# Programmes - Faculty of Sciences, fourth year of the undergraduate course

**1 MASTER'S DEGREE COURSES IN MATHEMATICS, INFORMATION TECHNOLOGY AND RELATED SUBJECTS**

1.a. Mathematics Prerequisites

Algebra

Elementary properties of integers. Division algorithm. Euclidean algorithm. Prime numbers. Existence and uniqueness of the prime decomposition.

Binomial coefficients. Congruences and their properties. Congruence classes. Euler’s function. Euler's theorem. The complex field. Trigonometric representation of complex numbers. Modulus and argument of a complex number, and its trigonometric representation. Modulus and argument of a product. Roots of a complex number. Permutations. Decomposition as a product of cycles. Sign of a permutation. Groups and their properties. normal subgroups. Quotients. Lagrange’s theorem. The symmetric group and the alternating group. Group actions, and applications to the structure of finite groups. Sylow's theorems. Rings and ideals. Domains and fields. Euclidean domains. Domains with principal ideals. Domains with unique factorization. Gauss' lemma. Polynomials with coefficients in a field, and their arithmetic. Roots. Irreducible polynomials. Real irreducible polynomials. Eisenstein's criterion for the irreducibility of rational polynomials. Field extensions. Finite extensions, degree of an extension. Breaking fields. Structure of finite fields. Automorphism groups of finite extensions. Galois extensions. Galois correspondence. Modules over commutative rings. Structure theorem for finitely generated modules on principal ideal domains. Commutative Noetherian rings. Hilbert’s basis theorem.

Geometry

Elementary examples of curves and surfaces in a plane and in space. Flat and conical.

Matrix calculation. Systems of linear equations. Gaussian elimination. Determinant. Rank. Vector spaces. Dimension. Matrices and linear transformations. Nucleus and image of a linear transformation. Subspaces. Sums and direct sums. Eigenvectors and eigenvalues. Autospaces. Characteristic polynomials. Trace. of The Jordan Canonical form. Jordan's theorem. The Cayley–Hamilton theorem. Bilinear forms and Hermitian forms. Quadratic forms, and their diagonalization. Definite and semi-definite forms.

Real and Hermitian internal products. Schwarz inequality. Orthogonal and orthonormal bases. Gram–Schmidt procedure. Orthogonal complement of a subspace.

Orthogonal and unitary matrices. Symmetric and Hermitian matrices. Isometries between Euclidean spaces. Spectral theorem. Projective spaces. Linear subspaces. Projective references. Projectivity. Fundamental theorem of projective geometry: given two projective references in Pn, there exists a unique projectivity of Pn which leads one into the other. Classification of real and complex quadrics, in affine and projective domains. Metric spaces. Topological spaces. Continuous applications. Hausdorff spaces. Connected spaces; connection of the image of a connected space. Compact spaces; compactness of the image of a compact space.

Locally connected spaces. Locally compact spaces. Connected spaces for arcs.

Locally connected spaces by arcs. Product spaces. Product of connected spaces and compact spaces. Quotient spaces. Topology of real and complex projective spaces. Fundamental group. Coverings. Connection between coverings and fundamental group. Curves parametrized in space. Length. Frenet's formulas. Surfaces parametrized in space. Orientation. Application of Gauss. Fundamental forms. Curvature. Theorema Egregium. Geodesics.

Analysis

Construction and axiomatization of real and complex numbers, Dedekind's axioms, lower and upper bound.

Sequences, limits, operations with limits, Cauchy’s criterion. Numerical series and convergence criteria. Power series, radius of convergence. The Bolzano–Weierstrass theorem. Functions, graphs, compound function and inverse function. Continuity, discontinuity, monotone functions. Image using continuous functions of intervals. Uniform continuity. Invertible continuous functions. Derivative and its properties. Derivative of elementary functions. Differential and successive derivatives. Relative maximum and minimum, mean value theorem, Taylor's formulas and series expansions of elementary functions. Convex and concave functions. Riemann integral and integrability of continuous and monotonic functions. Fundamental theorem of integral calculus. Methods for calculating the primitive: integration by parts, integration by substitution, integration of elementary and rational functions. Generalized integrals and convergence criteria.

***

Metric spaces. Completeness and completion, principle of contractions. Compactness in metric spaces, totally bounded sets. Sequences and series of functions. Completeness of spaces Ck. Ascoli–Arzela' theorem. Classical theory of trigonometric Fourier series. Differential calculus in several variables. Partial and differential derivatives, derivation of the composite function, Taylor's formulas. Theorems of implicit functions and local invertibility, Lagrange’s multiplier method. Gauss–Green and Stokes formulas. Basic notions on differential forms.

