Extremal set theory tries to answer questions about the maximal or minimal size of subsets of some universal set, while respecting certain imposed restrictions. In this talk we will discuss one such example, which is known as the Frankl-Wilson theorem. This theorem turns out to have applications in geometry, providing lower bounds on both the chromatic number of Euclidean space and the number of parts one needs to subdivide a bounded region in Euclidean space into smaller regions.

Harnessing disruptive innovations dynamics is theoretically presented based on Dirac-Solow-Swan model. The quantum view leads into the conclusion that hyperfine splitting of capital is occurred due to the disruption and as a consequence is the excitation of capital and labour from the old into the new industry of disruption. The bifurcation of capital dynamics occurs due to Christensen effect, after market symmetry breaking. It is shown that harnessing disruptive innovations relies on controlling expansion factor of capital accumulation on mainstream market.

$\indent$The classical congruence subgroup problem asks whether every finite
quotient of $G={\rm GL}_{n}(\mathbb{Z})$ comes from a finite quotient
of $\mathbb{Z}$. I.e. whether every finite index subgroup of $G$ contains a principal
congruence subgroup of the form
$G(m)= ker(G\to{\rm GL}_{n}(\mathbb{Z}/m\mathbb{Z})$ for some $m\in\mathbb{N}$? If the answer is affirmative we say that $G$ has the congruence subgroup property (CSP). It was already known in the $19^{\underline{th}}$ century that ${\rm GL}_{2} (\mathbb{Z})$ has many finite quotients which do
not come from congruence considerations. Quite surprising, it was proved in the sixties
that for $n\geq3, {\rm GL}_{n} (\mathbb{Z})$ does have the CSP.

Observing that ${\rm GL}_{n} (\mathbb{Z}) \cong {\rm Aut} (\mathbb{Z}^{n})$, one can generalize the congruence subgroup problem as follows: Let $\Gamma$ be a group. Does every finite index subgroup of $G = {\rm Aut}(\Gamma)$ contain a principal congruence subgroup of the form $G(M) = \ker(G\to{\rm Aut}(\Gamma/M)$ for some finite index characteristic subgroup $M\leq\Gamma$? Very few results are known when $\Gamma$ is not abelian. For example,
we do not know if ${\rm Aut} (F_{n})$ for $n\geq3$ has the CSP. But, in 2001 Asada proved, using tools from algebraic geometry, that ${\rm Aut} (F_{2})$ does have the CSP, and later, Bux-Ershov-Rapinchuk gave a group theoretic version of Asada's proof (2011).

On the talk, we will give an elegant proof to the above theorem, using
basic methods of profinite groups and free groups.

Subsampling methods as well as general sampling methods appear as natural tools
to handle very large database (big data in the indivual dimension) when
traditional statistical methods or statistical learning algorithms fail to be
implemented on too large datasets. The choice of the weights of the survey
sampling sheme may reduce the loss implied by the choice of a much more smaller
sampling size (according to the problem of interest).

I will first review some asymptotic results for general survey sampling based
empirical processes indexed by class of functions (Bertail and Clemencon, 2017),
for Poisson type and conditional Poisson (rejective) survey samplings. These
results may be extended to a large class of survey sampling plans via the notion
of negative association of most survey sampling plans (Bertail and Rebecq, 2017).
Then, in the perspective to generalize some statistical learning tasks to sampled
data, we will obtain exponential bounds for the probabilities of deviation of a
sample sum from its expectation when the variables involved in the summation are
obtained by sampling in a finite population according to a rejective scheme,
generalizing sampling without replacement and using an appropriate normalization.

In contrast to Poisson sampling, classical deviation inequalities in the i.i.d.
setting do not straightforwardly apply to sample sums related to rejective schemes
due to the inherent dependence structure of the sampled points. We show here how
to overcome this difficulty by combining the formulation of rejective sampling
as Poisson sampling conditioned upon the sample size with the Escher transformation.
In particular, the Bennet/Bernstein type bounds established highlight the effect of
the asymptotic variance of the (properly standardized) sample weighted sum, and are
shown to be much more accurate than those based on the negative association property.

The numerical analysis of finite element methods in computational
electromagnetism can be developed elegantly and comprehensively if commuting
projection operators between de Rham complexes are available. Hence the
construction of such commuting projection operators is central but has been
elusive in several practical relevant settings of low regularity. In this talk
we describe how to close this gap: we construct smoothed projections over
weakly Lipschitz domains and extend the theory to mixed boundary conditions.

We study random constructions in incidence structures using a general theorem on set systems. Our main result applies to a wide variety of well-studied problems in finite geometry to give almost tight bounds on the sizes of various substructures.

