- Home
- About the Department
- Research
- People
- Undergraduate Program
- Introduction
- Course Information
- Concentration in Mathematics
- First-year Information
- Paris Math Program
- Opportunities and Extracurricular Activities
- Junior Tutors, VCAs, and Readers
- Directed Reading Program
- Urban Teacher Education Program
- Association for Women in Mathematics
- Employment Resources
- Contact/Help

- Graduate Program
- Outreach
- Resources

## Department Colloquia and Special Talks

## Forthcoming colloquia and special talks (in chronological order)

### February 2015

#### Department Colloquium

**Date**: Monday, February 23, 2015

**Time**: 3:00pm--4:00pm

**Venue**: Eckhart 202

**Speaker**: Jordan Ellenberg (UW-Madison)

**Title**: Topology and counting

**Abstract**:
Weil wrote about the theory of function fields of curves over
finite fields as a God-given “bridge” between number theory and topology. In
modern terms, that bridge is provided by the Grothendieck-Lefschetz trace
formula, which allows us to say something about the number of points on a
variety X over a finite field (resp. the average value of an interesting
function on those points) in terms of the etale cohomology of X (resp. etale
cohomology with coefficients in an interesting sheaf.) In particular, this
bridge allows us to connect conjectures and theorems in topology (concerning
stable cohomology of moduli spaces) with conjectures and theorems in number
theory (concerning distribution and asymptotics of arithmetic functions.)
I’ll give a survey of results and ideas in this direction, including some
subset of: the Cohen-Lenstra conjectures on class groups of random number
fields, the Poonen-Rains conjectures on Selmer groups of random elliptic
curves, distribution of numbers of primes in short intervals, the
Batyrev-Manin conjectures, etc. The overall theme is that ideas from topology
provide a consistent, geometrically principled machine for generating
conjectures in number theory, and sometimes for proving these conjectures in
the global function field case.

## Past colloquia and special talks (in reverse chronological order)

### January 2015

#### Department Colloquium

**Date**: Wednesday, January 7, 2015

**Time**: 3:00pm--4:00pm

**Venue**: Eckhart 206

**Speaker**: Henry Cohn (Microsoft Research, New England)

**Title**: Sums of squares, correlation functions, and exceptional geometric structures

**Abstract**:
Some exceptional structures such as the icosahedron or E_8 root system have
remarkable optimality properties in settings such as packing, energy
minimization, or coding. How can we understand and prove their
optimality? In this talk, I'll interweave this story with two other
developments in recent mathematics (without assuming familiarity with
either): how semidefinite optimization and sums of squares have expanded
the scope of optimization, and how representation theory has shed light on
higher correlation functions for particle systems.

### November 2014

#### Department Colloquium

**Date**: Wednesday, November 19, 2014

**Time**: 3:00pm--4:00pm

**Venue**: Eckhart 206

**Speaker**: Miguel Walsh (University of Oxford)

**Title**: The algebraicity of sieved sets and rational points on curves

**Abstract**:
We will discuss some connections between the polynomial method, sieve theory, inverse problems in arithmetic combinatorics and the estimation of rational points on curves. Our motivating questions will be to classify those sets that are irregularly distributed in residue classes and to understand how many rational points of bounded height can a curve of fixed degree have.

#### Special Colloquium

**Date**: Thursday, November 6, 2014 (UNUSUAL DAY)

**Time**: 1:30pm--2:30pm (UNUSUAL TIME)

**Venue**: Ryerson 352 (Barn)

**Speaker**: Charles Smart (Cornell)

**Title**: The scaling limit of the Abelian sandpile on infinite periodic graphs

**Abstract**:
Joint work with Lionel Levine and Wesley Pegden. The
Abelian sandpile is a deterministic diffusion process on graphs which,
at least on periodic planar graphs, generates striking fractal
configurations. The scaling limit on the two dimensional integer
lattice can be described in terms of an Apollonian circle packing.
General periodic graphs share some of this structure, but appear to
have more complicated scaling limits in general. I will discuss both
the square lattice and general cases.