Ordinary differential equations and systems. Solving methods for linear equations and systems, exponential of a matrix. The Cauchy–Lipschitz theorem of local existence and uniqueness, maximal solution and global existence criteria.

Qualitative study of solutions.

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Power series in the complex range. Radius of convergence. Uniform convergence. Analytic functions. Complex derivative. Cauchy–Riemann equations. Holomorphic functions. Complex logarithm, and its determinations. Integral of a differential form along a path in C. Cauchy's formula. Maximum modulus principle. Open application theorem. Singularity of holomorphic functions. Removable singularities. Riemann theorem. Laurent’s series expansions. Essential poles and singularities. Meromorphic functions. Residues. The index of a point with respect to a path. Residue theorem. Logarithmic derivative. Rouche's theorem and applications.

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Lebesgue’s integral, Fatou's lemma, monotonic and dominated convergence.

The Fubini–Tonelli theorem, H¨older and Minkowski inequalities. Probability

Basic notions of measure theory and of integration in measure spaces.

Random variables, expectation, variance. laws of Probability. The notion of independence and Kolmogorov's dichotomy laws. Convergence of random variables ( law, probability, almost certain). Characteristic functions, law of large numbers. Central limit theorem.

**2. MASTER'S DEGREE COURSES IN PHYSICS AND RELATED SUBJECTS**

2.a. Mathematics Prerequisites

Elements of Theory of Analytic Functions:

- Complex plane, Cauchy-Riemann conditions, conformal transformations;
- Curvilinear integrals, Cauchy's theorem, Cauchy's formula;
- Series of functions, Weierstrass's theorem, Cauchy-Hadamard's theorem;
- Taylor series and analytic extension, Laurent series, isolated singularities and theorem of residues, calculus of integrals and sum of series with the method of residues.

Partial differential equations of classical Mathematical Physics

- Poisson and Laplace equations, hints of potential theory, boundary conditions.
- Wave equation, boundary conditions, solutions.
- Heat equation, boundary condition, solutions.

Elementary theory of distributions

- Support, derivatives, multiplications, topology of distributions.
- Tensor product and convolution of distributions
- Fourier series, Fourier and Laplace transforms of functions and distributions.
- Inversion of Fourier series, Fourier and Laplace transforms.

Functions of Mathematical Physics

- The Gamma function, the B(p,q) function, Stirling's formula, psi function, and their main properties.
- Bessel, Neumann, Hankel functions: definition and main properties.

2.b. Physics prerequisites

Classical mechanics

- Lagrange equations;
- canonical variables, Poisson brackets, Hamilton equations;
- oscillations;
- rigid body;
- variational principles and Lagrange equations for continuum mechanics;
- Newtonian gravity;
- special relativity.

Electromagnetism

- electrostatics;
- magnetostatics;
- electromagnetic properties of matter;
- Maxwell's equations;
- electromagnetic waves;
- radiation;
- diffusion of electromagnetic waves;
- diffraction and interference.

Thermodynamics

- first law;
- second law
- entropy;
- thermodynamic potentials and Maxwell's relations.

Statistical physics

- kinetic theory: the Boltzmann equation;
- the Maxwell-Boltzmann distribution;
- microcanonical, canonical, grand canonical sets;
- partition function;
- quantum statistics;
- blackbody;
- ideal gases of Fermi and Bose.

Quantum mechanics

- crisis of classical physics;
- wave/particle dualism and uncertainty principle;
- Schroedinger equation;
- operators and representations (or mathematical formalism);
- quantum particle in potential field;
- angular momentum ;
- hydrogen atom;
- theory of perturbations and transitions;
- spin;
- systems of identical particles;
- collisions;
- emission and absorption of radiation by atoms;
- semiclassical approximation

**3. MASTER'S DEGREE COURSES IN CHEMISTRY, GEOLOGICAL SCIENCES AND TECHNOLOGIES AND RELATED SUBJECTS**

3.a. Chemistry prerequisites

Structure of matter. Experiments by Thomson, Rutherford and Millikan.

Periodic table. Physical and chemical properties of the elements. Ideal gases. Equation of state of gases. Elements of kinetic theory of gases. Real gases. The Van der Waals equation for real gases. The nature of light. absorption and emission spectra. Planck's equation. Wave nature of the electron. The spectrum of the hydrogen atom. Bohr's atomic model. Quantum atomic model. Orbitals and quantum numbers. Stability of noble gases. Octet rule. Ionic bond. Energy balance of ionic bond formation.

The covalent bond. common covalence. Multiple bonds.

