# Upcoming Seminars & Events

## Primary tabs

##### Towards de-mystification of deep learning: function space analysis of the representation layers

**PLEASE NOTE DIFFERENT DAY (TUESDAY).** We propose a function space approach to Representation Learning [1] and the analysis of the representation layers in deep learning architectures. We show how to compute a `weak-type' Besov smoothness index that quantifies the geometry of the clustering in the feature space. This approach was already applied successfully to improve the performance of machine learning algorithms such as the Random Forest [2] and tree-based Gradient Boosting [3].

##### A variety with non-finitely generated automorphism group

If X is a projective variety, then Aut(X)/Aut^0(X) is a countable group, but little is known about what groups can occur. I will construct a projective variety for which this group is not finitely generated, and discuss how the construction can adapted to give an example of a complex projective variety with infinitely many non-isomorphic real forms.

##### Almost Rigidity of the Positive Mass Theorem

The Positive Mass Theorem states that an asymptotically flat Riemannian manifold, $M^3$, with nonegative Scalar curvature has nonnegative ADM mass, $m_{ADM}(M)\ge 0$, and if the ADM mass is 0 then we have rigidity: the manifold is isometric to Euclidean space. It has long been known that if one has a sequence of such manifolds $M^3_j$ with $m_{ADM}(M_j) \to 0$ then $M_j$ need not converge smoothly to Euclidean space. To avoid bubbling, one forbids the sequence of manifolds to have closed interior minimal surfaces, but allows the manifolds to have minimal boundaries. Dan Lee and I conject

##### Local Universality of Random Functions

In this talk, we discuss local universality results for a general class of random functions that includes random trigonometric polynomials and random orthogonal polynomials. We then apply these results to obtain estimates for the number of real roots and prove, in some cases, that this number satisfies the Central Limit Theorem. This is joint work with Van Vu.

##### Stability results in graphs of given circumference

In this talk we will discuss some Turan-type results on graphs with a given circumference. Let W_{n,k,c} be the graph obtained from a clique K_{c-k+1} by adding n-(c-k+1) isolated vertices each joined to the same k vertices of the clique, and let f(n,k,c)=e(W_{n,k,c}). Improving the Erdos-Gallai theorem, Kopylov proved in 1977 that for c<n, any 2-connected graph G on n vertices with circumference c has at most max (f(n,2,c),f(n,[c/2],c)) edges, with equality if and only if G equals W_{n,2,c} or W_{n,[c/2],c}. Recently, Furedi et al. obtained a stability version of Kopylov's theorem.

##### TBA - Jonathan Hanselman

##### Cohomology of p-adic Stein spaces

I will discuss a comparison theorem that allows us to recover p-adic (pro-)etale cohomology of p-adic Stein spaces with semistable reduction over local rings of mixed characteristic from complexes of differential forms. To illustrate possible applications, I will show how it allows us to compute cohomology of Drinfeld half-space in any dimension and of its coverings in dimension one. This is a joint work with Pierre Colmez and Gabriel Dospinescu.

##### Global well-posedness for the 2D Muskat problem

The Muskat problem was originally introduced by Muskat in order to model the interface between water and oil in tar sands. In general, it describes the interface between two incompressible, immiscible fluids of different constant densities in a porous media. In this talk I will prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the initial data is monotonic or has slope strictly less than 1. The curvature of these solutions solutions decays to 0 as $t$ goes to infinity, and they are unique when the initial data is $C^{1,\epsilon}$.

##### TBA - Amir Ali Ahmadi

##### TBA - Dmitry Batenkov

##### Min-max theory for constant mean curvature hypersurfaces

We describe the construction of closed constant mean curvature (CMC) hypersurfaces using min-max methods. In particular, our theory allows us to show the existence of closed CMC hypersurfaces of any prescribed mean curvature in any closed Riemannian manifold. This work is joint with Xin Zhou.

##### TBA - Ron Aharoni

##### TBA - Kyle Hayden

##### Kloosterman sums and Siegel zeros

Kloosterman sums arise naturally in the study of the distribution of various arithmetic objects in analytic number theory. The 'vertical' Sato-Tate law of Katz describes their distribution over a fixed field F_p, but the equivalent 'horizontal' distribution as the base field varies over primes remains open. We describe work showing cancellation in the sum over primes if there are exceptional Siegel-Landau zeros. This is joint work with Sary Drappeau, relying on a fun blend of ideas from algebraic geometry, the spectral theory of automorphic forms and sieve theory.

##### TBA - Yuval Peres

##### TBA - Yuval Peres

##### TBA-Lionel Levine

##### Compactification of the configuration space for constant curvature conical metrics

In this joint work with Rafe Mazzeo, we would like to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. We construct a resolution of the configuration space, and prove a new regularity result that the family of constant curvature conical metrics has a nice compactification as the cone points coalesce. This is one key ingredient to understand the full moduli space of such metrics with positive curvature and cone angles bigger than $2\pi$.

##### Unlikely intersections for algebraic curves in positive characteristic

Please follow this link for the abstract: http://www.math.ias.edu/seminars/abstract?event=131079