math is art
My first play was performed in LA directed by my amazing friend (Jacques Manjarrez)[https://jacquesmanjarrez.com/] as part of Nothing to See Here. The last night was recorded and can be seen (here)[https://youtu.be/YNYGaUns5wY?si=r–t-V_5jJw9tXTR&t=3602]. Impaction by Rin Ray Cast: Main Character Sick Creature (a human wrapped in blankets to look vaguely unidentifiable but certainly alive) Doctor…
I’ve felt for a long time that automorphisms of curves should control or at least exert serious force on the slopes on their Jacobians. Symmetry forces height, as I’ve written about previously in Models of Formal Groups Laws of Every Height, and Endomorphisms Directly Control Slope.
Here’s something enticing and strange: there are two “cut and paste” invariants of the same group which are equal to dual zeta values! the euler characteristic \( \chi(SL_2(\mathbb{Z})) = \zeta(-1), \) and the tamagawa measure of \( \mu(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}) = \zeta(2)\).
The Bernoulli numbers were defined by Faulhaber in terms of the following generating series. \[\frac{x}{e^x-1} = \sum_{k \geq 0} B_k \frac{x^k}{m!}\] But why? Where did this come from? Well, the mathematicians of the time were contemplating the following sorts of patterns: \[\begin{aligned} 1+2+\cdots+n &= \frac{n(n+1)}{2} \\ 1^2+2^2+\cdots+n^2 & = \frac{n(n+1)(2n+1)}{6} \\ 1^3+2^3+\cdots+n^3 & =…
Cech Covers (click the link to listen to us). I wrote this song with my beloved old room mate Christian Gorski in my last year of grad school while I was wrapping up my thesis. For weeks, I was doing nothing but computing etale sheaf cohomologies of ramified covers of the projective plane. I would decompose sheaves on these curves by cutting around the neighborhood of ramification point (the…
This drawing is an old drawing I made when I was preparing for my qualifying exam in my second year of grad school at Northwestern. It is the crystalline period map. The tower to the left is the “Lubin-Tate” tower, the deeper it goes the more level structure. In the upper right corner there is projective space, and the “cat like” creatures below are moduli stacks of curves. I always draw the…
I have been working recently to counter the writers block that has formed insidiously from an unhealthy creeping perfectionism. In order to do this, I will post some old art and music which at the time I felt was “not good enough to share” or “inappropriate for a professional mathematician to have associated to them.” But all of these labels and caring too much about what other people think – I am…
We outline connections between the gut microbiome, autoimmune conditions, neuropathic pain, eye pain, chemical intolerance, and a specific set of “overactive” mental illnesses. All seem to be connected to a sensory processing disorder. This is joint work with Luca Estinto.
This post is written together with Josh Mundinger. Last time, we compared the Harish-Chandra isomorphism \(Z(U\mathfrak g) \cong (\text{Sym} \mathfrak h)^{W,\cdot}\) for \(\mathfrak g= \mathfrak{sl}_2\) to the Duflo isomorphism \(Z(U\mathfrak g) \cong (\text{Sym } \mathfrak g)^{\mathfrak g} \cong (\text{Sym} \mathfrak h)^W\), and found that they differ exactly by a translation by \(\rho\). In this…
This post is written together with Josh Mundinger. We seek to understand the relations between \(1/2\)’s that appear across mathematics. From the Riemann Hypothesis to the L2 norm, we aim to see the myriad and enticing ways this unfurls; each instance of \(1/2\) connected in an anarchic network of equals. In this blog post, we examine a specific example arising in representation theory: the center…