Properties of covalent bonds. Bond polarity. Lewis molecular structures. Resonance hybrids. Exceptions to the octet rule: incomplete octet, unpaired electrons, magnetic properties of matter. Octet expansion. role of d orbitals. Shape of the molecules. VSEPR theory. Shape and geometry of molecules. Valence bond theory. Sigma bonds. Hybridization. Valence bond theory. Pi bonds. Resonance and delocalized pi bonds. Molecular orbital theory. Sigma and pi orbitals. Bonding and antibonding orbitals. Nature of intermolecular forces. States of aggregation of matter. Boiling and melting temperatures. Hydrogen bond. Phase diagrams of pure compounds (water and carbon dioxide). Phase transitions, triple point, critical point. The equilibrium reactions. Balance shift. Le Chatelier’s principle. Spontaneity of a reaction. Entropy and second law of thermodynamics. Free energy and spontaneity of a reaction. Solution equilibria. The dissociation of water. Weak acids and bases. The pH. Buffer solution. Acidic and basic hydrolysis. Titrations. Solution equilibria. Low solubility salts. Saturated solutions. Solubility and solubility product. Properties of solutions. Electrochemistry. Reduction potentials. Quantitative relationships in redox reactions. Electrolysis. Equilibrium constant of a redox reaction and standard potentials. Solution equilibria. Use of indicators in titrations.

Organic chemistry

Stereochemistry: definitions of chirality, asymmetry and dissymmetry; recognition of chiral compounds. Meso compounds. Erythro/threum nomenclature. Optical activity and enantiomeric excess. Cis/trans isomerism. Prostereoisomerism: enantio- or diastereotopic atoms or groups.

Radical halogenation of alkanes. Wurtz reaction and alkylsodium derivatives. Oxidative metalation and formation of lithium alkyl and Grignard reagents. Transmetallation reactions. Coupling reactions between organic halides and organometallic compounds. (Strong) oxygen nucleophiles: preparation of alkoxides and their use in the Williamson synthesis of ethers. Dehydrohalogenation reactions. Nitrogen nucleophiles and halide ammonolysis reactions. Gabriel synthesis. The cyanide ion as a carbon nucleophile: synthesis of nitriles. Acidity of acetylene and terminal alkynes: preparation of alkynes. Weak nucleophiles: monomolecular substitution and elimination reactions (SN1 and E1). Carbonyl as a functional group with electrophilic characteristics. Nucleophilic addition reactions to carbonyl. Carboxylic acid derivatives and nucleophilic acyl substitution reactions. Hydrolysis of esters in acidic and alkaline environments (saponification) - Transesterification reactions. Hydrolysis reactions in acid and alkaline environment of amides and nitriles. Preparation of sulphonic esters from alcohols and sulfonyl chlorides (Mesylates, triflates, tosylates). Halogenation reactions of alcohols with phosphorus halides and thionyl chloride.

Metal hydrides and hydrogen nucleophiles.

Reduction of carbonyl compounds and carboxylic acid derivatives with metal hydrides. Formation of aldehydes from carboxylic acid derivatives. Oxidation reactions of primary and secondary alcohols. Reactivity of the alcoholic function. Dehydration reactions of alcohols - hydration of alkenes. Electrophilic addition reactions to carbon-carbon double bonds. Regioselective hydration of carbon double bonds. Synthesis of ethers. Addition of halogen acids to alkenes. Halogenation of carbon-carbon double bonds. Allyl halogenation reactions of unsaturated systems. Reactivity of 1,2-diols in acid medium: formation of carbonyl compounds. Catalytic hydrogenation of double and triple bonds. Heat of hydrogenation and stability of alkenes. Methods of preparing olefins: beta elimination reactions. Hoffman's pyrolysis. Wittig synthesis. The carbon-carbon triple bond and the reactivity of alkynes towards electrophiles. Preparation of acetylenes. Preparation of reactivity of vinyl bromides. Conjugated dienes: heats of hydrogenation and stability. Resonance limit structures. Reactivity 1,2 and 1,4 towards electrophiles. The structure of benzene and the concept of aromaticity. Resonance structures and molecular orbital theory. Aromatic, non-aromatic and antiaromatic systems, Huckel's rule. Electrophilic aromatic substitution. Halogenation, nitration and sulphonation reactions of benzene. Electrophilic aromatic substitution reaction on anilines. Synthesis of phenols. Friedel-Kraft alkylation and acylation reactions. Reactivity of alkyl groups linked to the aromatic ring. Tautomeric equilibrium in carbonyl compounds. Enolate ions (kinetic and thermodynamic enolates). Acid and base catalyzed halogenation of carbonyl compounds. Aldol condensation reaction. Condensation of Claisen esters. Acetic vinegar synthesis. Nitrogen compounds: structure, characteristics, preparations and reactivity of imines, oximes and hydrazones, nitriles, carbodiimides, isocyanates, enamines, nitrates, nitrosoderivatives, azo compounds, diazo compounds, azides and related compounds. Organic sulfur compounds: basic reactivity and main methods of preparation of thiols, sulfides, disulfides, sulfoxides, sulfones, sulfonic acids and sulfenic acids; thioketals; alpha sulfur carbanions. Organosilicon compounds. Organometallic reagents. Types and denomination. Reactivity and preparation: oxidative additions, hydrogen-metal, halogen-metal and metal-metal exchanges, hydrometallations and carbometallations. Polycyclic aromatic hydrocarbons: reactions with electrophiles and nucleophiles. Heteroaromatic compounds; reactivity and synthesis properties.Carbohydrates: properties and reactivity.Polysaccharides: main characteristics of cellulose, amylose, amylopectin and glycogen.

