An element $a$ of a Lie algebra $L$ is called a Fredholm element if the adjoint operator $\mathrm{ad}_a:L \to L$ is a Fredholm linear map. That is: its kernel is a finite-dimensional space and its range $\mathrm{ad}_a(L)$ is a finite-codimensional subspace. Is there an infinite-dimensional Lie a...

I have two vector bundles $E_1$, $E_2$ over $M$ and an embedding of the smooth sections $\lambda : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1 \oplus E_2)$. I consider a Fredholm differential operator $D_1 : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1)$ which can easily be lifted to the Fredholm differ...

Let $K\in L^2((0,1)\times(0,1))$ and consider the operator defined in $L^2(0,1)$ by $$Lu(x):=u(x)-\int_0^1K(s,x)u(s)ds.$$ What kind of assumption might I impose on $K$ such that $\lambda=1$ will be not an eigenvalue of the operator $L$?. Any ideas?. Thank you.

Sard-Smale theorem holds for Fredholm maps $f:M\rightarrow B$ between separable Banach manifolds $M,N$. There are some constrains relating the Fredholm index $\operatorname{ind}(f)$ of $f$ to its differentiablity class. More precisely, we need to require $f\in C^{r}$, where $r>\max{(\operatorname...

An operator $T:X\rightarrow Y$ is said to be upper semi-Fredholm if its range is closed and its kernel is finite-dimensional. M. Schechter (1972) introduced a quantity $$\nu(T):=\sup_{\operatorname{codim} M<\infty}\inf_{x\in M, \,\|x\|=1}\|Tx\|$$ and proved that $T$ is upper semi-Fredholm if and ...

Let $N,n>0$. Suppose $$D: C^\infty_c(\mathbb{R}^n, \mathbb{R}^N)\to C_c^\infty(\mathbb{R}^n,\mathbb{R}^N)$$ is an elliptic differential operator of degree one. Construct its corresponding degree zero pseudo-differential operator $$(1+DD^*)^{-\frac{1}{2}}D :C^\infty_c(\mathbb{R}^n,\mathbb{R}^N)\...

In Sewall Wright's Evolution of Mendalian Population, the equation for the nonrecurrent mutation is $$\frac{\phi(x)}n = \binom n {nx}\int_0^1 q^{nx}(1-q)^{n(1-x)}\phi(q)\,dq,\quad \forall x\in[0,1],$$ where $n>0$ and $\binom a b$ is the binomial coefficient. We are to solve for a nonzero function...

Let $X$ be a complex infinite dimensional Banach space, and let $T \in B(X)$ be nonscalar. The Fredholm spectrum of $T$ is defined by: $$ \sigma_{\Phi} (T) := \lbrace \lambda \in \mathbb{C} : T- \lambda Id \not \in \Phi \rbrace $$ (an operator is Fredholm iff $\dim \ker T, \dim \ker T^* < \infty$...

Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space. For a continuous map $T\colon X\to F$, there exists a closed subspace $W\subseteq H$ with $\dim H/W<\infty$ such th...

(I am posting this in my capacity as chair of the ICM programme committee.) ICM 2022 will feature a number of "special lectures", both at the sectional and plenary level, see last year's report of the ICM structure committee. The idea is that these are lectures that differ from the traditional IC...

We just got a question tagged icm-2014 and we have icm-2010 (with two closed questions tagged with it). I think it is not necessary to have separate tags here and one ICMs tag should suffice. Actually, I do not even think this ICM tag is necessary, conferences should suffice. The numbers o...

The next International Congress of Mathematicians (ICM) will be next year in Rio de Janeiro, Brazil. The present question is the 2018 version of similar questions from 2014 and 2010. Can you, please, for the benefit of others give a short description of the work of one of the plenary speakers? L...

Is there a place to watch ICM 2018 plenary lectures (and other lectures if possible)? Here is the official Youtube channel of the ICM but they don't seem to be posting the lectures. https://www.youtube.com/channel/UCnMLdlOoLICBNcEzjMLOc7w Update: The public lectures of Etienne Ghys, Cedric Vil...

Is there a possibility to watch ICM 2014 opening ceremony and the big talks online? I hope there is since it was possible for the previous meeting.

The next International Congress of Mathematicians (ICM) will take place in 2014 in Seoul, Korea. The present question is meant to gather brief overviews of the work of the plenary speakers for the ICM 2014. More precisely, anybody who feels qualified to give a short description of the work of on...

Articles from the Proceedings of the International Congress of Mathematicians, Seoul, 2014 don't appear to be on Mathscinet. Why is this? (Someone pointed this out to me recently, and I was reminded of it today when I tried to cite a lecture.)

The ICM is approaching. It would be nice for everybody who feels qualified to give a brief overview of the work of one of the plenary speakers. If anything, this would serve to make all of us a little more cultured. On that same note - I would be especially interested in getting to know more abou...

The logo for this year's ICM show the inequality $ |\tau(n)| \leq n^{11/2} d(n)$ where $\sum \tau(n)q^n = q \prod_{n \neq 1} (1 - q^n)^{24}$ is the tau function. Wikipedia says this bound was conjectured by Ramanujan (appropriate for a conference in Hyderabad) and proven by Deligne in '74 in the...