# Open Neighborhood Seminar

### Harvard University Math Department

## Fall 2021

### September 1 - MT/ONS Opener

**Speaker:**Cliff Taubes (Harvard)**Title:**Math and the universe**Abstract:**It is likely that all known aspects of our universe are amenable to mathematical analysis. Are there universes that arenâ€™t? If you have thought about this, come and join the discussion.

### September 8

**Speaker:**Laura DeMarco (Harvard)**Title:**Prime numbers and Julia sets**Abstract:**Interesting patterns of prime numbers can arise in recursively defined sequences (such as the Fibonacci sequence). For non-linear recursions, there are intriguing connections with complex dynamics and the Mandelbrot set. This is just one of the many ways that Number Theory and Chaotic Dynamical Systems come together. I'll present a few examples and a glimpse into some of my own research in this direction.

### September 22

**Speaker:**Mihnea Popa (Harvard)**Title:**Projective vs. abelian geometry**Abstract:**Projective space and complex tori are two of the simplest types of manifolds we encounter, and in many ways they seem very different from each other. I will try to convince you however that, at least if we consider a special (but at the same time very common) class of tori called principally polarized abelian varieties, then the geometry of their subvarieties exhibits surprising, and to date mostly unexplained, similarities to the geometry of subvarieties in projective space.

### October 6

**Speaker:**Mark Shusterman (Harvard)**Title:**Alternative twin prime problems**Abstract:**It is conjectured that there are infinitely many pairs of primes that differ by 2. After presenting some motivation, results toward the conjecture, and obstructions to further progress, we will consider an analogous problem where the primes are replaced by irreducible polynomials with coefficients in the integers modulo 3 (or modulo 5). Alternative problems of this type often boil down to counting solutions to algebraic equations. Work of Grothendieck and Deligne reduces the latter to (topological) questions about the shape of the geometric figures cut out by our equations. I will report on joint work with Will Sawin obtaining some control on the number of higher-dimensional holes inside the figures in question. This allows us to make progress on some alternative twin prime problems, similar to the ones mentioned above.

### October 20

**Speaker:**Christopher Eur (Harvard)**Title:**Why would a sequence be unimodal?**Abstract:**Suppose we have a combinatorial sequence of numbers, for example from considering the number of forests in a graph, or the number of faces in a polytope, or the number of partitions of an integer. A basic question we can ask is whether it is unimodal---does the sequence monotonically increase and then decrease? We survey some methods, old and new, that are elementary but inspired from the so-called "Kahler package" property enjoyed by cohomology rings of smooth projective complex manifolds.

### November 3

**Speaker:**Ronen Mukamel (Harvard Medical School)**Title:**The postgenomic era, as viewed by a mathematician**Abstract:**Recent technological developments now allow for the accurate, high resolution measurement of millions of human genomes. I will describe some of the challenges and opportunities of this postgenomic era, and how these technologies stand poised to revolutionize our understanding of human biology and human health.

### November 17

**Speaker:**Kavita Ramanan (Brown)**Title:**Tales of random projections: where probability meets geometry**Abstract:**In several areas of mathematics, including probability theory, asymptotic functional analysis, statistics and data science, one is interested in high-dimensional objects, such as measures, data or convex bodies. One common theme is to try to understand what lower-dimensional projections can say about the corresponding high-dimensional objects. I will describe several results that address this question, starting with classical results and moving on to more recent breakthroughs, my own research and some open questions. The talk will be self-contained and accessible to undergraduate students.

**Organizers:**Ana Balibanu (ana@math.harvard.edu) and Dori Bejleri (bejleri@math.harvard.edu). Please drop us an email if you're curious about the seminar!