### Octobor 2014

#### AWM Colloquium

**Date**: Friday, Octobor 3, 2014 (UNUSUAL DAY)

**Time**: 3:00pm--4:00pm

**Venue**: Eckhart 206

**Speaker**: Sylvie Méléard (Ecole Polytechnique)

**Title**: Stochastic Modeling for Darwinian Evolution

**Abstract**:
We are interested in modeling the Darwinian evolution resulting from
the interplay of phenotypic variation and natural selection through
ecological interactions, in the specific scales of the biological
framework of adaptive dynamics. We consider a multi-resources
chemostat model, where the competition between bacteria results from
the sharing of several resources which have their own dynamics.
Starting from a stochastic birth and death process model, we prove
that, when advantageous mutations are rare, the population behaves on
the mutational time scale as a jump process moving between
equilibrium states. An essential technical ingredient is the study of
the long time behavior of a multi-resources dynamical system. In the
small mutational steps limit this process in turn gives rise to a
differential equation in phenotype space called canonical equation of
adaptive dynamics. From this canonical equation and still assuming
small mutation steps, we give a characterization of the evolutionary
branching points.

### May 2014

#### AWM Colloquium

**Date**: Wednesday, May 21, 2014

**Time**: 3:00pm--4:00pm

**Venue**: Eckhart 206

**Speaker**: Lauren Williams (Berkeley)

**Title**: The positive Grassmannian

**Abstract**:
The positive Grassmannian is a remarkable subset of the real
Grassmannian which has recently arisen in diverse contexts such as
integrable systems, scattering amplitudes, and free probability. I will
provide an elementary introduction to the positive Grassmannian, focusing on a
few intriguing aspects: what it ``looks like," and its connection to shallow
water waves (via the KP hierarchy).

#### Department Colloquium

**Date**: Wednesday, May 7, 2014

**Time**: 2:00pm--3:00pm

**Venue**: Eckhart 206

**Speaker**: Alice Guionnet (MIT)

**Title**: Free probability and random matrices; from isomorphisms to universality

**Abstract**:

### March 2014

#### Department Colloquium

**Date**: Wednesday, March 19, 2014

**Time**: 3:00pm--4:00pm

**Venue**: Eckhart 206

**Speaker**: William P. Minicozzi II (Johns Hopkins University)

**Title**: Uniqueness of blowups and Lojasiewicz inequalities

**Abstract**:
Mean curvature flow starting from any smooth closed hypersurface is smooth for short time, but always becomes singular as the nonlinearities dominate.
Once one knows that singularities occur, one naturally wonders what the
singularities are like. For minimal varieties the first answer, by
Federer-Fleming in 1959, is that they weakly resemble cones. For mean
curvature flow (MCF), by the combined work of Huisken, Ilmanen, and White,
singularities weakly resemble shrinkers. Unfortunately, the simple proofs
leave open the possibility that a minimal variety or a MCF looked at under a
microscope will resemble one blowup, but under higher magnification, it might
microscope will resemble one blowup, but under higher magnification, it might
(as far as anyone knows) resemble a completely different blowup. Whether this
ever happens is perhaps the most fundamental question about singularities. We
will discuss the proof of this long standing open question for MCF at all
generic singularities and for mean convex MCF at all singularities. This is
joint work with Toby Colding.

#### Department Colloquium (Joint with CS Theory Seminar)

**Date**: Friday, March 7, 2014

**Time**: 1:30pm--2:30pm

**Venue**: Eckhart 133

**Speaker**: Assaf Naor (Courant Institute)

**Title**: Vertical versus horizontal Poincare inequalities

**Abstract**:
A classical result of Pansu and Semmes asserts that the
Heisenberg group does not admit a bi-Lipschitz embedding into any
Euclidean space. Several alternative proofs of this fact have been
subsequently found, yielding the non-embeddability of the Heisenberg group
into a variety of spaces, including uniformly convex spaces and L_1. These
proofs rely on metric differentiation methods, i.e., the use of a limiting
procedure to show that it suffices to rule out certain more structured
embeddings. For certain applications, which will be explained in this
talk, it is important to get quantitative bounds, in which case turning
the metric differentiation arguments into quantitative statements is quite
challenging and yields sub-optimal bounds. This talk, which assumes no
prerequisites, will start by explaining the approaches to the Heisenberg
embeddability question based on metric differentiation, and then present a
new and different approach based on Littlewood-Paley theory that yields
asymptotically sharp distortion bounds for embeddings of balls in the
Heisenberg group into uniformly convex spaces. We will end with a
conjectural isoperimetric inequality that is motivated by the new
Littlewood-Paley approach, and explain its implications to approximation
algorithms.

