Jan 9 
No Seminar 

Jan 16 At 1:00 in 509 Lake 
Gabriel Cunningham (U. Mass, Boston) 
Small Polyhedra with Prescribed Symmetry
abstract±
What is the smallest convex polyhedron with exactly 7 symmetries? What is the smallest abstract polyhedron with triangular faces, meeting 7 at every vertex? This talk will be an exploration of problems like these, spanning convex geometry, graph theory, and group theory. There are many open problems just waiting for an audience member to solve them.
Poster.

Jan 23 
Max Engelstein (MIT) 
Minimizers, Critical Points, Energies and Equations
abstract±
Minimizers of energies show up all over mathematics and physics. We will discuss two approaches to studying them; the first, developed through the 20th century, relies on the fact that minimizers are critical points and thus satisfy an equation. The second approach, the subject of more recent research, uses the minimization property more directly, often omitting PDE completely. Throughout the talk we will compare the two approaches keeping two key examples in mind; minimal surfaces and harmonic functions.
Poster.

Jan 30 
No Seminar 

Feb 6 
No Seminar 

Feb 13 
Benjamin Schmidt (UT Austin) 
The Halphen Problem and Related Questions
abstract±
A 19th century problem in algebraic geometry is to understand the relation between the genus and the degree of a curve in complex projective space. This is easy in the case of the projective plane, but becomes quite involved already in the case of three dimensional projective space. In this lecture series I will give an introduction to the topic. From a modern perspective there is no reason to restrict to the case of projective space, and I will discuss some recent progress on the question in the case of abelian threefolds in joint work with Emanuele Macri.
Poster.

Feb 20 
David Rolnick (MIT) 
Why is deep learning so deep?
abstract±
The field of deep learning has gained great popularity in recent years due to the ability of artificial neural networks to fit unknown functions of input data. Neural networks are compositions of linear and nonlinear functions, with the "depth" of the network measuring the number of successive compositions. In practice, deeper networks have proven to be more powerful at expressing natural functions. In this seminar (joint work with Henry Lin and Max Tegmark), we will see (1) that arbitrarily good approximations to polynomials are possible with a fixed network size and (2) this fixed size decreases exponentially as the depth increases. These results justify behavior observed in practice and suggest rules of thumb for picking the depth of a network.
Poster.

Feb 27 
Rahul Singh (NEU) 
The Conormal Variety of a Schubert Variety
abstract±
Let X, Y be compact homogeneous spaces for a semisimple group G, and let O be an orbit under the diagonal action of G on XxY.
We study the conormal variety N of O.
Suppose first that Y is a cominuscule Grassmannian.
We construct a vector bundle on a BottSamelson variety resolving the singularities of N.
In type A, this allows us to identify the equations defining N as a subvariety of the cotangent bundle of XxY.
This suggests some natural conjectures and proof strategies regarding the equations of N for general G and Y, which we discuss.
Poster.

Mar 6 
No Seminar 
Spring Break

Mar 13 
Will Boney (Harvard) 
Compactness, elementary and nonelementary
abstract±
Compactness is one of the most powerful tools in model theory. It states that, when searching to find a structure satisfying an infinite list of axioms written in firstorder logic, it suffices to find a structure satisfying each of the finite subsets of this list. Depending on interest and background of the audience, we will discuss the statement, proof, and applications of this theorem, as well as extensions beyond firstorder logic.
Poster.

Mar 20 
Daniel Glasscock (NEU) 
Topological dynamics as a tool in Ramsey theory
abstract±
In 1927, Bartel Leendert van der Waerden proved that no matter how the natural numbers are finitely partitioned, one of the pieces contains arbitrarily long arithmetic progressions. In 1978, Hillel Furstenberg and Benjamin Weiss proved that given any homeomorphism T of a compact metric space X, there exists at least one point (x, . . . , x) whose iterates under the product map T^n × ··· × T^n return arbitrarily close to (x, . . . , x). Remarkably, these theorems are equivalent! In this talk, I will explain how topological dynamics is used as a tool in Ramsey theory and how we have used it recently to make progress on one of my favorite open problems.
Poster.

Mar 27 
Simone Cecchini (NEU) 
Positive scalar curvature and minimal hypersurfaces: old and new.
abstract±
It is a classical result of Schoen and Yau that minimal hypersurfaces can be used to construct obstructions against metrics of positive scalar curvature on compact manifolds of dimension at most seven. This is so far the most powerful method to construct obstructions to such metrics on nonspin manifolds. The dimension restriction has been recently removed by the same authors. In this talk I will discuss both the classical and the new results of Schoen and Yau on minimal hypersurfaces. In the final part I will discuss a question recently raised by Gromov about the relationship between the new results of Schoen and Yau and the notion of enlargeability due to Gromov and Lawson.
Poster.

Apr 3 
Pablo Soberón (NEU) 
Different approaches to Tverberg's colored theorem
abstract±
The colored version of Tverberg's theorem is a conjecture in combinatorial geometry that has resisted proof for over 25 years. However, through a wide range of different techniques, several cases and related results have been proven. During this talk I will showcase some of my favorite.
Poster.

Apr 10 
Xiaolei Zhao (NEU) 
How to represent cohomology classes by subvarieties?
abstract±
The celebrated Lefschetz’ Theorem on (1,1)classes states that for a smooth projective complex variety, its integral cohomology classes satisfying a natural condition (of type (1,1)) can be represented by divisors (i.e. linear combinations of subvarieties of codimension 1). We will revisit the original proof of Lefschetz in the surface case, which dates back to almost a century ago. The idea behind this proof governed many later developments in Hodge theory, and finally led to a series of works in the past decade, in which a convincing approach to attack the Hodge conjecture was provided. We will also survey these developments if time permits.
Poster.

Apr 17 
No Seminar 

Apr 24 
Emanuele Ventura (Texas A&M) 
Tensor and Waring ranks
abstract±
Recently the thriving of multilinear algebra in both pure and applied mathematics has revitalized classical geometric questions in the framework of tensor rank.
In analogy with matrices, the tensor rank of a tensor T is the smallest number of rankone tensors whose sum coincides with T.
When a tensor is invariant under the symmetry group, it can be regarded as homogeneous polynomial. In this case, the classical notion of its Waring rank,
going back to Sylvester, is the symmetric analogue of the rank above. A vast generalization of these ranks is that of Xrank, for any given nondegenerate projective variety X.
The aim of this talk is to give an introduction to these topics.
Poster.



End of Semester (Seminar Resumes Fall 2018) 