Inorganic Chemistry

The periodic table. Periodic properties. Dimensions, ionization energy, electron affinity, electronegativity.

Molecular symmetry. Symmetry elements and operations. Assignment of symmetry classes. VSEPR rules: validity and limits. Criterion for chirality. Solids. Forces acting in a crystal lattice. Sphere packing. Close packed structures. Ionic solids. Lattice energy and Madelung constant. Born-Haber cycle. Metallic bond: conductors, insulators, semiconductors.

Hydrogen and hydrides: properties, reactivity, main compounds. Hydrogen bond. Alkali and alkaline-earth metals and their oxides, peroxides, superoxides. Properties and reactivity.

Group 13: generalities. Boron chemistry, main compounds and structural consequences of electron deficiency. aluminum chemistry, generalities, reactivity and main compounds.

Group 14: generalities. Carbon and its inorganic compounds. Properties and reactivity of silicon and silicates, aluminosilicates and zeolites.

Group 15: general information. Derivatives of nitrogen in positive and negative oxidation states. phosphorus and derivatives.

Group 16: general information. Oxygen and sulfur chemistry. Halogens: general information, reactivity and main compounds. Noble gases: general information, chemical-physical characteristics, main compounds.Transition metals: periodic properties, general information, reactivity, main oxidation states.

Crystal Field Theory: tetrahedral, octahedral and square planar coordination. High and low spin complexes. Consequences of orbital separation d. Strong and weak ligands and spectrochemical series of ligands. Pi acid ligands and pi donor ligands. 18 electron rule. Chelating effect. Thermodynamic aspects of coordination complexes. Molecular orbital theory applied to coordination complexes and ligand field theory. HSAB theory. Chemical kinetics of coordination compounds and reaction mechanisms. Ligand substitution reactions: associative and dissociative mechanism. Trans effect and cis-Platinum synthesis. Spectra of coordination compounds. Electronic d-d transitions and charge transfer. Orgel diagrams. Crystalline field stabilization energies (CFSE) and trend along a period of the periodic table. Use of coordination compounds in catalysis.

Analytical Chemistry

Acid-base equilibrium in water. Strong acids and bases. Weak acid. Weak base. Salts of weak acids and bases. Buffer solutions. Titration curves for simple acid-base systems. Visual indicators. Solutions of ampholytes. Distribution curves. Acid-base equilibria in complex systems. Titration curves of polyprotic acids and bases.

Ion exchangers. Ion exchange equilibria. Cationic and anionic resins. Main analytical applications. Potentiometry. Measuring instrument. Reference electrodes. Membrane electrodes with particular reference to the glass electrode. Internal and external sphere assemblies. Complex formation equilibria. Solvation and electrostatic interaction of ions. Formation of EDTA complexes. Constants and apparent constants of formation. Use of the auxiliary complexing agent. Formation equilibria of complexes. Titration curves. Metallochromic indicators. Direct, back and displacement titrations. Precipitation equilibria. Gravimetric analysis. Characteristics of the precipitates. Effects of temperature, ionic strength, common ion, pH and presence of complexing agents on the solubility of precipitates. Precipitation titrations. Argentometry. Titration curves. Instrumental techniques for following precipitation titrations. Potentiometric titration of a mixture of halides. Redox equilibria. Electrode potentials. Electromotive force of a cell. The Nernst equation. Equilibrium constant of the cell reaction. Titration curves for redox equilibria. Symmetrical and asymmetrical titrations. Preventive oxidizers and reducers. Different ways of visual indication of the end point in redox titrations. Instrumental techniques suitable for following redox titrations. Iodometry.

