Algebraic varieties are common solutions of bunch of multi-variable polynomials equations, for example, straight line, circle, cuspidal curve, nodal curve, sphere, etc. Classifying all algebraic varieties up to isomorphism is the ultimate goal of algebraic geometry. Of course, this is nearly impossible achieve, so we consider various weaker form of classification, and classifying varieties ‘Birationally’ is of those tools. In this talk I will explain what it means to classify varieties birationally, what are the difficulties in higher dimensions and the role of Minimal Model Program (MMP) in birational classification.

The classical Bertini theorem in algebraic geometry says that a general hyperplane section of a smooth quasi-projective subvariety of a projective space over an algebraically closed field is also smooth. It was already known long time ago that such a result holds over any infinite field. However, this turned out to be false over finite field, as Katz showed. Poonen then showed that Bertini theorem can be salvaged over finite fields by allowing hypersurfaces of large degree rather than just hyperplanes. In this talk, we shall revisit these Bertini theorems. In particular, we shall prove new Bertini theorems for normal and integral schemes over finite fields. This is based on a joint work with Mainak Ghosh.

In this talk, I will discuss the structure of the free boundary in the obstacle problem for fractional powers of the heat operator. Our results are derived from the study of a lower dimensional obstacle problem for a class of local, but degenerate, parabolic equations. The analysis will be based on new Almgren, Weiss and Monneau type monotonicity formulas and the associated blow-up analysis. This is a joint work with D. Danielli, N. Garofalo and A. Petrosyan.

Let a finite abelian group G act on a normal local domain R with residue field of R of char. 0. Assume that R^G is a UFD. Then R is a free R^G- module. In particular, if R^G is regular then R is Cohen Macaulay. We will start preparation for P. Samuel's descent theory.

Macdonald polynomials are a remarkable family of orthogonal symmetric polynomials in several variables. An enormous amount of combinatorics, group theory, algebraic geometry and representation theory is encoded in these polynomials. It is known that the characters of level one Demazure modules are non-symmetric Macdonald polynomials specialized at t=0. In this talk, I will define a class of polynomials in terms of symmetric Macdonald polynomials and using representation theory we will see that these polynomials are Schur-positive and are equal to the graded character of level two Demazure modules for affine sl_{n+1}. As an application we will see how this gives rise to an explicit formula for the graded multiplicities of level two Demazure modules in the excellent filtration of Weyl modules. This is based on joint work with Vyjayanthi Chari, Peri Shereen and Jeffrey Wand.

We prove a version of the mountain pass theorem for lower semicontinuous and lambda-geodesically convex functionals on the space of probability measures P(M) equipped with the W_2 Wasserstein metric, where M is a compact Riemannian manifold or R^d. As an application of this result, we show that the empirical process associated to a system of weakly interacting diffusion processes exhibits a form of noise-induced metastability. The result is based on an analysis of the associated McKean–Vlasov free energy, which for suitable attractive interaction potentials has at least two distinct global minima at the critical parameter value b = b_c. Joint work with Andre Schlichting.

In Clauset et. al. (2009), PLFit estimation procedure has been proposed for the power-law index and became popular immediately for its versatile applicability. This has been used in many areas including scale-free networks, energy networks, preferential attachment model, teletrafic data etc. But the theoretical support for this estimation procedure is still lacking. In this talk, consistency of PLFit procedure will be addressed under semiparametric assumption. This is an ongoing joint work with Bohan Chen, Remco van der Hofstad and Bert Zwart.

We shall introduce smooth and formally smooth morphisms and study their basic properties. We shall complete the proof of CST (Cohen’s structure theorem for complete local rings).

Suppose $0<\alpha<1$. The problem of determining the size of a maximum set of lines (through the origin) in R^d s.t. the angle between any two of them is arccos(\alpha) has been one of interest in combinatorial geometry for a while now (since the mid 60s). Recently, Yufei Zhao and some of his students settled this in a strong form. We will see a proof of this result. The proof is a linear algebraic argument and should be accessible to all grad students.

Let M be a finitely generated seminormal submonoid of the free monoid \mathbb Z_+^n and let k be a field. Then Anderson conjectured that all finitely generated projective modules over the monoid algebra k[M] is free. He proved this in case n=2. Gubeladze proved this for all n using the geometry of polytopes. In a series of 3 lectures, we will outline a proof of this theorem.