial properties of the (hyper)graph. In this talk, we will present a characterization of some edge ideals which have linear resolutions as well as some results on their regularity and Betti numbers.