Properties of electromagnetic radiation. Quantitative aspects of absorption measures. Lambert-Beer law and limits of its applicability. Absorption measurement instrumentation. Single and double beam UV visible spectrophotometers. Source. Prism and grating monochromators. Spectrophotometric instrumentation. Photomultipliers. Analytical errors in absorption analysis.

Analytical protocol. Errors in an analytical procedure. White analytical chemistry. Reference materials and their use. The treatment of samples. Physical separations.

Sample treatment: acid/base digestions, microwave oven, liquid/liquid and solid/liquid extractions. Calibration diagrams, atomic absorption techniques. factors influencing atomization, production of free atoms and ions, graphite furnace. Source of radiation in AAS, analysis with and without preliminary atomization. Background signal correction in AAS. Separation techniques and chromatography. Theory of chromatographic plates. Band broadening: the Van Deemter equation. The Giddings equation. Calculation of the number of theoretical plates. The instrumentation in GC. The detectors in GC. Mass spectrometer as a detector in GC. Acquisition in TIC and in SIM. Quantitative analysis by GC/MS. Interpretation of MS.TLC spectra and HPLC instrumentation. Types of detectors for HPLC. Electrochemical methods: potentiometry and voltammetry. Behaviour of an electrochemical cell in response to an applied potential. Operation of the working electrode at Hg. Current/potential curves. Amperometry. DC polarography, impulse voltammetry, DPASV, cyclic voltammetry. Tests of significance and theory of errors. Elements of Chemometrics.

Physical Chemistry

The thermodynamic model. System definition. Open, closed and isolated systems.

Thermodynamic state of a system: thermodynamic equilibrium. State variables. Extensive and intensive variables. Unit of measure. Pressure and mechanical balance. Temperature and thermal equilibrium. Zeroth law of thermodynamics. Temperature scales. The equation of state. Boyle's law. Law of Charles and Gay Lussac. The ideal gas temperature scale. Avogadro's principle. Equation of state of the perfect gas. Gas mixtures and Dalton's law. Real gases. Compression factor and virial equation of state. The isotherms of real gas. Critical variables. The Van der Waals equation. Critical variables and reduced variables. The first law of thermodynamics. The concepts of work, heat and energy. Sign Convention. Mechanical work and expansion work. Maximum work of expansion. The reversible expansion process. Heat. Heat measurement. The first law of thermodynamics and the internal energy function. The principle of conservation of energy. State functions and their properties. Thermal capacity. Heat capacity at constant volume and constant pressure. The enthalpy function. Relations between thermal capacity and state functions. Relations between heat capacity at constant pressure and heat capacity at constant volume. Applications of the first principle. Reversible isothermal expansion of an ideal gas. Internal energy and enthalpy variations in heating processes at constant volume and constant pressure. Applications of the first principle. Adiabatic expansion of an ideal gas. Relations between state variables in reversible adiabatic transformations.

Examples of applications of the first law. Comparison between isothermal expansion processes and adiabatic expansion processes. Irreversible processes of adiabatic expansion.

The enthalpy function. Enthalpy as a state function. Enthalpy of physical transformations. Definition of the standard state for physical transformations. Enthalpy of reaction. Definition of the standard reaction state. Exothermic reactions and endothermic reactions. Combustion enthalpy. Standard molar enthalpy of formation. Applications of the first principle. Dependence of enthalpy on temperature. Dependence of heat capacity on temperature. Hess' law. Dependence of reaction enthalpy on temperature. Kirkhoff's law. Bond dissociation enthalpy. Lattice enthalpy. Born-Haber cycle. Solution enthalpy. Enthalpy of hydration. Calls on the properties of state functions. internal pressure. Properties of internal energy. Expandability. Expandability coefficient. Euler relation and reciprocal identity rule. The properties of enthalpy. Isothermal compressibility coefficient. Joule Thomson's isoenthalpy coefficient. Joule Thomson's isothermal coefficient. Linde's refrigeration machine. Difference between Cp and Cv. The second law of thermodynamics. The direction of spontaneous changes. The dispersion of energy. The concept of entropy. Formulation of the second principle. The Clausius statement and the Kelvin statement. Entropy as a state function. The second law of thermodynamics. The Carnot cycle. Thermodynamic definition of temperature. The second law of thermodynamics. Carnot's theorem. The Clausius inequality. Calculation of entropy changes: state transitions; isothermal expansion of an ideal gas; isothermal and isobaric mixing of ideal gases; isobaric and isochoric heating. Entropy as a state function. Calculation of entropy changes in irreversible processes. The measurement of entropy. The third law of thermodynamics. The Nernst theorem. Entropy of reaction. The fundamental equation of thermodynamics. The auxiliary functions. Helmholtz Free Energy and Gibbs Free Energy. Consequences of the Clausius inequality. Equilibrium and evolution criteria in thermodynamic systems. The fundamental equation of thermodynamics. Free energy and maximum work. Standard free energies of reaction. Dependence of Gibbs free energy on temperature and pressure. The Gibbs Helmholtz equation. Properties of internal energy. Maxwell's relations. The physical transformations of substances. Phase diagrams. Phase stability and phase boundaries. Phase diagram of carbon monoxide. Phase diagram of water. Phase diagram of helium. Phase transitions. The thermodynamic criterion of equilibrium and the concept of chemical potential. The chemical potential. Effect of temperature and pressure on phase stability. Effect of external pressure on vapour pressure. Applications and examples. Phase balance. Clapeyron's equation. The Calusius Clapeyron equation. Solutions and mixtures. The thermodynamic model. Partial molar properties.

