24 APR 2024
16:00 - 17:00

A polynomial vector field on a complex plane C^2 defines a foliation of the plane and one would like to know when the leaves of this foliation are algebraic, i.e. when the analytic integral curves are algebraic. This is a hard unsolved problem. I will suggest a conjectural approach to this problem which uses the reduction of the vector field modulo primes p. It turns out that the corresponding problem in characteristic p is easy to solve. We then conjecture that the original vector field is algebraic if and only if its reduction modulo a prime p is algebraic for almost all p. We have some partial results towards proving this conjecture. This is a work in progress with D. Leshchiner.


https://zoom.us/join
Meeting ID: 908 611 6889
Passcode: 362880

Subscribing the Mathematics Colloquium mailing list:
https://groups.google.com/g/ipm-math-colloquium
Venue: , (Online)

24 APR 2024
12:45 - 13:45

Let $B_{1}$ be the unit ball in $mathbb{R}^{n}$. We analyze the positive solutions to the problem egin{equation}label{aaa} left{egin{array} {ll} - Delta u + u = u vert uvert ^{p - 2}, & mbox{ in } B_{1}, \%u>0, & mbox{ in } Omega, \dfrac{partial u}{partial u} = 0, & mbox{ on } partial B_{1} , end{array} ight. end{equation} where $p > 2$ and $frac{partial}{partial u}$ is the outward normal derivative. This problem, sometimes referred to as the Lane-Emden equation with Neumann boundary conditions, arises for instance in mathematical models which aim to study pattern formation, and more specifically in those governed by diffusion and cross-diffusion systems. In this talk, I will discuss about the existence of multiple solutions of problem eqref{aaa}. In particular, I focus on the existence of non-constant solutions for $p$ close to $2$. Finally, I will explain some results that extend these multiplicity results to non-homogeneous Neumann problems.
Venue: Niavaran, Lecture Hall 1

24 APR 2024
14:00 - 15:00

Entropy of a random variable is a measure of how much a random variable can be losslessly compressed. When certain symbols are allowed to be confused, the notion of graph entropy is used. A central problem in this subject is the question of maximum entropy of a graph and the distribution achieving it. As observed by Changiz-Rezaei and Godsil, this problem is closely related to the notion of fractional chromatic number of graphs and uniform cover of the vertices by maximal independent sets. Simony et al. generalized the notion of graph entropy to other geometric objects known as convex corners. In this talk, we consider the problem of maximum entropy of convex corners. In particular we investigate maximum entropy problem for independent polytopes of matroids and graphs.

Zoom room information is:
https://us06web.zoom.us/j/89600366073?pwd=RMHt0eGdvGtkaDBG1Me3y8bOLgooGh.1
Meeting ID : 896 0036 6073
Passcode : 290109
Venue: Niavaran, Lecture Hall 1

17 APR 2024
14:00 - 15:00

The classical Tverberg's theorem of 1966 states that any set of size (d+1)(q-1)+1 in the Euclidean space of dimension d can be partitioned into q parts whose convex hulls intersect. In this talk, I will talk about many variants of this theorem and a conjecture by Engstrom and Noren that was recently proved with the aid of Najafian, ALipour, and Mazidi.

Zoom room information is:
https://us06web.zoom.us/j/89600366073?pwd=RMHt0eGdvGtkaDBG1Me3y8bOLgooGh.1
Meeting ID : 896 0036 6073
Passcode : 290109
Venue: Niavaran, Lecture Hall 1

17 APR 2024
17:30 - 19:00

The geometry of moduli spaces and stacks of vector bundles on curves have been intensively studied from different perspectives; for example, via point counting over finite fields by Harder and Narasimhan, and gauge theoretically by Atiyah and Bott over the complex numbers. Following Grothendieck's vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some properties of this category, I will present a formula for the motive of the moduli stack of vector bundles on a smooth projective curve; this formula is compatible with classical computations of invariants of this stack due to Harder, Atiyah--Bott and Behrend--Dhillon. The proof involves rigidifying this stack using Flag-Quot schemes parametrising Hecke modifications as well as a motivic version of an argument of Laumon and Heinloth on the cohomology of small maps, which is closely related to the Grothendieck-Springer resolution. I will explain how to extend this to a formula for the stack of coherent sheaves and, if there is time, I will give an overview of other motivic descriptions of closely related moduli spaces. This is joint work with Simon Pepin Lehalleur.

https://zoom.us/join
Meeting ID: 9086116889
Passcode: 362880
Venue: , (Zoom Lecture)

3 APR 2024
14:00 - 15:00

We address selected topics in the area of identification problems on graphs and other discrete structures. In this type of problems, we wish to select a small set of elements (usually, vertices) with the goal of distinguishing a set of structures (usually, the vertices or the edges of the graph) by means of membership, neighborhood, or distances to the elements in the solution set. We introduce the topic in general and highlight a few recent interesting results, problems and conjectures in the area.
Venue: , (Zoom Lecture)

7 MAR 2024
14:00 - 16:00

Class Field Theory is a profound and significant branch of algebraic number theory, primarily concerned with the study of abelian extensions of local and global fields by utilizing intrinsic objects from the ground field. Through the e x a m i n a t i o n o f c o m p l e x interrelations and principles such as reciprocity laws, conductors, and Lfunctions, this theory illuminates the elegant structures governing global fields. In the forthcoming lecture, our focus will be on number fields, aiming to elucidate the core concepts and methodologies with a keen emphasis on motivational examples and historical context. It is hard to do justice to the field in one lecture, so expect a lot of bias and injustice!!
Venue: Niavaran, Lecture Hall 1

6 MAR 2024
14:00 - 15:00

For an arbitrary graph $G$, a hypergraph $mathcal{H}$ is called Berge-$G$ if there is an injection $i: V(G)longrightarrow V(mathcal{H})$ and a bijection $psi :E(G)longrightarrow E(mathcal{H})$ such that for each $e=uvin E(G)$, we have ${i(u),i(v)}subseteq psi (e)$. We denote by $mathcal{B}^rG$, the family of $r$-uniform Berge-$G$ hypergraphs. For families $mathcal{F}_1, mathcal{F}_2,ldots, mathcal{F}_t$ of $r$-uniform hypergraphs, the Ramsey number $R(mathcal{F}_1, mathcal{F}_2,ldots, mathcal{F}_t)$ is the minimum integer $n$ such that in every hyperedge coloring of the complete $r$-uniform hypergraph on $n$ vertices with $t$ colors, there exists $i$, $1leq ileq t$, such that there is a monochromatic copy of a hypergraph in $mathcal{F}_i$ of color $i$.In this lecture, we will first provide a brief history of Ramsey numbers and then focus on Ramsey numbers involving $3$-uniform Berge cycles.

Zoom room information:
https://us06web.zoom.us/j/89600366073?pwd=RMHt0eGdvGtkaDBG1Me3y8bOLgooGh.1 Meeting ID: 896 0036 6073
Passcode: 290109
Venue: Niavaran, Lecture Hall 1