Abstract: In the late 70s Sarkozy proved the following theorem: Given a polynomial f(T) over the integers with f(0)=0, there exists a constant c_f such that for any set $A\subset [n]$ of size at least $n/(log n)^{c_f}$ there exist distinct $a,b\in A$ such that $a-b=f(x)$ for some $x$. In 2016, Ben Green proved a function field analog of the same result but with a much better bound for $|A|$: Given a polynomial $F\in\bF_q[T]$ of degree $k$ with $F(0)=0$, there exists $0 q^{(1-c)n}$ there exist $\alpha(T)\neq\beta(T)$ in $A$ such that $\alpha(T)-\beta(T)=F(\gamma(T))$ for some $\gamma(T)\in\bF_q[T]$. We will see a proof of this result.

Abstract: This talk will be based on the following elementary and nice exposition http://www.math.stonybrook.edu/~roblaz/Reprints/Green.Laz.Simple.Pf.Petri.pdf Using some simple facts about projective space, cohomology, cohomology of line bundles on projective space, we shall prove the following theorems: 1. Noether's theorem - Projective normality of the canonical embedding of non-hyperelliptic curves. 2. Petri's -theorem - Let X be a smooth and projective curve of genus g \geq 5. Assume that X carries a line bundle A of degree g-1 with h^0(A)=2. Further assume that both A and \Omega_X\otimes A^* are generated by their global sections. Then the homogeneous ideal of X in its canonical embedding is generated by degree 2 elements.

Abstract: In this talk, I will prove a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. I will show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed formula for certain root multiplicities. As an application, using the combinatorics of Lyndon words, we construct a basis for the root spaces corresponding to these roots and determine the Hilbert series in the case when all simple roots are imaginary. In last ten minutes, We will talk about chromatic discriminant of a graph: The absolute value of the coefficient of q in the chromatic polynomial of a graph G is known as the chromatic discriminant of G and is denoted $\alpha(G)$. We start with a brief survey on many interesting algebraic and combinatorial interpretations of $\alpha(G)$. We use two of these interpretations (in terms of acyclic orientations and spanning trees) to give two bijective proofs for a recurrence formula of $\alpha(G)$ which comes from the Peterson recurrence formula for root multiplicities of Kac-Moody algebras.

Abstract: We shall discuss the Cartan-Dieudonne theorem which establishes that every orthogonal transformation of the n-dimensional Euclidean space is a composition of at most n reflections. We shall show how to construct these n reflections using the Householder matrices.

Abstract ​: ​ I will start the talk from the classical "Cayley transform" for the special orthogonal group SO(n) defined by Arthur Cayley in 1846. A connected linear algebraic group G over C is called a *Cayley group* if it admits a *Cayley map*, that is, a G-equivariant birational isomorphism between the group variety G and its Lie algebra Lie(G). For example, SO(n) is a Cayley group. A linear algebraic group G is called *stably Cayley* if G x S is Cayley for some torus S. I will consider semisimple algebraic groups, in particular, simple algebraic groups. I will describe classification of Cayley simple groups and of stably Cayley semisimple groups. (Based on joint works with Boris Kunyavskii and others.)

Abstract : In his first memoir, Galois gave a criterion for an irreducible equation of prime degree to be solvable by radicals. In the second memoir, he defined primitive equations and showed that if a primitive equation is solvable by radicals, then its degree is the power of a prime. His results can be reformulated in terms of extensions of fields. We will show how to extend this reformulation and parametrise all primitive solvable extensions of an arbitrary field. (An extension is called primitive if there are no intermediate extensions, and it is called solvable if the Galois group of its Galois closure is a solvable group). All these concepts will be recalled and illustrated through examples. If time permits, we will discuss an arithmetic application. The talk should be accessible to a wide audience, including students.

Abstract : In his first memoir, Galois gave a criterion for an irreducible equation of prime degree to be solvable by radicals. In the second memoir, he defined primitive equations and showed that if a primitive equation is solvable by radicals, then its degree is the power of a prime. His results can be reformulated in terms of extensions of fields. We will show how to extend this reformulation and parametrise all primitive solvable extensions of an arbitrary field. (An extension is called primitive if there are no intermediate extensions, and it is called solvable if the Galois group of its Galois closure is a solvable group). All these concepts will be recalled and illustrated through examples. If time permits, we will discuss an arithmetic application. The talk should be accessible to a wide audience, including students.

Abstract: This is the first of two lectures where we'll cover Groebner bases of toric ideals. We start with an introduction to toric ideals and then study their Grobener bases. Our main goal will be a theorem of Bernd Sturmfels from 1991 that relates (certain) initial ideals of toric ideals to regular triangulations of an associated point configuration. The lectures are based on Chapters 4 and 8 of the book ``Groebner Bases and Convex Polytopes'' by Sturmfels.