Partial molar volume. Applications and examples

Partial molar free energy. A broader definition of chemical potential The Gibbs-Duhem equation and the thermodynamics of mixing. Mixing free energy of an ideal gas. Mixing functions. Chemical potential in ideal solutions. Raoult's law. Ideal dilute solutions and Henry's law. Solubility of gases. Chemical equilibrium and colligative properties. Boiling point elevation. Freezing point lowering and ideal solubility. The phenomenon of osmosis and osmotic pressure. Component phases and degrees of freedom. The phase rule. Phase diagrams of binary systems. Composition temperature phase diagrams. Liquid vapour phase diagrams. Liquid-liquid phase diagrams and phase separation. Critical temperature of solution. Solid-liquid phase diagrams. Real solutions and the concept of activity. Solvent activity and solute activity. Real solutions. Activity coefficients and related standard states. The choice of the standard state. Reaction equilibrium. Spontaneous reactions and free energy of reaction. Endergonic and exergonic reactions ΔG of reaction and ΔG° of reaction. ΔG° of reaction and equilibrium constant. The thermodynamic constant of reaction equilibrium. Dependence of the equilibrium on pressure and temperature. The rate of chemical reactions. Introduction and experimental study methods. Definition of the reaction rate. The kinetic equation. Determination of the kinetic law and integration of the kinetic equations. Zero order kinetic equations. First order equations. Second order kinetic equations. The half-life reaction. Determination of the order of reaction. The method of excess. The initial velocity method. Dependence of the reaction rate on the temperature. The Arrhenius equation. Examples of kinetic studies. Consecutive reactions. Approximation of the steady state. Study of enzymatic kinetics. Elementary reactions and determination of the reaction mechanism. Chain reactions.

Light-matter interaction in the various spectral regions; Some inadequacies of classical physics and the need to introduce quantum descriptions.

The postulates of Quantum Mechanics. Some simple cases in Quantum Mechanics: the free particle, the particle in the one- and three-dimensional box. Transition probability, radiative and non-radiative transitions. Harmonic oscillator. Schrodinger equation Hamiltonian operator and its formal construction; Stationary solutions of the Schrodinger equation and their implications. Wave functions for the harmonic oscillator. Orders of magnitude of electronic, vibrational, rotational and translational energies and their implications. Rigid rotor. Rotation in the plane and in three-dimensional space. Rotational energy and wave functions. Rotational motion of nonlinear molecules, rotational spectra and their characteristics. Vibrational motion in polyatomic molecules and normal modes Hydrogen atom. Wave functions and energies.

Electronic contribution to the energy of polyelectronic atoms. Construction of the Hamiltonian. Selection rules.