In its classical setting, the Congruence
Subgroup Problem (CSP) asks whether every finite index subgroup of
$GL_{n}(\mathbb{Z})$ contains a principal congruence subgroup of the
form $\ker(GL_{n}(\mathbb{Z})\to GL_{n}(\mathbb{Z}/m\mathbb{Z}))$ for
some $m\in\mathbb{Z}$. It was known already in the 19th century that
for $n=2$ the answer is negative, and actually $GL_{2}(\mathbb{Z})$
has many finite index subgroups which do not come from congruence
considerations. On the other hand, quite surprisingly, in the sixties
it was found out by Mennicke, and separately by Bass-Lazard-Serre,
that the answer for $n>2$ is affirmative. This result was a
breakthrough that led to a rich theory which generalized the problem
to matrix groups over rings.

Viewing $GL_{n}(\mathbb{Z})\cong Aut(\mathbb{Z}^{n})$ as the automorphism
group of of $\Delta=\mathbb{Z}^{n}$, one can generalize the CSP to
automorphism groups as follows: Let $\Delta$ be a group, does every
finite index subgroup of $Aut(\Delta)$ contain a principal congruence
subgroup of the form: $\ker(Aut(\Delta)\rightarrow Aut(\Delta/M))$
for some finite index characteristic subgroup $M\leq\Delta$? Considering
this generalization, there are very few results when $\Delta$ is
non-abelian. For example, only in 2001 Asada proved, using tools from
Algebraic Geometry, that $Aut(F_{2})$ has an affirmative answer to
the CSP, when $F_{2}$ is the free group on two generators. For $Aut(F_{n})$
when $n>2$ the problem is still unsettled. On the talk, I will present
the problem from a few aspects, and introduce some recent results
for non-abelian groups. The main result will assert that while the
dichotomy in the abelian case is between $n=2$ and $n>2$, when $\Delta$
is the free metabelian group on n generators, we have a dichotomy
between $n=2,3$ and $n>3$.

A variety is called rational if it is birational to projective space. For example, the only compact, smooth, and rational Riemann surface is the Riemann sphere. A general compact Riemann surface carries two natural invariants which measure its complexity & non-rationality from both a topological and an algebraic perspective: the genus and the gonality. Both of these invariants have classically played a very important role in the study of curves. In higher dimensions there are a number generalizations of these birational invariants which measure the irrationality of an algebraic variety. I will discuss the computation of one of these invariants — the degree of irrationality — and I will pose a number of open problem about these measures of irrationality.

The conditions which determine whether or not a matrix is special orthogonal are polynomial in the matrix entries and thus give an explicit description of SO(n) as an embedded algebraic variety. We give a formula for the degree of this variety for any n which is interpretable as counting non-intersecting lattice paths. This degree also contributes to the degree of low-rank semidefinite programming. We explain how to verify the formula explicitly using numerical algebraic geometry (for $n\leq7$) and how numerical computations aid in the study of the real locus of this variety.

Multi-strain microbial communities often exhibit complex spatial organization that emerges due to the interplay of various cooperative and competitive interaction mechanisms. One strong competitive mechanism is contact-dependent neighbor killing, such as that enabled by the type VI secretion system (T6SS). It has been previously shown that contact-dependent killing can result in bistability of bacterial mixtures, so that only one strain survives and displaces the other. However, it remains unclear whether stable coexistence is possible in such mixtures. Using a population dynamics model for a mixture of two bacterial strains, we found that coexistence can be made possible by combining contact-dependent killing with long-range growth inhibition, leading to the formation of various cellular patterns. These patterns emerge in a much broader parameter range than that required for the Turing instability, suggesting this may be a more robust mechanism for pattern formation.

I will discuss the theory of theta representations for the degree n cover
of Glr and in particular those distinguished ones that have unique Whittaker
models. I will concentrate on the study of the know cuspidal distinguished
representations and possible generalizations.

I will explain how to build Koornwinder polynomials at q = t from moments of Askey-Wilson polynomials.
I will use the combinatorial theory of Viennot for orthogonal polynomials and their moments. An extension of this theory allows to build multivariate orthogonal polynomials.
The key step for this construction area Cauchy identity for Koornwinder polynomials and a Jacobi-Trudi formula for the 9th variation of Schur functions. This gives us an elegant path model for these polynomials. I will also explain a positivity conjecture for these polynomials that we can prove in several special cases. For this, we link them to the stationary distribution of an exclusion process and prove positivity by exhibiting a combinatorial model called rhombic staircase tableau. This talk is based on joint work with Olya Mandelshtam (Brown) and Lauren Williams (Berkeley).

Given a smooth projective variety over an algebraically closed field of positive characteristic, can we always dominate it by another smooth projective variety that lifts to characteristic 0? We give a negative answer to this question.