#### Department Colloquium

**Date**: Wednesday, March 5, 2014

**Time**: 3:00pm--4:00pm

**Venue**: Eckhart 206

**Speaker**: Bhargav Bhatt (Columbia)

**Title**: Hodge theory in the p-adic world

**Abstract**:
Hodge theory, which is the study of "hidden" structures on
the cohomology groups of complex manifolds, is a cornerstone of
complex geometry. My talk will discuss the analogous theory in p-adic
algebraic geometry. More precisely, after reviewing some relevant
aspects of the complex story, I will explain the p-adic analog of de
Rham's theorem (identifying de Rham and singular cohomology for
manifolds), emphasizing the crucial role played by the arithmetic of
p-adic fields. The presentation will be based on recent work of
Beilinson and myself.

### February 2014

#### Department Colloquium/Joint Seminar

**Date**: Wednesday, Friday, February 26, 28, 2014

**Time**: 2:30pm--4:00pm

**Venue**: Eckhart 206

**Speaker**: Frank Calegari (Northwestern)

**Title**: The stable homology of congruence subgroups

**Abstract**:
Let $F$ be a number field. A stability result due to Charney and Maazen says that the homology groups $H_d(\mathrm{SL}_N(\mathcal{O} _F),\mathbf{Z})$ (for $d$ fixed) are independent of $N$ for $N$ sufficiently large. The resulting stable cohomology groups are intimately related to the algebraic $K$-theory of $\mathcal{O}_F$. In these talks, we shall explore the homology of the $p$-power congruence subgroups of $\mathrm{SL}_N(\mathcal{O}_F)$ in fixed degree $d$ as $N$ becomes large. We show that the resulting homology groups consist of two parts: an ``unstable'' part which depends only on local behavior concerning how the prime $p$ splits in $F$, and a ``stable'' part which contains global information concerning $p$-adic regulator maps. Our argument consists of two parts. The first part (which is joint work with Matthew Emerton) explains how to modify the homology of congruence subgroups in a suitable way (using completed homology) to obtain groups which are literally stable for large $N$. We prove this by combining recent results in representation stability with the representation theory of $p$-adic groups. The second part consists of relating these stable homology groups to $K$-theory using ideas from algebraic topology. We give some applications of our results. The first concerns the computation of $H_2$ of the $p$-congruence subgroup of $\SL_N(\Z)$ for sufficiently large $N$. The second concerns describing the extent to which the stable classes in characteristic zero constructed by Borel become more and more divisible by $p$ when one passes to higher and higher $p$-power level.

#### Department Colloquium

**Date**: Wednesday, February 12, 2014

**Time**: 3:00pm--4:00pm

**Venue**: Eckhart 206

**Speaker**: Theodore A. Slaman (Berkeley)

**Title**: On Normal Numbers

**Abstract**:
A real number is simply normal in base b if in its base-b expansion each digit
appears with asymptotic frequency 1/b. It is normal in base b if it is simply
normal in all powers of b, and absolutely normal if it is simply normal in
every integer base. By a theorem of E. Borel, almost every real number is
absolutely normal. We will present three main results. We will give an
efficient algorithm, which runs in nearly quadratic time, to compute the
binary expansion of an absolutely normal number. We will demonstrate the full
logical independence between normality in one base and another. We will give a
necessary and sufficient condition on a set of natural numbers M for there to
exist a real number X such that X is simply normal to base b if and only if b
is an element of M. If time allows, we will discuss normality for Liouville
numbers.

### November 2013

#### Department Colloquium

**Date**: Wednesday, November 20, 2013

**Time**: 3 - 4 PM

**Venue**: Eckhart 206

**Speaker**: Nick Nozenblyum (Northwestern)

**Title**: Derived Geometry and Quantum Invariants

**Abstract**:
Over the last several decades, quantum field theory has been a source of many new ideas in mathematics and particularly in geometry. The Feynman path integral has been used to define many interesting invariants of manifolds and algebraic varieties. For instance, the so called B-model suggests that there should be natural operations on the Hodge cohomology of a projective Calabi-Yau manifold parametrized by Riemann surfaces with boundary.
I will describe how derived algebraic geometry, which is roughly the study of solutions of equations up to homotopy, provides a natural setting for studying these types of invariants. From this point of view, the Feynman integral can be interpreted as an integral over a derived space. I will describe how the B-model, as well as a plethora of other interesting invariants such as the Todd genus and the Witten genus, can be constructed using these ideas.