Spin-orbit coupling. Zeeman effect. Electronic spin. The Pauli principle. Spin functions and selection rules. UV-vis spectroscopy, electronic transitions. Quantum mechanical derivation of the selection rules and the Franck-Condon factor. Fluorescence spectroscopy: the Jablonki diagram, Stokes’ shift, Kasha’s rule, quantum yield, kinetics of excited and quenching states. IR and Raman spectroscopy. Rotational spectroscopy. Nuclear magnetic resonance spectroscopy. Introduction to statistical thermodynamics. Need for a statistical description of constituted systems from many particles; Quantum states accessible to a macroscopic system. Probability of a quantum state accessible to an isolated macroscopic system. Boltzmann entropy and number of accessible states of an isolated system. Irreversible processes and variation of the number of accessible states, with examples. Macroscopic system in contact with a thermostat and probability of a definite state of the system in equilibrium with the thermostat itself. Calculation of the number of states accessible to the system and to the thermostat for schematic situations. Boltzmann canonical function, relative and absolute probabilities; Canonical partition function and its meaning. Dependence of the partition function on the zero of the energies; Representative Gibbs set; Canonical Boltzmann distribution and average values. Average energy as internal energy of the system; Heat capacity at constant volume from the partition function; Entropy through the statistical approach Gibbs and Helmholtz free energies through statistics; Equation of state from the partition function. Standard deviation of the energy from the mean value; Application of the canonical distribution to the case of distinguishable and independent particles. Paramagnetism of ideal systems and magnetic energy; Considerations on the third law of thermodynamics; Thermal capacity of a perfect crystal as a function of temperature; Overcoming the Einstein model: outline of Debye's mode.l Indistinguishable particles, meaning of indistinguishability and explicit examples of calculation of the partition function of schematic systems; Canonical partition function for a monatomic perfect gas; Order of magnitude of the translational partition function, Thermodynamic properties of a monatomic gas. Partition function of a diatomic perfect gas and corresponding thermodynamic properties; Energy equipartition and its approximate validity; Symmetry number and its meaning; frozen degrees of freedom. Polyatomic perfect gas and corresponding partition function; Chemical equilibrium and equilibrium constant from a statistical point of view

3.b. Mathematics Prerequisites

Mathematics programmes for the degree courses in Chemistry

Prerequisites for the first year admission exam

3.c. Physics prerequisites

Mechanics

Kinematics. The laws of motion. Work and energy. Friction forces. Gravitational force. Potential energy. Momentum. collisions. Angular momentum. Motion in central field. Rigid body dynamics. Swings and waves. Fluid mechanics. Lagrange equations. Hamilton's equations.

Thermodynamics and Statistical Physics

Temperature. Ideal gases. Heat and the first law of thermodynamics. Entropy and the second law of thermodynamics. Thermodynamic potentials. Thermal machines. Kinetic theory of gases. Boltzmann distribution. Partition function. Quantum statistics.

Electrodynamics

Coulomb's law and electric field. Gauss's law. Electric potential. Electric current, circuits, resistance and capacitance. Magnetic field and Lorentz force. Ampere's law. Faraday's law, inductance. Maxwell's equations. Electromagnetic waves. Radiation. Reflection, refraction, geometric optics. Interference and diffraction. Polarization and magnetization of material media. Dielectric function.

Quantum mechanics

Crisis of classical physics. Wave/particle dualism and the uncertainty principle. The Schroedinger equation. Quantum particle in potential field. Angular momentum. Hydrogen atom. Perturbation theory and transitions. Spin. Systems of identical particles. Helium atom. Hydrogen molecule. Emission and absorption of radiation by atoms.

**4. MASTER'S DEGREE COURSES IN MOLECULAR AND CELLULAR BIOLOGY AND RELATED SUBJECTS**

4.a. Biology prerequisites

Genetics and Molecular Biology

Macromolecules of biological importance

Molecular recognition processes through non-covalent bonds Nucleic acids

Protein structure

Proteins as catalysts

Protein functions

Enzymes, structural proteins and signal transduction

Protein assembly and degradation

Molecular genetics

RNA and proteins

DNA repair

DNA replication

Genetic recombination

Quantitative genetics

Inheritance, linkage disequilibrium and penetrance

Genetic drift and bottleneck

Fitness and selection

Sexual reproduction and evolutionary theories

Neutral mutations, molecular clock and macromolecule evolution

Recombinant DNA

DNA separation, fragmentation and sequencing

Nucleic acid hybridization

DNA cloning

DNA engineering

Quantitative PCR, ChIP

DNA sequencing

Technologies for studying the transcriptome

The cell nucleus

Chromosomal DNA and its packaging

The global structure of chromosomes (centromeres and telomeres)

Duplication of chromosomes

RNA synthesis and its processing

Gene expression control

DNA-binding motifs in transcription regulatory proteins

Activation and repression of gene expression

mRNA stability and turnover

Chromatin structure and epigenetic control of transcription

Post-transcriptional control and microRNA

Intracellular compartments and protein sorting

The compartmentalization of eukaryotic cells

The transport of molecules in and out of the nucleus

The transport of proteins in the mitochondria

Peroxisomes, lysosomes and proteasomes

The endoplasmic reticulum

Vesicular trafficking in the secretory pathways and endocytosis

Transport from the endoplasmic reticulum through the Golgi apparatus

Transport from the trans Golgi to lysosomes

Transport from the plasma membrane via the endosome: endocytosis

Transport from the trans Golgi to the cell surface: exocytosis

The molecular mechanisms of vesicular transport and the maintaining compartmental diversity

Signal transduction

G-Protein-coupled surface receptors

Surface receptors with enzymatic activity

Second messengers and signal transduction to the nucleus Receptors for hydrophobic signals (e.g. steroid hormones)

The cytoskeleton

Intermediate filaments

Microtubules

Cilia and centrosomes

Actin filaments

Actin-associated proteins

The cell cycle

The early embryonic cell cycle and the role of mitosis-promoting factor (MPF)

Yeasts and the molecular genetics of cell cycle control

Control of cell division in multicellular animals

The mechanics of cell division

Mitosis

Cytokinesis

Cell junctions, cell adhesion, and the extracellular matrix

Cell junctions

Cell-cell adhesion

The extracellular matrix of animals

Extracellular matrix receptors on animal cells: integrins

Cellular mechanisms of development

Morphogenetic movements that define the body plan

The diversification of animal cells in the early embryo

Cellular memory. Cell determination. The concept of Drosophila place values and the molecular genetics of pattern formation. I. Genesis of the Drosophila body plan and the molecular genetics of pattern formation. II. Homeotic genes and body regionalization, Neural development: neural induction and neural tube patterning

Cancer

Cancer as a multistage mutagenic process

Oncogenes and tumor suppressors

Diffusible factors and neoangiogenesis

Neurobiology

Neuron cell biology.

Organelles, dendrites, axon, axonal transport, microtubules and intermediate filaments

The cellular elements of nervous tissue

Neurons, Astrocytes, Oligodendrocytes, myelin, blood brain barrier

Passive electrotonic properties of axons and dendrites

Cell membrane structure

Ion pumps and genesis of potential difference

Nernst potential

RC circuit and electrotonic propagation

Action potential genesis and propagation

Voltage-gated and rectifying channels

Refractory period

Hodgkin & Huxley equation

Synapse structure and function

Electrical synapses: ultrastructure and molecular composition Chemical synapses: ultrastructure

The SNARE complex and vesicle-associated proteins

Postsynaptic density

Neurotransmitters and their receptors

Neurotransmitters and their biosynthesis

Mechanisms of vesicular reuptake of neurotransmitters

The neuromuscular plaque

Ionotropic receptors

Metabotropic receptors

Postsynaptic potentials and synaptic integration

EPSP. IPSP and quantum transmission, endplate and central synapses

Information processing in dendrites;

Spatial and temporal summation, dendritic spines and their function

Development of the nervous system

Formation and elimination of synapses, programmed cell death, neutrophic factors

Neuronal plasticity

Synaptic mechanisms of plasticity, Hebb's law

Molecular mechanisms of plasticity: second messengers and retrograde messages.

Habituation

Long-term potentiation

Critical periods and plasticity during development

Sensory systems

Sensory transduction; chemical senses: taste and smell;

The somatosensory system

Somatosensory pathways, somatotopic map, electrophysiological properties of cortical neurons (columnar functional organization)

Vision

Phototransduction and adaptation

The retina and the visual pathways

Primary visual cortex, visuotopic map, properties of visual neurons and organization in columns.

Motor system

Myofiber structure, molecular mechanisms of contraction

Spinal cord organization

Motor unit

Spinal motor control, reflexes and locomotion;

Descending voluntary control, primary motor cortex and premotor cortices

Circadian rhythm

Molecular bases (cell-autonomous) of the circadian rhythm Synchronization of the circadian rhythm with light stimuli

Learning and memory: mechanisms and basic definitions

Operational, short- and long-term memory

Role of protein synthesis in memory

Protocols for studying memory in experimental animals

Overview of neurodegenerative diseases

Alzheimer, Parkinson and Huntington: molecular basis

Epilepsy and ischemia: role of ecitotoxicity

4.b. Mathematics Prerequisites

Mathematics programmes for the degree courses in Biology

Prerequisites for the first year admission exam

4.c. Physics prerequisites

Mechanics

Kinematics. The laws of motion. Work and energy. Friction forces. Gravitational force. Potential energy. Momentum. Collisions. Angular momentum. Motion in central field. Swings and waves. Fluid mechanics.

Thermodynamics

Temperature. Ideal gases. Heat and the first law of thermodynamics. Entropy and the second law of thermodynamics. Thermal machines. Kinetic theory of gases.

Electrodynamics

Coulomb's law and electric field. Gauss's law. Electric potential. Electric current, circuits, resistance and capacitance. Magnetic field and Lorentz force. Ampere's law. Faraday's law, inductance. Maxwell's equations. Electromagnetic waves. Reflection, refraction, geometric optics. Interference and diffraction.