#### Special Representation Theory Seminar

**Date**: Tuesday, November 12, 2013

**Time**: 3 - 4 PM

**Venue**: Eckhart 308

**Speaker**: Steven
Sam (University of California, Berkeley)

**Title**: Twisted commutative algebras and representation stability

**Abstract**:
Twisted commutative algebras are commutative rings with an action of the
infinite general linear group and are connected with recent studies of
"representation stability". The simplest nontrivial example of a twisted
commutative algebra is the infinite polynomial ring C[x_1, x_2, ...] with
the natural action of GL(infinity). Its module category is equivalent to
the category of (complex) FI-modules, which was recently introduced by
Church-Ellenberg-Farb to study cohomology of configuration spaces and many
other examples. In this talk I will explain joint work with Andrew Snowden
on the structural properties of this category and the connection to
FI-modules. Time permitting, I will explain what we know about more
complicated twisted commutative algebras and their connections with
representation stability.

This colloquium is jointly sponsored by the Association for Women in Mathematics:

**Date**: Tuesday, November 5, 2013

**Time**: 4:30PM -- 5:30 PM

**Venue**: Eckhart 206

**Speaker**: Wei Ho
(Columbia University)

**Title**: Arithmetic invariant theory and applications

**Abstract**:
The origins of "arithmetic invariant theory" come from the
work of Gauss, who used integer binary quadratic forms to study ideal
class groups of quadratic fields. The underlying
philosophy---parametrizing arithmetic and geometric objects by orbits
of group representations---has now been used to study higher degree
number fields, curves, and higher-dimensional varieties. We will
discuss some of these constructions and highlight the applications to
topics such as bounding ranks of elliptic curves and dynamics on K3
surfaces.

This talk is intended for a general mathematical audience.

### May 2013

**Date**: Friday, May 24, 2013

**Time**: 3-4PM

**Venue**: Eckhart 206

**Speaker**: Thomas Scanlon

**Title**: Diophantine geometry of algebraic dynamics

**Abstract**: Generalizing from the case of sequences defined by
linear recurrence relations, one may consider sequences obtained by
iteratively applying polynomials. Several authors have raised a
conjecture about iterates of polynomial functions inspired by
Mordell's conjecture (Faltings' Theorem) about rational points on
curves ... (full abstract at http://math.uchicago.edu/research/abstracts/Cohenlecture2013.pdf).

**Date**: Thursday, May 23, 2013

**Time**: 4:30-5:30 PM

**Venue**: Eckhart 206

**Speaker**: Luis Caffarelli

**Abstract**: We will discuss several moments involving
segregation: adjacent ones, where particles of different kind
annihilate on contact, and models where the segregation occurs at a
distance We will describe some simple mathematical models and their
analytical and geometric properties.

This colloquium is jointly sponsored by the Association for Women in Mathematics:

**Date**: Friday, May 10, 2013

**Time**: 4:30 - 5:30 PM

**Venue**: Eckhart 206 (Room capacity: 72)

**Speaker**: Lai-Sang Young (Courant Institute/NYU)

**Talk title**: Measuring dynamical complexity

**Abstract**: I will discuss three ways to capture dynamical
complexity: (A) hyperbolicity, a geometric characterization of
instability, (B) entropy, an information-theoretical approach to
capturing the randomness of dynamical events, and (C) correlation
decay or memory loss as a function of time. I will review these ideas,
discuss how they are related, and give a very brief (and somewhat
personal) survey of the progress made in the last decades. For
illustration, I will use a concrete example, namely that of
shear-induced chaos in periodically kicked oscillators. The idea of
this example goes back to van der Pol nearly 100 years ago, but the
mathematics was done only recently.

### April 2013

This colloquium is jointly sponsored by the Association for Women in Mathematics:

**Date**: Friday, April 19, 2013

**Time**: 4 - 5 PM

**Venue**: Eckhart 206 (Room capacity: 72)

**Speaker**: Laura DeMarco (UIC, works in dynamics and complex analysis)

**Talk title**: Complex and Arithmetic Dynamics.

**Abstract**: Questions about complex dynamical systems have
traditionally been approached with techniques from analysis (complex
or geometric). In the last 5 years or so, methods from arithmetic and
algebraic geometry have played a central role -- and the result is an
active new research area, the "arithmetic of dynamical systems" (to
borrow the title of Silverman's textbook on the subject). The
questions themselves have evolved, inspired by results from arithmetic
geometry. In this talk, I will present joint work with Matt Baker,
where we study "special points" within the moduli space of complex
polynomial dynamical systems.

**Date**: Wednesday, April 3, 2013

**Time**: 3 - 4 PM

**Venue**: Eckhart 206 (Room capacity: 72)

**Speaker**: Marston Conder. He is the AMS-NZMS Maclaurin lecturer.

**Talk title**: Some unexpected theoretical consequences of computations involving groups

**Abstract**: In this talk I will give some instances of
experimental computations involving groups (in the context of their
actions on graphs and maps) that have led to unexpected theoretical
discoveries. These include new presentations for 3-dimensional special
linear groups, a closed-form definition for the binary reflected Gray
codes, a new theorem on groups expressible as a product of an abelian
group and a cyclic group, and revealing observations about the genus
spectra of particular classes of regular maps on surfaces. Such
examples highlight the value of computational algebra as a tool for
experimental work, which can have surprising outcomes.

### March 2013

**Date**: Tuesday, March 19, 2013

**Time**: 3 - 4 PM

**Venue**: Eckhart 206 (Room capacity: 72)

**Speaker**: Jeff Harvey (University of Chicago Physics Department)

**Talk title**: Mock Modular Forms, K3 surfaces, and Umbral Moonshine

**Abstract**: Monstrous moonshine is a phenomenon linking
modular forms with the representation theory of the Monster, the
largest sporadic group. Mock modular forms are generalizations of
modular forms introduced by Ramanujan, but a coherent definition of
them appeared only in the last ten years due to work of Zwegers
connecting them to indefinite theta functions, Appell-Lerch sums and
meromorphic Jacobi forms. Recently evidence for a new kind of
moonshine, dubbed Umbral Moonshine, has emerged involving connections
between certain of these mock modular forms and a sequence of finite
groups that arise as subgroups of the Conway group. The first of these
examples, discovered by Eguchi, Ooguri and Tachikawa, involves a
connection between the largest Mathieu group M24 and a mock modular
form which is related to the elliptic genus of K3 surfaces. The origin
and explanation of Umbral Moonshine are at the moment a mystery. I
will survey these new developments and suggest that the eventual
explanation is likely to involve aspects of string theory and the
AdS/CFT correspondence. This talk is based on arXiv preprint 1204.2779
with Miranda Cheng and John Duncan.

This colloquium is jointly sponsored by the Association for Women in Mathematics:

**Date**: Wednesday, March 6, 2013

**Time**: 3 - 4 PM

**Venue**: Eckhart 206 (Room capacity: 72)

**Speaker**: Karin Melnick (University of Maryland)

**Talk title**: A rigidity theorem in conformal geometry and beyond

**Abstract**: The classical exponential map in Riemannian
geometry has the following very important implications: if an isometry
f fixes a point and has trivial derivative there, then f is trivial;
moreover, the differential gives a simple normal form for all
isometries fixing a given point. Conformal transformations of a
Riemannian manifold are required only to preserve angles, not
distances. These have no exponential map. Nontrivial conformal
transformations can have differential equal the identity at a fixed
point, but this occurrence has very strong implications for the
underlying manifold.

I will present this rigidity phenomonenon in conformal geometry and a wide range of generalizations. The key to these results is the notion of Cartan geometry, which infinitesimally models a manifold on a homogeneous space. This point of view leads to a normal forms theorem for conformal Lorentzian flows. It also leads to a suite of results on a seemingly widespread rigidity phenomenon for flows on parabolic geometries, a rich family of geometric structures whose homogeneous models include flag varieties and boundaries of symmetric spaces.

### February 2013

**Date**: Tuesday, February 19, 2013

**Time**: 4:30 - 5:30 PM

**Venue**: Eckhart 206 (Room capacity: 72)

**Speaker**: Leonid Polterovich (Tel Aviv University; visiting
the University of Chicago)

**Title**: Symplectic topologist's tale of quantum mechanics

**Abstract**: We focus on constraints on the Poisson brackets
found within Symplectic Topology. Their interpretation and proof are
related to Quantum Mechanics. In the talk we discuss an exchange of
ideas between these fields.

### November 2012

**Date**: Wednesday, November 28, 2012

**Time**: 5:00-6:00 PM

**Venue**: Eckhart 202 (Room capacity: 72)

**Speaker**: Luis Silvestre (University of Chicago)

**Title**: On the continuity of solutions to drift-diffusion equations

**Abstract**: A drift-diffusion equation is like the heat
equation with an extra first order term. In some cases, the Laplacian
is replaced by a fractional Laplacian. There are several nonlinear
models in a variety of contexts that fit into this scheme. In order to
understand the solvability of the non linear models, it is essential
to obtain a priori estimates on the smoothness of the solution for
linear drift-diffusion equations when the first order term is given by
a very irregular vector field times the gradient. We will analyze
different smoothness estimates in different situations. It is
particularly important to understand the consequences of assuming that
the drift is divergence free, given their applications to models
related to incompressible fluids.

### October 2012

**Date**: Wednesday, October 3, 2012

**Time**: 4:30-5:30 PM

**Venue**: Eckhart 133 (Room capacity: 106)

**Speaker**: Maryanthe Malliaris (University of Chicago)

**Title**: Comparing the complexity of unstable theories

**Abstract**: Model theory brings a unique and powerful point of
view to bear on many fundamental structural questions in
mathematics. This talk will have two interrelated aims. The first is
to motivate and present some recent progress of Malliaris on Keisler's
order: a suggested program of comparing the complexity of theories
from asymptotic (ultrapower) points of view. These results, the first
in decades in this program, build towards a true classification of the
unstable theories. The stable theories have been well understood
thanks to Morley's 1962 Chicago PhD thesis and to Shelah's significant
generalizations of Morley's work.

The second is to indicate how our work arising from this model theoretic program has led to theorems in Szemeredi regularity and in set theory and general topology. Cantor proved in 1874 that the continuum is uncountable, and Hilbert's first problem asked whether it is the smallest uncountable cardinal. A program developed over the course of the last century to measure the continuum in terms of various related cardinals. By Godel 1939 and Cohen 1963, Hilbert's first problem was proved independent of ZFC. However, several basic questions about these cardinal invariants of the continuum remained open despite much investigation. The oldest and probably the most famous of these is whether "p = t", proved in a special case by Rothberger 1948, building on Hausdorff 1934. Very recently, Malliaris and Shelah resolved this question, in a surprising way, using model-theoretic tools developed for the study of Keisler's order.

### AWM Colloquium, April 2012

The colloquium is jointly sponsored by the Department of Mathematics and the Association for Women in Mathematics.

**Date**: Friday, April 6, 2012

**Time**: 3 - 4 PM

**Venue**: Eckhart 206 (Room capacity: 72)

**Speaker**: Melanie Wood (University of Michigan)

**Title**: Counting polynomials and motivic stabilization

**Abstract**: We will begin with the problem of counting
polynomials modulo a prime p with a given pattern of root
multiplicity. Here we will discover phenomena that point to vastly
more general patterns in configuration spaces of points. To see these
patterns, one has to work in the ring of motives — so we will describe
this place where a space is equivalent to the sum of its pieces. We
will then be able to describe how these patterns in the ring of
stability of configuration spaces. This talk is based on joint work
with Ravi Vakil.

### November 2011

**Date**: Friday, November 11, 2011

**Time**: 4 PM onward

**Venue**: Eckhart 206 (Room capacity: 72)

**Speaker**: Viatcheslav Kharlamov (IRMA, Strasbourg)

**Title**: First steps in real enumerative geometry.

**Abstract**: Surprisingly, in a quite a number of real
enumerative problems the number of real solutions happens to be
comparable (for example, in logarithmic scale) with the number of
complex ones. For the moment, such a phenomenon is studied quite in
depth in the case of interpolation of real points on a real rational
surface by real rational curves. The key tool here are the Welschinger
invariants (which can be seen as a real analogue of genus zero
Gromov-Witten invariants). In this talk, based on joint works with
I. Itenberg and E. Shustin, I will remind how Welschinger invariants
look like in the case of surfaces, point certain modifications, and
present some recursive formulas that allow to control Welschinger
invariants in the case of Del Pezzo surfaces and by these means to
establish some basic properties of the invariants that imply, in
particular, the abundance of real solutions.